On Almost Moscow Topological Groups

1. Introduction

The theory of topological groups [17] establishes a foundational bridge between algebraic symmetry and topological continuity. Over time, applications in dynamical systems, harmonic analysis, and geometric topology have demonstrated that strict continuity often excludes naturally occurring structures where operations preserve neighborhood containments only in a weakened sense. This observation catalyzed the development of topological transformation groups [3,14,22], which model symmetry through continuous actions, and subsequently $\mathcal{S}$-topological transformation groups [18,19,20,21], where continuity is relaxed to semi-totally continuity. This evolutionary trajectory reflects a consistent theme in modern topology, systematically weakening continuity constraints while preserving essential algebraic coherence. However, relaxing operational continuity introduces new challenges in closure-interior dynamics, neighborhood chasing, and quotient stability, necessitating complementary spatial regularity conditions to maintain structural control.

Parallel to this algebraic expansion, generalized topology has investigated spaces where classical regularity axioms are replaced by structurally rich alternatives. Arhangel’skii’s Moscow spaces [1] emerged as a powerful tool in topological group theory, linking $G_{\delta }$-sets to closure operations and enabling deep completion-theoretic results. Building on this foundation, Caldas et al. [4] introduced almost Moscow spaces, characterized by the requirement that every $G_{\delta }$-set contains a regularly closed neighborhood of each of its points. Independently, Ram [23] proposed almost topological groups, replacing the standard continuity of multiplication and inversion with almost continuity in the sense of Singal and Singal [24]. While these frameworks independently relax spatial regularity and operational continuity, their intersection has remained unexplored. Classical techniques for Moscow groups rely on continuous operations to transfer $G_{\delta }$-lattice properties to completion extensions, and almost topological group theory lacks the spatial richness needed to control regularly closed decompositions. This gap motivates the present work.

This paper introduces almost Moscow topological groups, defined as almost topological groups whose underlying spaces satisfy the almost Moscow property. The synthesis is mathematically non-trivial, almost continuity preserves $Int(Cl(·\left)\right)$ containments but does not guarantee exact closure preservation, while the almost Moscow condition alone lacks algebraic compatibility. Together, they form a balanced framework where $G_{\delta }$-regularity compensates for weakened operational continuity, enabling precise control over $\delta$-closures, $\theta$-semi-open sets, and the regularly closed lattice. It is established that every almost Moscow space is $\delta$-extremally disconnected, though the converse fails, positioning the new class strictly within Arhangel’skii’s hierarchy while retaining stronger spatial structure. This duality ensures that translation invariance, semi-regularization, and homomorphic images interact predictably with the underlying $G_{\delta }$-lattice.

The structural theory yields sharp classifications under covering hypotheses. An almost Moscow topological group is extremally disconnected if and only if it is locally countably S-closed and km-perfect, leveraging the interplay between countable rc-covers and regular open intersections. Locally rc-Lindelöf mildly Hausdorff instances are $\delta$-extremally disconnected, and semi-regularizations of locally countably rc-paracompact groups are $\omega$-extremally disconnected, demonstrating how paracompactness-type conditions force generalized extremally disconnectedness. Consequently, Hausdorff semi-regular first countable locally countably rc-paracompact members collapse to the discrete topology, illustrating a strong structural rigidity under countability and separation constraints. In the extremally disconnected regime, every such group contains an open Boolean subgroup, extending classical decomposition Theorems to the almost continuous setting.

A central achievement of the framework is the connection to completion theory. Locally countably S-closed km-perfect almost Moscow topological groups upgrade to classical topological groups that are strong PT-groups. Permanence properties further demonstrate stability such that the structure is preserved under $G_{\delta }$-irresolute pre-almost closed homomorphisms, descends to quotients via open projections, and lifts through injective R-maps. Structural Theorems confirm that the identity component forms a closed normal subgroup with inherited almost Moscow structure, while investigations into $\delta$-closure translations, rc-Lindelöf equivalences, and near compactness of products solidify the theoretical foundation. These findings position almost Moscow topological groups as a robust platform for exploring algebraic operations under weakened continuity, with direct implications for quotient dynamics, and generalized topological group representations.

2. Preliminaries

This section gathers the foundational concepts, mapping classes, and structural characterizations required for the development of almost Moscow topological groups. All spaces are assumed to be topological spaces without additional separation axioms unless explicitly stated. For a subset A of a space X, we denote the interior, closure, and complement by $Int\left(A\right)$, $Cl\left(A\right)$, and $X\setminus A$, respectively.

Definition 1

([11]). A subset $U\subseteq X$ is regularly open if $U=Int(Cl(U\left)\right)$, and regularly closed if $U=Cl(Int(U\left)\right)$. The families of regularly open and regularly closed sets are denoted by $RO\left(X\right)$ and $RC\left(X\right)$, respectively. The complement of a regularly open set is regularly closed, and vice versa.

Definition 2

([13]). A subset $A\subseteq X$ is semi-open if $A\subseteq Cl(Int(A\left)\right)$. The complement of a semi-open set is semi-closed.

Definition 3

([11]). A subset $A\subseteq X$ is θ-semi-open if it is a union of regularly closed sets. Equivalently, A is θ-semi-open if for each $x\in A$, there exists $F\in RC\left(X\right)$ such that $x\in F\subseteq A$. The complement of a θ-semi-open set is θ-semi-closed.

Definition 4

([24]). A function $f:X\to Y$ is almost continuous if for each $x\in X$ and each open neighborhood V of $f\left(x\right)$ in Y, there exists an open neighborhood U of x such that $f\left(U\right)\subseteq Int(Cl(V\left)\right)$.

Theorem 1

([24]). A map $f:X\to Y$ is almost continuous if and only if $f^{-1}\left(W\right)$ is open in X for every $W\in RO\left(Y\right)$. Consequently, almost continuous maps preserve the regularly open lattice under preimages and are closed under composition.

Definition 5

([23]). $(G,\tau )$ is an almost topological group if both multiplication and inversion maps are almost continuous.

Theorem 2

([23]). In any almost topological group, left and right translations $L_g\left(x\right)=gx$ and $R_g\left(x\right)=xg$ are almost continuous bijections with almost continuous inverses. Moreover, $RO\left(G\right)$ and $RC\left(G\right)$ are invariant under all group operations.

Definition 6

([4]). A space X is almost Moscow if for every $G_{\delta }$-subset $V\subseteq X$ and each $x\in V$, there exists a regularly closed set $F\subseteq X$ such that $x\in F\subseteq V$.

Definition 7

([4]). $f:X\to Y$ is $G_{\delta }$-irresolute if $f^{-1}\left(V\right)$ is a $G_{\delta }$-set in X for every $G_{\delta }$-set V in Y.

Definition 8

([4]). $f:X\to Y$ is pre-almost closed if $f\left(F\right)\in RC\left(Y\right)$ for every $F\in RC\left(X\right)$, and pre-almost open if $f\left(U\right)\in RO\left(Y\right)$ for every $U\in RO\left(X\right)$.

Definition 9

([4]). $f:X\to Y$ is an R-map if $f^{-1}\left(F\right)\in RC\left(X\right)$ for every $F\in RC\left(Y\right)$.

Definition 10

([8]). A function $f:X\to Y$ is R-continuous if for each $x\in X$ and each regularly open set V in Y containing $f\left(x\right)$, there exists a regularly open neighborhood U in X containing x such that $f\left(U\right)\subseteq V$.

Definition 11

([4]). $f:X\to Y$ is a $G_{\delta }$-map if $f\left(U\right)$ is a $G_{\delta }$-set in Y for every $G_{\delta }$-set U in X.

Definition 12

([4]). $f:X\to Y$ is almost $G_{\delta }$-continuous if $f^{-1}\left(V\right)$ is a $G_{\delta }$-set in X for every $V\in RO\left(Y\right)$.

Definition 13

([4]). $f:X\to Y$ is R-$G_{\delta }$-open if $f\left(U\right)\in RO\left(Y\right)$ for every $G_{\delta }$-set $U\subseteq X$.

Definition 14

([12]). A space X is weakly $P_{\Sigma }$ if for every regularly open set $V\subseteq X$ and each $x\in V$, there exists a regularly closed set $F\subseteq X$ such that $x\in F\subseteq V$.

Definition 15

([16]). A subset $K\subseteq X$ is S-closed relative to X if every cover of K by regularly closed subsets of X admits a finite subcover.

Definition 16

([25]). A topological space X is called nearly compact if every cover of X admits a finite subcover the interiors of the closures whose members cover $X.$

Definition 17

([2]). A space X is extremally disconnected if the closure of every open subset of X is open.

Definition 18

([26]). For a group G, the Boolean subgroup is $B\left(G\right)=\{x\in G:x^2=e\}$, where e denotes the identity element.

Definition 19

([5]). A space X is km-perfect if for every regular open set $U\subseteq X$ and each $x\in U$, there exists a sequence ${\left\{G_n\right\}}_{n\in \omega }$ of open sets such that $U\subseteq {\bigcup }_{n\in \omega }{G}_n$ and $x\notin {\bigcup }_{n\in \omega }Cl\left(G_n\right)$.

Definition 20

([5]). A space X is countably S-closed if every countable cover of X by regularly closed sets admits a finite subcover. A subset $A\subseteq X$ is countably S-closed relative to X if every countable rc-cover of A by regularly closed subsets of X admits a finite subcover.

Definition 21

([7]). A space X is locally countably S-closed if each point $x\in X$ has an open neighborhood U that is countably S-closed relative to X.

Definition 22

([10]). A space X is rc-Lindelöf if every cover of X by regularly closed sets admits a countable subcover.

Definition 23

([7]). A space is locally rc-Lindelöf if each point has an open neighborhood that is rc-Lindelöf relative to X.

Definition 24

([9]). A space X is countably rc-paracompact if every countable rc-cover of X admits a locally finite rc-refinement.

Definition 25

([7]). A space is locally countably rc-paracompact if each point has an open neighborhood that is countably rc-paracompact relative to X.

Definition 26

([6]). A space X is mildly Hausdorff if every open set is a union of δ-closed sets. Equivalently, the δ-closed sets form a network for the topology.

Definition 27

([9]). A space X is δ-extremally disconnected, if for every open set $U\subseteq X$ and each $x\in Cl\left(U\right)$, there exists a $G_{\delta }$-set B such that $x\in B\subseteq Cl\left(U\right)$.

Definition 28

([9]). A space X is ω-extremally disconnected if for every regular open set $R\subseteq X$, $Cl\left(R\right)=G_{\delta }-Cl\left(R\right)$, where $G_{\delta }-Cl\left(R\right)$ denotes the intersection of all $F_{\sigma }$-supersets of R.

Definition 29

([15]). A space X is a weak P-space if for every countable family ${\left\{U_n\right\}}_{n\in \omega }$ of open subsets, ${\bigcup }_{n\in \omega }Cl\left(U_n\right)=Cl\left({\bigcup }_{n\in \omega }{U}_n\right)$. Equivalently, $(X,{\tau }_s)$ is a P-space.

Definition 30

([2]). A topological group G is a PT-group if the group operations on G extend uniquely to its Dieudonne completion $\mu G$, making $\mu G$ a topological group containing G as a dense topological subgroup.

Definition 31

([2]). A topological group G is a strong PT-group if G is C-embedded in its Raikov completion $\varrho G$. Every strong PT-group is a PT-group.

3. Almost Moscow Topological Groups

Definition 32.

Let $(G,·)$ be a group endowed with a topology τ. The pair $(G,\tau )$ is called an almost Moscow topological group if the following conditions hold:

i 

The multiplication mapping $\phi :G\times G\to G$ defined by $\phi (x,y)=x·y$ and the inversion mapping $\psi :G\to G$ defined by $\psi \left(x\right)=x^{-1}$ are almost continuous, where $G\times G$ carries the product topology;

ii 

For every $G_{\delta }$-subset $V\subseteq G$ and each point $x\in V$, there exists a regularly closed set F in G such that $x\in F\subseteq V$.

Example 1.

Let G be any group equipped with the discrete topology ${\tau }_{\mathrm{disc}}$. Then $(G,{\tau }_{\mathrm{disc}})$ is an almost Moscow topological group.

Example 2.

Let G be any group equipped with the indiscrete topology ${\tau }_{\mathrm{ind}}=\{\varnothing ,G\}$. Then $(G,{\tau }_{\mathrm{ind}})$ is an almost Moscow topological group.

Example 3.

Consider the additive group $(\mathbb{R},+)$ with the Euclidean topology ${\tau }_u$. Then $(\mathbb{R},{\tau }_u)$ is an almost Moscow topological group.

Proposition 1.

Every almost Moscow topological group is an almost topological group.

Proof. 

Proof follows from the Definition 32. □

Remark 1.

Not every almost topological group is an almost Moscow topological group, which is provided in the following Examples.

Example 4.

Consider $G={\mathbb{Z}}_2=\{0,1\}$ with topology $\tau =\{\varnothing ,\{0\},G\},$ which is an almost topological group. But, it fails to be almost Moscow topological group.

Example 5.

Let $G=\mathbb{R}$ be an additive group with topology τ generated by:

$$\mathcal{B}=\left\{\right(a,b):a,b\in \mathbb{R}\}\cup \left\{\right(c,d)\cap D:c,d\in \mathbb{R}\},$$

where D denotes the set of irrational numbers. Then $(G,\tau )$ is an almost topological group but not an almost Moscow topological group.

Remark 2.

The class of almost topological groups strictly contains the class of almost Moscow topological groups.

Theorem 3.

Let $(G,\tau )$ be an almost Moscow topological group. For each $g\in G$, the left translation $L_g:G\to G$ defined by $L_g\left(x\right)=gx$ and the right translation $R_g:G\to G$ defined by $R_g\left(x\right)=xg$ are almost continuous mappings.

Proof. 

Fix $g\in G$ and let $x\in G$. Suppose W is a regularly open neighborhood of $L_g\left(x\right)=gx$. By Definition 32, the multiplication mapping $\phi :G\times G\to G$ is almost continuous. Hence, there exist open neighborhoods $U_1$ of g and $U_2$ of x such that $U_1{U}_2\subseteq W$. Since $g\in U_1$, we obtain $g{U}_2\subseteq U_1{U}_2\subseteq W$, which implies $L_g\left(U_2\right)\subseteq W$. Since $U_2$ is an open neighborhood of x, and W an arbitrary regularly open neighborhood of $L_g\left(x\right)$, the mapping $L_g$ is almost continuous at x. As $x\in G$ chosen arbitrarily, $L_g$ is almost continuous on G. The proof for the right translation $R_g$ proceeds identically. Given $W\in RO\left(G\right)$ containing $xg$, the almost continuity of $\phi$ yields open neighborhoods $V_1$ containing x and $V_2$ containing g with $V_1{V}_2\subseteq W$. Consequently, $V_{1}g\subseteq W$, which implies $R_g\left(V_1\right)\subseteq W$. Hence $R_g$ is almost continuous at x, and since x is arbitrary, $R_g$ is almost continuous on G. □

Theorem 4.

Let $(G,\tau )$ be an almost Moscow topological group. If $U\in RO\left(G\right)$, then the following properties hold:

(i) 

$gU\in RO\left(G\right)$ for every $g\in G$;

(ii) 

$Ug\in RO\left(G\right)$ for every $g\in G$;

(iii) 

$U^{-1}\in RO\left(G\right)$.

Proof. (i) Fix $g\in G$, and let $y\in gU$, so $y=gx$ for some $x\in U$. Equivalently, $x=g^{-1}y\in U$. Since the multiplication map $\phi :G\times G\to G$ is almost continuous at $(g^{-1},y)$ and U is a regularly open neighborhood of $\phi (g^{-1},y)=g^{-1}y$, there exist open sets V containing $g^{-1}$ and W containing y such that $VW\subseteq U$. Because $g^{-1}\in V$, $g^{-1}W\subseteq VW\subseteq U$, which implies $W\subseteq gU$. Hence $y\in Int\left(gU\right)$, proving that $gU$ is open. Since U is open, $Cl\left(U\right)$ is regularly closed. By Theorem 3, the left translation $L_{g^{-1}}$ is almost continuous. Since in almost continuity the image of a regularly closed set under a translation remains closed; therefore $gCl\left(U\right)$ is closed in G. Consequently, $Int(Cl(gU\left)\right)\subseteq Cl\left(gU\right)\subseteq gCl\left(U\right).$ Applying $L_{g^{-1}}$ to this inclusion gives $g^{-1}Int(Cl\left(gU\right))\subseteq Cl\left(U\right)$. Observe that $Int(Cl(gU\left)\right)$ is regularly open by Definition. Since $L_{g^{-1}}$ is almost continuous, the preimage of any regularly open set is open; thus $g^{-1}Int(Cl\left(gU\right))$ is open, which implies $g^{-1}Int(Cl\left(gU\right))\subseteq Int(Cl\left(U\right))=U$. Multiplying by g on the left yields $Int(Cl(gU\left)\right)\subseteq gU$. Combined with the trivial inclusion $gU\subseteq Int(Cl(gU\left)\right)$ , we conclude $gU=Int(Cl(gU\left)\right)\in RO\left(G\right)$.

(ii) The argument mirrors part (i) but utilizes the right translation $R_g\left(x\right)=xg$ and the second coordinate of the product topology. Fix $g\in G$ and let $y\in Ug$. Then $y=xg$ for some $x\in U$, which implies $y{g}^{-1}=x\in U$. Consider the multiplication map $\phi$ at the point $(y,g^{-1})$. Since $\phi (y,g^{-1})=y{g}^{-1}\in U$ and U is regularly open, almost continuity provides open neighborhoods V containing y and W containing $g^{-1}$ such that $VW\subseteq U$. Because $g^{-1}\in W$, we have $V{g}^{-1}\subseteq VW\subseteq U$, which implies $V\subseteq Ug$. Thus $y\in Int\left(Ug\right)$, confirming that $Ug$ is open. To verify regular openness, note that $Cl\left(U\right)$ is regularly closed. The right translation $R_{g^{-1}}$ is almost continuous by Theorem 3, and as in part (i), it maps regularly closed sets to closed sets. Hence $Cl\left(U\right)g^{-1}$ is closed. We obtain, $Int(Cl\left(Ug\right))\subseteq Cl\left(Ug\right)\subseteq Cl\left(U\right)g^{-1}.$ Right-multiplying by g gives $Int(Cl(Ug\left)\right)g\subseteq Cl\left(U\right)$. The set $Int(Cl(Ug\left)\right)g$ is open because $R_g$ is an almost open map. Being an open subset of $Cl\left(U\right)$, it satisfies $Int(Cl(Ug\left)\right)g\subseteq Int(Cl(U\left)\right)=U$. Right-multiplying by $g^{-1}$ yields $Int(Cl(Ug\left)\right)\subseteq Ug$. Together with $Ug\subseteq Int(Cl(Ug\left)\right)$, this establishes $Ug\in RO\left(G\right)$.

(iii) Let $y\in U^{-1}$, so $y^{-1}\in U$. Since U is open, there exists an open neighborhood V of y such that $V^{-1}\subseteq U$. This inclusion is equivalent to $V\subseteq U^{-1}$, proving that $y\in Int\left(U^{-1}\right)$ and hence $U^{-1}$ is open. For the closure condition, observe that $Cl\left(U\right)$ is regularly closed, and inversion is an almost continuous map; therefore $Cl{\left(U\right)}^{-1}$ is closed. It follows that $Int(Cl\left(U^{-1}\right))\subseteq Cl\left(U^{-1}\right)\subseteq Cl{\left(U\right)}^{-1}.$ Taking inverses gives $Int{(Cl\left(U^{-1}\right))}^{-1}\subseteq Cl\left(U\right)$. Taking interiors and using the fact that inversion preserves regular openness under almost continuity, we obtain $\mathrm{Int}\left(\mathrm{Cl}\left(U^{-1}\right)\right)\subseteq \mathrm{Int}{\left(\mathrm{Cl}\left(U\right)\right)}^{-1}=U^{-1}$. □

Corollary 1.

Let $(G,\tau )$ be an almost Moscow topological group. If F is a regularly closed subset of G, then $gF\in RC\left(G\right)$ and $F^{-1}\in RC\left(G\right)$ for every $g\in G$.

Proof. 

Since F is regularly closed, $G\setminus F$ is regularly open. By Theorem 4 (i) and (iii), $g(G\setminus F)$ and ${(G\setminus F)}^{-1}$ are regularly open. Noting that $g(G\setminus F)=G\setminus gF$ and ${(G\setminus F)}^{-1}=G\setminus F^{-1}$, the complements $gF$ and $F^{-1}$ are regularly closed. □

Theorem 5.

Let $(G,\tau )$ be an almost Moscow topological group and let $U\in RO\left(G\right)$. Then for every $g\in G$, the following closure identities hold:

(i) 

$\mathrm{Cl}\left(gU\right)=g\mathrm{Cl}\left(U\right)$;

(ii) 

$\mathrm{Cl}\left(Ug\right)=\mathrm{Cl}\left(U\right)g$;

(iii) 

$\mathrm{Cl}\left(U^{-1}\right)=\mathrm{Cl}{\left(U\right)}^{-1}$.

Proof.

(i) Let $z\in g\mathrm{Cl}\left(U\right)$. Then $z=gx$ for some $x\in \mathrm{Cl}\left(U\right)$. Consider an arbitrary open neighborhood W of z. By the almost continuity of $\phi$ at $(g,x)$, there exist open neighborhoods A of g and B of x such that $AB\subseteq \mathrm{Int}\left(\mathrm{Cl}\right(W\left)\right)$. Since $x\in \mathrm{Cl}\left(U\right)$, we have $U\cap B\ne ⌀$. Choose $u\in U\cap B$. Then $gu\in gU\cap AB\subseteq gU\cap \mathrm{Cl}\left(W\right)$, which implies $gU\cap \mathrm{Cl}\left(W\right)\ne ⌀$. Because $gU$ is open, it follows that $gU\cap W\ne ⌀$. As W arbitrary, $z\in \mathrm{Cl}\left(gU\right)$. Also, let $y\in \mathrm{Cl}\left(gU\right)$ and set $x=g^{-1}y$. Let W be any open neighborhood of x. Applying the almost continuity of $\phi$ at $(g^{-1},y)$, we obtain open neighborhoods A of $g^{-1}$ and B of y such that $AB\subseteq \mathrm{Int}\left(\mathrm{Cl}\right(W\left)\right)$. Since $y\in \mathrm{Cl}\left(gU\right)$, there exists $gu\in gU\cap B$ for some $u\in U$. Consequently, $u=g^{-1}\left(gu\right)\in A(gU\cap B)\subseteq AB\subseteq \mathrm{Int}\left(\mathrm{Cl}\left(W\right)\right)$. Thus $u\in U\cap \mathrm{Cl}\left(W\right)$, and since U is open, $U\cap W\ne ⌀$. This shows $x\in \mathrm{Cl}\left(U\right)$, whence $y=gx\in g\mathrm{Cl}\left(U\right)$. Combining both directions yields $\mathrm{Cl}\left(gU\right)=g\mathrm{Cl}\left(U\right)$.

(ii) Let $z\in \mathrm{Cl}\left(U\right)g$, so $z=xg$ with $x\in \mathrm{Cl}\left(U\right)$. Let W be an open neighborhood of z. By almost continuity of $\phi$ at $(x,g)$, there exist open neighborhoods A of x and B of g such that $AB\subseteq \mathrm{Int}\left(\mathrm{Cl}\right(W\left)\right)$. Since $x\in \mathrm{Cl}\left(U\right)$, choose $u\in U\cap A$. Then $ug\in Ug\cap AB\subseteq Ug\cap \mathrm{Cl}\left(W\right)$. The openness of $Ug$ implies $Ug\cap W\ne ⌀$, proving $z\in \mathrm{Cl}\left(Ug\right)$. Also, let $y\in \mathrm{Cl}\left(Ug\right)$ and set $x=y{g}^{-1}$. Let W be an open neighborhood of x. By almost continuity of $\phi$ at $(y,g^{-1})$, there exist open neighborhoods A of y and B of $g^{-1}$ such that $AB\subseteq \mathrm{Int}\left(\mathrm{Cl}\right(W\left)\right)$. Since $y\in \mathrm{Cl}\left(Ug\right)$, choose $ug\in Ug\cap A$ with $u\in U$. Then $u=\left(ug\right)g^{-1}\in (Ug\cap A)B\subseteq AB\subseteq \mathrm{Int}\left(\mathrm{Cl}\left(W\right)\right)$. Hence $u\in U\cap \mathrm{Cl}\left(W\right)$, and openness of U gives $U\cap W\ne ⌀$. Thus $x\in \mathrm{Cl}\left(U\right)$, implying $y=xg\in \mathrm{Cl}\left(U\right)g$. Therefore, $\mathrm{Cl}\left(Ug\right)=\mathrm{Cl}\left(U\right)g$.

(iii) Let $z\in \mathrm{Cl}{\left(U\right)}^{-1}$, so $z^{-1}\in \mathrm{Cl}\left(U\right)$. Let W be an open neighborhood of z. By almost continuity of the inversion map $\psi$ at $z^{-1}$, there exists an open neighborhood V of $z^{-1}$ such that $V^{-1}\subseteq \mathrm{Int}\left(\mathrm{Cl}\left(W\right)\right)$. Since $z^{-1}\in \mathrm{Cl}\left(U\right)$, choose $u\in U\cap V$. Then $u^{-1}\in U^{-1}\cap V^{-1}\subseteq U^{-1}\cap \mathrm{Cl}\left(W\right)$. As $U^{-1}$ is open, $U^{-1}\cap W\ne ⌀$, which yields $z\in \mathrm{Cl}\left(U^{-1}\right)$. On the other hand, let $y\in \mathrm{Cl}\left(U^{-1}\right)$ and set $x=y^{-1}$. Let W be an open neighborhood of x. By almost continuity of $\psi$ at y, there exists an open neighborhood V of y such that $V^{-1}\subseteq \mathrm{Int}\left(\mathrm{Cl}\left(W\right)\right)$. Since $y\in \mathrm{Cl}\left(U^{-1}\right)$, choose $u^{-1}\in U^{-1}\cap V$ with $u\in U$. Then $u={\left(u^{-1}\right)}^{-1}\in V^{-1}\subseteq \mathrm{Int}\left(\mathrm{Cl}\left(W\right)\right)$, so $u\in U\cap \mathrm{Cl}\left(W\right)$. Openness of U implies $U\cap W\ne ⌀$, hence $x\in \mathrm{Cl}\left(U\right)$ and $y=x^{-1}\in \mathrm{Cl}{\left(U\right)}^{-1}$. Therefore $\mathrm{Cl}\left(U^{-1}\right)=\mathrm{Cl}{\left(U\right)}^{-1}$. □

Theorem 6.

Let $(G,\tau )$ be an almost Moscow topological group and $F\in RC\left(G\right)$. Then for every $g\in G$:

(i) 

$\mathrm{Int}\left(gF\right)=g\mathrm{Int}\left(F\right)$;

(ii) 

$\mathrm{Int}\left(Fg\right)=\mathrm{Int}\left(F\right)g$;

(iii) 

$\mathrm{Int}\left(F^{-1}\right)=\mathrm{Int}{\left(F\right)}^{-1}$.

Proof. 

Since $(G,\tau )$ be an almost Moscow topological group, translations and inversion preserve $RO\left(G\right)$ and $RC\left(G\right)$.

(i). As $F\in RC\left(G\right)$, $\mathrm{Int}\left(F\right)\in RO\left(G\right)$. Translation invariance gives $g\mathrm{Int}\left(F\right)\in RO\left(G\right)$, hence open. From $\mathrm{Int}\left(F\right)\subseteq F$ we get $g\mathrm{Int}\left(F\right)\subseteq gF$, so $g\mathrm{Int}\left(F\right)\subseteq \mathrm{Int}\left(gF\right)$. Conversely, $gF\in RC\left(G\right)$ implies $\mathrm{Int}\left(gF\right)\in RO\left(G\right)$. Then $g^{-1}\mathrm{Int}\left(gF\right)\in RO\left(G\right)$ and $g^{-1}\mathrm{Int}\left(gF\right)\subseteq F$, whence $g^{-1}\mathrm{Int}\left(gF\right)\subseteq \mathrm{Int}\left(F\right)$. Left-multiplying by g yields $\mathrm{Int}\left(gF\right)\subseteq g\mathrm{Int}\left(F\right)$.

(ii). Identical to (i) using right translation $R_g$.

(iii). Since $F^{-1}\in RC\left(G\right)$, $\mathrm{Int}\left(F^{-1}\right)\in RO\left(G\right)$. Inversion preserves $RO\left(G\right)$, so $\mathrm{Int}{\left(F\right)}^{-1}\subseteq F^{-1}$ gives $\mathrm{Int}{\left(F\right)}^{-1}\subseteq \mathrm{Int}\left(F^{-1}\right)$. Conversely, ${\left(\mathrm{Int}\left(F^{-1}\right)\right)}^{-1}\subseteq F$ implies ${\left(\mathrm{Int}\left(F^{-1}\right)\right)}^{-1}\subseteq \mathrm{Int}\left(F\right)$, and inverting again yields $\mathrm{Int}\left(F^{-1}\right)\subseteq \mathrm{Int}{\left(F\right)}^{-1}$. □

Theorem 7.

Let $(G,\tau )$ be an almost Moscow topological group and A a semi-open subset of G. Then for every $g\in G$:

(i) 

$\mathrm{Cl}\left(gA\right)\subseteq g\mathrm{Cl}\left(A\right)$,

(ii) 

$\mathrm{Cl}\left(Ag\right)\subseteq \mathrm{Cl}\left(A\right)g$,

(iii) 

$\mathrm{Cl}\left(A^{-1}\right)\subseteq \mathrm{Cl}{\left(A\right)}^{-1}$.

Proof. 

Since A is semi-open, its closure $\mathrm{Cl}\left(A\right)$ is a regularly closed set. In any almost topological group, left and right translations preserve regularly closed sets. Consequently, $g\mathrm{Cl}\left(A\right)$ is closed in G. Because $gA\subseteq g\mathrm{Cl}\left(A\right)$, taking closures yields $\mathrm{Cl}\left(gA\right)\subseteq g\mathrm{Cl}\left(A\right)$. The remaining inclusions follow by same arguments using right translation and inversion. □

Theorem 8.

Let $(G,\tau )$ be an almost Moscow topological group and let $A\subseteq G$ be both semi-open and semi-closed. Then for every $g\in G$:

(i) 

$\mathrm{Cl}\left(gA\right)=g\mathrm{Cl}\left(A\right)$ and $\mathrm{Cl}\left(A^{-1}\right)=\mathrm{Cl}{\left(A\right)}^{-1}$,

(ii) 

$\mathrm{Int}\left(gA\right)=g\mathrm{Int}\left(A\right)$ and $\mathrm{Int}\left(A^{-1}\right)=\mathrm{Int}{\left(A\right)}^{-1}$.

Proof. 

Semi-openness gives $\mathrm{Cl}\left(A\right)\in RC\left(G\right)$, yielding $\mathrm{Cl}\left(gA\right)\subseteq g\mathrm{Cl}\left(A\right)$ by translation invariance. Semi-closedness implies $\mathrm{Int}\left(A\right)\in RO\left(G\right)$, so $\mathrm{Cl}\left(A\right)=\mathrm{Cl}\left(\mathrm{Int}\right(A\left)\right)$. Since closure commutes with translation for regularly open sets, $g\mathrm{Cl}\left(A\right)=g\mathrm{Cl}\left(\mathrm{Int}\right(A\left)\right)=\mathrm{Cl}\left(g\mathrm{Int}\right(A\left)\right)\subseteq \mathrm{Cl}\left(gA\right)$, establishing equality. The inverse and interior identities follow identically using inversion invariance and the duality $\mathrm{Int}\left(S\right)={\left(\mathrm{Cl}\left(S^c\right)\right)}^c$. □

Theorem 9.

Let $(G,\tau )$ be an almost Moscow topological group.

(i) 

If U is open, then $gU\subseteq \mathrm{Int}\left(g\mathrm{Int}\right(\mathrm{Cl}\left(U\right)\left)\right)$ for all $g\in G$.

(ii) 

If B is closed, then $\mathrm{Cl}\left(g\mathrm{Cl}\right(\mathrm{Int}\left(B\right)\left)\right)\subseteq gB$ for all $g\in G$.

Proof. (i) $U\subseteq \mathrm{Int}\left(\mathrm{Cl}\right(U\left)\right)$ implies $gU\subseteq g\mathrm{Int}\left(\mathrm{Cl}\right(U\left)\right)$. Since $\mathrm{Int}\left(\mathrm{Cl}\right(U\left)\right)\in RO\left(G\right)$, its translate is open, hence contained in its own interior. (ii) Closedness of B gives $\mathrm{Int}\left(B\right)\subseteq B$, so $\mathrm{Cl}\left(\mathrm{Int}\right(B\left)\right)\subseteq B$. As $\mathrm{Cl}\left(\mathrm{Int}\right(B\left)\right)\in RC\left(G\right)$, translation preserves closure, $\mathrm{Cl}\left(g\mathrm{Cl}\right(\mathrm{Int}\left(B\right)\left)\right)=g\mathrm{Cl}\left(\mathrm{Int}\right(B\left)\right)\subseteq gB$. □

Theorem 10.

Let $(G,\tau )$ be an almost Moscow topological group and $A\subseteq G$. Then for every $g\in G$:

(i) 

$\mathrm{Cl}\left(gA\right)\subseteq g{\mathrm{Cl}}_{\delta }\left(A\right)$,

(ii) 

$g{\mathrm{Int}}_{\delta }\left(A\right)\subseteq \mathrm{Int}\left(gA\right)$.

Proof. (i) Let $x\in \mathrm{Cl}\left(gA\right)$ and W be open with $g^{-1}x\in W$. Almost continuity of multiplication yields open neighborhoods $U\ni g^{-1}$, $V\ni x$ with $UV\subseteq \mathrm{Int}\left(\mathrm{Cl}\right(W\left)\right)$. Since $gA\cap V\ne \varnothing$, pick $p\in gA\cap V$. Then $g^{-1}p\in A\cap UV\subseteq A\cap \mathrm{Int}\left(\mathrm{Cl}\left(W\right)\right)$, so $g^{-1}x\in {\mathrm{Cl}}_{\delta }\left(A\right)$. (ii) If $y\in g{\mathrm{Int}}_{\delta }\left(A\right)$, then $y=gx$ with $x\in P\subseteq A$ for some regularly open P. Then $gP$ is regularly open and $y\in gP\subseteq gA$, so $y\in \mathrm{Int}\left(gA\right)$. □

Theorem 11.

Let $(G,\tau )$ be an almost Moscow topological group and U open. Then ${\mathrm{Cl}}_{\delta }\left(gU\right)=g{\mathrm{Cl}}_{\delta }\left(U\right)$ for all $g\in G$.

Proof. 

${\mathrm{Cl}}_{\delta }\left(U\right)\in RC\left(G\right)$, so $g{\mathrm{Cl}}_{\delta }\left(U\right)$ is closed, giving ${\mathrm{Cl}}_{\delta }\left(gU\right)\subseteq g{\mathrm{Cl}}_{\delta }\left(U\right)$. Conversely, let $y=gx\in g{\mathrm{Cl}}_{\delta }\left(U\right)$ and $W\ni y$ be open. Almost continuity yields open $O_1\ni g$, $O_2\ni x$ with $O_1{O}_2\subseteq \mathrm{Int}\left(\mathrm{Cl}\left(W\right)\right)$. Since $x\in {\mathrm{Cl}}_{\delta }\left(U\right)$, $U\cap \mathrm{Int}\left(\mathrm{Cl}\left(O_2\right)\right)\ne ⌀$. Pick $h\in U\cap O_2$. Then $gh\in gU\cap O_1{O}_2\subseteq gU\cap \mathrm{Int}\left(\mathrm{Cl}\left(W\right)\right)$, so $y\in {\mathrm{Cl}}_{\delta }\left(gU\right)$. □

Theorem 12.

Let $G,H$ be almost Moscow topological groups and $f:G\to H$ a homomorphism. If f is R-continuous at $e_G$, then f is almost continuous on G.

Proof. 

Fix $x\in G$ and let $V\in RO\left(H\right)$ contain $f\left(x\right)$. Then $f{\left(x\right)}^{-1}V\in RO\left(H\right)$ contains $e_H$. By R-continuity at $e_G$, $\exists U\in RO\left(G\right)$ with $e_G\in U$ and $f\left(U\right)\subseteq f{\left(x\right)}^{-1}V$. Then $f\left(xU\right)\subseteq V$. Since $xU$ is open, f is almost continuous at x. □

Theorem 13.

Let $(G,\tau )$ be an almost Moscow topological group. If $A,B\subseteq G$ are compact, then $AB$ is nearly compact.

Proof. 

The multiplication mapping $\phi :G\times G\to G$ is almost continuous. The product $A\times B$ is compact in $G\times G$. Since almost continuous images of compact spaces are nearly compact, $\phi (A\times B)=AB$ is nearly compact. □

Theorem 14.

Let $(G,\tau )$ be an almost Moscow topological group. Then the following conditions are equivalent:

(i) 

The underlying space G is almost Moscow.

(ii) 

For every $G_{\delta }$-subset $V\subseteq G$ and each $x\in V$, there exists a regularly closed set $F\subseteq G$ such that $x\in F\subseteq V$.

(iii) 

Every $G_{\delta }$-subset of G is a union of regularly closed sets.

(iv) 

Every $F_{\delta }$-subset of G is an intersection of regularly open sets.

(v) 

Every $F_{\delta }$-subset of G is θ-semi-closed.

Proof. 

The equivalence between (i) and (ii) is immediate from the Definition of an almost Moscow space. Assuming (ii), let V be an arbitrary $G_{\delta }$-subset of G. For each $x\in V$, there exists a regularly closed set $F_x$ with $x\in F_x\subseteq V$. The family $\{F_x:x\in V\}$ then satisfies $V={\bigcup }_{x\in V}{F}_x$, establishing (iii). To obtain (iv), consider an $F_{\delta }$-subset A. Its complement $G\setminus A$ is a $G_{\delta }$-set, which by (iii) admits a decomposition $G\setminus A={\bigcup }_{\alpha \in \Lambda }{F}_{\alpha }$ into regularly closed sets. Taking complements yields $A={\bigcap }_{\alpha \in \Lambda }(G\setminus F_{\alpha })$, where each $G\setminus F_{\alpha }$ is regularly open, thereby confirming (iv). Suppose (iv) holds and let A be an $F_{\delta }$-subset. By hypothesis, $A={\bigcap }_{\alpha \in \Lambda }{U}_{\alpha }$ with each $U_{\alpha }$ regularly open. The complement $G\setminus A={\bigcup }_{\alpha \in \Lambda }(G\setminus U_{\alpha })$ is consequently a union of regularly closed sets. Since any union of regularly closed sets forms a $\theta$-semi-open set, $G\setminus A$ is $\theta$-semi-open, which implies that A is $\theta$-semi-closed, verifying (v). Finally, assume (v). Then every $G_{\delta }$-subset of G is $\theta$-semi-open, as complements of $\theta$-semi-closed $F_{\delta }$-sets are $\theta$-semi-open $G_{\delta }$-sets. A well-known characterization of almost Moscow spaces states that a topological space is almost Moscow if and only if each of its $G_{\delta }$-subsets is $\theta$-semi-open. Thus, G satisfies the almost Moscow property, closing the cycle and establishing (i). These equivalences hold intrinsically for the underlying space of any almost Moscow topological group. □

Theorem 15.

Let G and H be almost topological groups, and let $f:G\to H$ be a surjective group homomorphism. If f is $G_{\delta }$-irresolute and pre-almost closed, and if G is an almost Moscow topological group, then H is also an almost Moscow topological group.

Proof. 

Let $V\subseteq H$ be a $G_{\delta }$-set. The $G_{\delta }$-irresoluteness of f yields $f^{-1}\left(V\right)\in G_{\delta }$-subset of G. Since G is almost Moscow, $f^{-1}\left(V\right)={\bigcup }_{\alpha \in \Delta }{F}_{\alpha }$ for a family $\left\{F_{\alpha }\right\}\subseteq RC\left(G\right)$. Pre-almost closedness of f implies $f\left(F_{\alpha }\right)\in RC\left(H\right)$ for each $\alpha$. Surjectivity gives $V=f\left(f^{-1}\left(V\right)\right)={\bigcup }_{\alpha \in \Delta }f\left(F_{\alpha }\right)$, establishing that H is an almost Moscow space. Now, consider ${\phi }_H:H\times H\to H$. Fix $(h_1,h_2)\in H\times H$ and let $W\in RO\left(H\right)$ contain $h_1{h}_2$. Choose $g_1,g_2\in G$ with $f\left(g_i\right)=h_i$. Since f is $G_{\delta }$-irresolute, $f^{-1}\left(W\right)\in G_{\delta }$-subset of G and $g_1{g}_2\in f^{-1}\left(W\right)$. By almost Moscow of G, there exists $F\in RC\left(G\right)$ such that $g_1{g}_2\in F\subseteq f^{-1}\left(W\right)$. The almost continuity of ${\phi }_G$ provides open neighborhoods $U_1\ni g_1$ and $U_2\ni g_2$ with $U_1{U}_2\subseteq Int(Cl\left(F\right))$. Then

$$f\left(U_1\right)f\left(U_2\right)=f\left(U_1{U}_2\right)\subseteq f(Int(Cl\left(F\right)))\subseteq Int(Cl\left(f\left(F\right)\right))\subseteq Int(Cl\left(W\right)),$$

where the second inclusion follows from the general topological fact $f(Int(Cl\left(F\right)\left)\right)\subseteq Int(Cl(f\left(F\right)\left)\right)$. Since f is a surjective homomorphism, $f\left(U_1\right)$ and $f\left(U_2\right)$ are neighborhoods of $h_1$ and $h_2$, proving ${\phi }_H$ is almost continuous at $(h_1,h_2)$. For inversion ${\psi }_H:H\to H$, let $h\in H$ and $W\in RO\left(H\right)$ contain $h^{-1}$. Choose $g\in G$ with $f\left(g\right)=h$. Then $g^{-1}\in f^{-1}\left(W\right)\in G_{\delta }$-subset of G. By almost Moscow of G, there exists $F\in RC\left(G\right)$ with $g^{-1}\in F\subseteq f^{-1}\left(W\right)$. Almost continuity of ${\psi }_G$ yields an open neighborhood $U\ni g$ such that $U^{-1}\subseteq Int(Cl\left(F\right))$. Consequently,

$$f{\left(U\right)}^{-1}=f\left(U^{-1}\right)\subseteq f(Int(Cl\left(F\right)))\subseteq Int(Cl\left(f\left(F\right)\right))\subseteq Int(Cl\left(W\right)),$$

establishing almost continuity of ${\psi }_H$ at h. Thus H is an almost Moscow topological group. □

Theorem 16.

Let G and H be almost topological groups, and let $f:G\to H$ be a surjective group homomorphism. If f is almost $G_{\delta }$-continuous and pre-almost closed, and if G is an Almost Moscow Topological Group, then H is a weakly $P_{\Sigma }$ almost topological group.

Proof. 

Let W be an arbitrary regular open subset of H. The almost $G_{\delta }$-continuity of f ensures that $f^{-1}\left(W\right)$ is a $G_{\delta }$-set in G. Since G is an almost Moscow topological group, its underlying space is almost Moscow, which characterizes every $G_{\delta }$-set as a union of regularly closed sets. Accordingly, there exists a family $\{F_{\alpha }:\alpha \in \Lambda \}$ of regularly closed subsets of G such that $f^{-1}\left(W\right)={\bigcup }_{\alpha \in \Lambda }{F}_{\alpha }$. The pre-almost closedness of f ensures that each image $f\left(F_{\alpha }\right)$ is regularly closed in H and surjectivity yields $W=f\left(f^{-1}\left(W\right)\right)={\bigcup }_{\alpha \in \Lambda }f\left(F_{\alpha }\right)$. Hence every regularly open set in H is a union of regularly closed sets, confirming that H is weakly $P_{\Sigma }$. Since H is an almost topological group by hypothesis, so that H is a weakly $P_{\Sigma }$ almost topological group. □

Theorem 17.

Let G and H be almost topological groups, and let $f:G\to H$ be an injective group homomorphism. If f is an R-map and a $G_{\delta }$-map, and if H is an almost Moscow topological group, then G is also an almost Moscow topological group.

Proof. 

Let $U\subseteq G$ be a $G_{\delta }$-set. By $G_{\delta }$-map, $f\left(U\right)\in G_{\delta }$-set of H. Since H is almost Moscow, $f\left(U\right)={\bigcup }_{\alpha \in \Lambda }{V}_{\alpha }$ for $\left\{V_{\alpha }\right\}\subseteq RC\left(H\right)$. Injectivity of f implies $U=f^{-1}\left(f\left(U\right)\right)={\bigcup }_{\alpha \in \Lambda }{f}^{-1}\left(V_{\alpha }\right)$. As f is an R-map, $f^{-1}\left(V_{\alpha }\right)\in RC\left(G\right)$ for each $\alpha$, so U is a union of regularly closed sets, thus G is almost Moscow. Now, consider ${\phi }_G:G\times G\to G$. Fix $x,y\in G$ and let $W\in RO\left(G\right)$ contain $xy$. Since f is an R-map, preimages of regularly closed sets are regularly closed; by complementation, $f^{-1}\left(V\right)\in RO\left(G\right)\subseteq {\tau }_G$ for every $V\in RO\left(H\right)$. Let $V\in RO\left(H\right)$ be any regularly open neighborhood of $f\left(x\right)f\left(y\right)=f\left(xy\right)$. The almost continuity of ${\phi }_H$ provides open sets $O_1,O_2\subseteq H$ with $f\left(x\right)\in O_1$, $f\left(y\right)\in O_2$, and $O_1{O}_2\subseteq Int(Cl\left(V\right))$. Define $U_1=f^{-1}\left(O_1\right)$ and $U_2=f^{-1}\left(O_2\right)$. Since $O_1,O_2$ are unions of regularly open sets and f pulls back regularly open sets to regularly open sets, $U_1$ and $U_2$ are open neighborhoods of x and y in G. Moreover,

$$U_1{U}_2=f^{-1}\left(O_1\right)f^{-1}\left(O_2\right)\subseteq f^{-1}\left(O_1{O}_2\right)\subseteq f^{-1}(Int(Cl\left(V\right))).$$

Because f is an R-map, $f^{-1}(Cl\left(V\right))$ is closed in G, so $f^{-1}(Int(Cl\left(V\right)))\subseteq Int\left(f^{-1}(Cl\left(V\right))\right)=Int(Cl\left(f^{-1}\left(V\right)\right))$. Taking V sufficiently small within the regular open neighborhood basis at $f\left(xy\right)$ ensures $Int(Cl\left(f^{-1}\left(V\right)\right))\subseteq Int(Cl\left(W\right))$. Thus $U_1{U}_2\subseteq Int(Cl\left(W\right))$, proving ${\phi }_G$ is almost continuous at $(x,y)$. For inversion ${\psi }_G$, let $x\in G$ and $W\in RO\left(G\right)$ contain $x^{-1}$. Choose $V\in RO\left(H\right)$ containing $f{\left(x\right)}^{-1}$. Almost continuity of ${\psi }_H$ gives an open neighborhood $O\ni f\left(x\right)$ with $O^{-1}\subseteq Int(Cl\left(V\right))$. Set $U=f^{-1}\left(O\right)$, which is open in G by R-map. Then

$$U^{-1}=f^{-1}{\left(O\right)}^{-1}=f^{-1}\left(O^{-1}\right)\subseteq f^{-1}(Int(Cl\left(V\right)))\subseteq Int(Cl\left(f^{-1}\left(V\right)\right))\subseteq Int(Cl\left(W\right)),$$

establishing almost continuity of ${\psi }_G$ at x. Hence G forms an almost Moscow topological group. □

Theorem 18.

Let G and H be almost topological groups, and let $f:G\to H$ be an injective group homomorphism. If f is an R-map and R-$G_{\delta }$-open, and if H is weakly $P_{\Sigma }$, then G is an almost Moscow topological group.

Proof. 

Let $U\subseteq G$ be an arbitrary $G_{\delta }$-set. The R-$G_{\delta }$-openness of f implies that $f\left(U\right)\in RO\left(H\right)$. Because H is weakly $P_{\Sigma }$, every regularly open set in H decomposes into a union of regularly closed sets; thus, there exists an index set $\Lambda$ and a family $\{V_{\alpha }:\alpha \in \Lambda \}\subseteq RC\left(H\right)$ such that

$$f\left(U\right)=\bigcup _{\alpha \in \Lambda }{V}_{\alpha }.$$

The injectivity of f yields an exact preimage pullback:

$$U=f^{-1}\left(f\left(U\right)\right)=f^{-1}\left(\bigcup _{\alpha \in \Lambda }{V}_{\alpha }\right)=\bigcup _{\alpha \in \Lambda }{f}^{-1}\left(V_{\alpha }\right).$$

Since f is an R-map, $f^{-1}\left(V_{\alpha }\right)\in RC\left(G\right)$ for each $\alpha \in \Lambda$. Consequently, U is expressed as a union of regularly closed subsets of G. Thus, G forms an almost Moscow Space. Since G is endowed with an almost topological group structure by hypothesis, almost continuity holds. Hence G forms an almost Moscow topological group. □

Theorem 19.

Let G and H be almost topological groups, and let $f:G\to H$ be a surjective group homomorphism. If f is $G_{\delta }$-irresolute and pre-almost open, and if $f^{-1}\left(h\right)$ is S-closed relative to G for each $h\in H$, and if G is an almost Moscow topological group, then H is also an almost Moscow topological group.

Proof. 

Let $V\subseteq H$ be an arbitrary $G_{\delta }$-set and let $h\in V$. The $G_{\delta }$-irresoluteness of f ensures that $f^{-1}\left(V\right)$ is a $G_{\delta }$-set in G. Since G is an almost Moscow topological group, its underlying space is almost Moscow; thus, for each $g\in f^{-1}\left(h\right)$, there exists a regularly closed set $F_g\subseteq G$ such that $g\in F_g\subseteq f^{-1}\left(V\right)$. The family $\{F_g:g\in f^{-1}\left(h\right)\}$ constitutes a cover of the fiber $f^{-1}\left(h\right)$ by regularly closed sets. By hypothesis, $f^{-1}\left(h\right)$ is S-closed relative to G, implying the existence of a finite subcover. Specifically, there exist points $g_1,\dots ,g_n\in f^{-1}\left(h\right)$ such that

$$f^{-1}\left(h\right)\subseteq {\displaystyle \bigcup _{i=1}^n}{F}_{g_i}.$$

Let $K={\bigcup }_{i=1}^n{F}_{g_i}$. Since the finite union of regularly closed sets is regularly closed, $K\in RC\left(G\right)$. Furthermore, $K\subseteq f^{-1}\left(V\right)$. We define the set $F\subseteq H$ by

$$F=H\setminus f(G\setminus K).$$

The complement $G\setminus K$ is regularly open in G. The pre-almost openness of f (Caldas et al., Definition 3) implies that the image of any regularly open set is regularly open; hence $f(G\setminus K)\in RO\left(H\right)$. Consequently, its complement F is regularly closed in H. Now, we verify the containment conditions for F. First, $h\in F$: if $h\notin F$, then $h\in f(G\setminus K)$, so there exists $x\in G\setminus K$ such that $f\left(x\right)=h$. This implies $x\in f^{-1}\left(h\right)$, contradicting the fact that $f^{-1}\left(h\right)\subseteq K$. Second, $F\subseteq V$: let $y\in F$. Then $y\notin f(G\setminus K)$, which implies $f^{-1}\left(y\right)\cap (G\setminus K)=\varnothing$. Thus $f^{-1}\left(y\right)\subseteq K\subseteq f^{-1}\left(V\right)$, yielding $y\in V$. We have constructed a regularly closed set $F\subseteq H$ such that $h\in F\subseteq V$, establishing that H is an almost Moscow space. Since, H is an almost topological group by hypothesis, which ensures that the multiplication and inversion maps on H are almost continuous. Hence H is an almost Moscow topological group.

□

Theorem 20.

Let G be an almost Moscow topological group and let H be a closed normal subgroup of G. If the canonical projection $\pi :G\to G/H$ is open, then the quotient space $G/H$ endowed with the quotient topology is an almost Moscow topological group.

Proof. 

Since H is normal, $G/H$ carries a natural group structure and the quotient topology, under which $\pi$ is continuous. Let $\overline{x},\overline{y}\in G/H$ and let $W\in RO(G/H)$ contain $\overline{x}\overline{y}$. Choose $x,y\in G$ with $\pi \left(x\right)=\overline{x}$ and $\pi \left(y\right)=\overline{y}$. The preimage ${\pi }^{-1}\left(W\right)$ is regularly open in G and contains $xy$. By the almost continuity of multiplication in G, there exist open neighborhoods $U_x\ni x$ and $U_y\ni y$ such that $U_x{U}_y\subseteq Int(Cl\left({\pi }^{-1}\left(W\right)\right))$. Applying $\pi$ and using its openness yields $\pi \left(U_x\right)\pi \left(U_y\right)=\pi \left(U_x{U}_y\right)\subseteq \pi (Int(Cl\left({\pi }^{-1}\left(W\right)\right)))\subseteq Int(Cl\left(\pi \left({\pi }^{-1}\left(W\right)\right)\right))=Int(Cl\left(W\right))$, where the final inclusion follows from $\pi (Int(Cl\left(A\right)\left)\right)\subseteq Int(Cl(\pi \left(A\right)\left)\right)$ for open continuous maps. Since $\pi \left(U_x\right)$ and $\pi \left(U_y\right)$ are open neighborhoods of $\overline{x}$ and $\overline{y}$, $\overline{\phi }:G/H\times G/H\to G/H$ is almost continuous at $(\overline{x},\overline{y})$. An identical argument applied to the inversion map $\overline{\psi }:G/H\to G/H$ confirms its almost continuity. Now, let $V\subseteq G/H$ be a $G_{\delta }$-set and $\overline{x}\in V$. The preimage ${\pi }^{-1}\left(V\right)$ is a $G_{\delta }$-set in G containing $x\in {\pi }^{-1}\left(\overline{x}\right)$. Since G is almost Moscow, there exists a regularly closed set $F\subseteq G$ with $x\in F\subseteq {\pi }^{-1}\left(V\right)$. The openness and continuity of $\pi$ imply $\pi \left(F\right)=\pi (Cl(Int\left(F\right)\left)\right)=Cl\left(\pi \right(Int\left(F\right)\left)\right)=Cl(Int(\pi \left(F\right)\left)\right)$, so $\pi \left(F\right)$ is regularly closed in $G/H$. Moreover, $\overline{x}\in \pi \left(F\right)\subseteq \pi \left({\pi }^{-1}\left(V\right)\right)=V$. Thus every $G_{\delta }$-set in $G/H$ contains a regularly closed neighborhood of each of its points. Hence, $G/H$ is an almost Moscow topological group. □

Theorem 21.

Let G be an almost Moscow topological group. The connected component $G_0$ containing the identity is a closed normal subgroup of G, and the quotient $G/G_0$ is an almost Moscow topological group.

Proof. 

Connected components are closed in any topological space, so $G_0$ is closed in G. The multiplication map $\phi :G\times G\to G$ is almost continuous, and almost continuous images of connected sets remain connected, thus $\phi (G_0\times G_0)=G_0{G}_0^{-1}$ is connected. Since $e\in G_0{G}_0^{-1}$ and $G_0$ is the maximal connected subset containing the identity, we obtain $G_0{G}_0^{-1}\subseteq G_0$, which implies $G_0$ is a subgroup. Normality follows from the fact that for any $g\in G$, the conjugation map $c_g\left(x\right)=gx{g}^{-1}$ is an almost continuous automorphism. The image $c_g\left(G_0\right)$ is therefore connected and contains e, so $c_g\left(G_0\right)\subseteq G_0$, that is $g{G}_0{g}^{-1}\subseteq G_0$. Hence $G_0⊴G$. Applying Theorem 20 with $H=G_0$ immediately yields that $G/G_0$ inherits the almost continuity and the almost Moscow property. Therefore, $G/G_0$ is an almost Moscow topological group. □

Theorem 22.

Let G and H be almost Moscow topological groups, and let $f:G\to H$ be a surjective homomorphism. If f is open and continuous, then the induced map $\overline{f}:G/kerf\to H$ defined by $\overline{f}(xkerf)=f\left(x\right)$ is an isomorphism of almost Moscow topological groups.

Proof. 

Let $K=kerf$. Since f is a homomorphism, K is a normal subgroup of G, and the quotient $G/K$ carries a natural group structure. Let $\pi :G\to G/K$ denote the canonical projection, which is continuous by the Definition of the quotient topology. Consider $f=\overline{f}\circ \pi$. Since f is surjective, the induced map $\overline{f}:G/K\to H$ is a bijective homomorphism of groups. Since $G/K$ is endowed with the quotient topology, a subset $U\subseteq G/K$ is open if and only if ${\pi }^{-1}\left(U\right)$ is open in G. Because f is continuous, for any open set $V\subseteq H$, $f^{-1}\left(V\right)={\pi }^{-1}\left({\overline{f}}^{-1}\left(V\right)\right)$ is open in G, which implies ${\overline{f}}^{-1}\left(V\right)$ is open in $G/K$, so $\overline{f}$ is continuous. Now, let U be an open subset of $G/K$. Then ${\pi }^{-1}\left(U\right)$ is open in G by the continuity of $\pi$. Since f is surjective and open, the image $f\left({\pi }^{-1}\left(U\right)\right)$ is open in H. Observing that $f\left({\pi }^{-1}\left(U\right)\right)=\overline{f}\left(\pi \left({\pi }^{-1}\left(U\right)\right)\right)=\overline{f}\left(U\right)$, we conclude that $\overline{f}\left(U\right)$ is open in H. Thus, $\overline{f}$ is an open map. Since $\overline{f}$ is a continuous, open bijection, it is a homeomorphism. Consequently, $\overline{f}$ preserves the topological structure of H on the quotient space $G/K$. Because H is an almost Moscow topological group, the space $G/K$ inherits the almost Moscow property and the almost continuity of group operations via the homeomorphism $\overline{f}$. Therefore, $\overline{f}$ is an isomorphism of almost Moscow topological groups. □

Lemma 1.

Let G be an extremally disconnected almost Moscow topological group. If there exists $U\in RO\left(G\right)$ such that $e\in U\subseteq B\left(G\right)$, where $B\left(G\right)=\{x\in G:x^2=e\}$, then G contains an open Boolean subgroup.

Proof. 

Let $U\in RO\left(G\right)$ satisfy $e\in U\subseteq B\left(G\right)$. Since G is extremally disconnected, $Cl\left(U\right)\in {\tau }_G$. The almost continuity of multiplication $\phi :G\times G\to G$ at $(e,e)$ implies the existence of open neighborhoods $V_1,V_2\ni e$ such that

$$V_1{V}_2\subseteq Int(Cl\left(U\right))=Cl\left(U\right),$$

where the equality follows from extremal disconnectedness. Set $W=V_1\cap V_2$. Then W is an open neighborhood of e and $W^2\subseteq Cl\left(U\right)$. Since G is extremally disconnected, $Cl\left(W\right)\in {\tau }_G$. For $x,y\in Cl\left(W\right)$, we have $xy\in Cl\left(W^2\right)\subseteq Cl(Cl\left(U\right))=Cl\left(U\right)$. In particular, for any $x\in Cl\left(W\right)$, we obtain $x^2\in Cl\left(U\right)\subseteq Cl\left(B\left(G\right)\right)=B\left(G\right)$, since $B\left(G\right)$ is a subgroup. Thus $x^4=e$ for all $x\in Cl\left(W\right)$. Now let $x\in Cl\left(W\right)$. Since $W\subseteq Cl\left(U\right)$ and $U\subseteq B\left(G\right)$, we have $W\subseteq Cl\left(B\right(G\left)\right)=B\left(G\right)$. Hence $Cl\left(W\right)\subseteq Cl\left(B\right(G\left)\right)=B\left(G\right)$. Therefore $Cl\left(W\right)$ is an open subset of G contained in $B\left(G\right)$, and since $Cl\left(W\right)$ is a subgroup, it is an open Boolean subgroup of G. □

Theorem 23.

Every extremally disconnected almost Moscow topological group contains an open Boolean subgroup.

Proof. 

Let G be an extremally disconnected almost Moscow topological group. The inversion map $\iota :G\to G$ defined by $\iota \left(x\right)=x^{-1}$ is almost continuous by Definition 32. Consider the set $B\left(G\right)=\{x\in G:\iota (x)=x\}$. Since G is almost Moscow and $e\in G$, there exists $F\in RC\left(G\right)$ such that $e\in F\subseteq G$. As G is extremally disconnected, $F=Cl(Int(F\left)\right)$ with $Int\left(F\right)\in RO\left(G\right)$. Let $U=Int\left(F\right)$. Then $U\in RO\left(G\right)$ and $e\in U$. Claim $U\subseteq B\left(G\right)$. Let $x\in U$. Since $\iota$ is almost continuous at x and U is a regularly open neighborhood of x, there exists an open neighborhood $V\ni x$ such that $\iota \left(V\right)\subseteq Int(Cl(U\left)\right)=Cl\left(U\right)$. Since G is extremally disconnected and U is regularly open, $Cl\left(U\right)$ is open and regularly closed. Thus $\iota \left(V\right)\subseteq Cl\left(U\right)$. For any $v\in V$, $v^{-1}\in Cl\left(U\right)$. Since $x\in V$ and $x\in U$, $x^{-1}\in Cl\left(U\right)\cap U=U$. Hence $x^{-1}=x$, which implies $x^2=e$. Therefore $U\subseteq B\left(G\right)$. By Lemma 1, G contains an open Boolean subgroup. □

Corollary 2.

If G is an extremally disconnected almost Moscow topological group, then the quotient $G/B_0$, where $B_0$ is the open Boolean subgroup, is a discrete almost Moscow topological group.

Proof. 

By Theorem 23, G contains an open Boolean subgroup $B_0$. Since $B_0$ is open, the canonical projection $\pi :G\to G/B_0$ is an open mapping. The quotient topology on $G/B_0$ is discrete because $B_0$ is open. By Theorem 20, $G/B_0$ is a discrete almost Moscow topological group. □

Proposition 2.

Every almost Moscow space is δ-extremally disconnected.

Proof. 

Let $(X,\tau )$ be an almost Moscow space. Fix an open set $U\subseteq X$ and a point $x\in Cl\left(U\right)$. Since X is almost Moscow, there exists a regular closed set $F_1\subseteq X$ such that $x\in F_1$. Setting $V_1=Int\left(F_1\right)$, we observe that $x\in Cl\left(V_1\right)$ and, because $x\in Cl\left(U\right)$, the intersection $V_1\cap U$ is nonempty with $x\in Cl(V_1\cap U)$. Applying the almost Moscow property to the open set $V_1\cap U$ yields a regular closed set $F_2$ satisfying $x\in F_2\subseteq Cl(V_1\cap U)\subseteq Cl\left(U\right)$. Proceeding inductively, suppose regular closed sets $F_1\supseteq \dots \supseteq F_n$ have been constructed with $x\in F_n\subseteq Cl\left(U\right)$; letting $V_n=Int\left(F_n\right)$, the same argument produces a regular closed set $F_{n+1}$ such that $x\in F_{n+1}\subseteq Cl(V_n\cap U)\subseteq Cl\left(U\right)$. The resulting decreasing sequence $\{F_n:n\in \mathbb{N}\}$ of regular closed sets yields the $G_{\delta }$-subset $B_x={\bigcap }_{n=1}^{\infty }{F}_n$, which contains x and satisfies $B_x\subseteq Cl\left(U\right)$. Consequently, every point in $Cl\left(U\right)$ is contained in a $G_{\delta }$-subset of $Cl\left(U\right)$, which implies $Cl\left(U\right)={\bigcup }_{x\in Cl\left(U\right)}{B}_x$ is a union of $G_{\delta }$-sets, thereby establishing that X is $\delta$-extremally disconnected. □

Remark 3.

Every almost Moscow space is a Moscow space by Proposition 2.

Remark 4.

The converse of the Proposition 2 is not true, which is provided in the following exmaple.

Example 6.

An uncountable set with the cocountable topology is δ-extremally disconnected but not almost Moscow.

Theorem 24.

Let G be an almost Moscow topological group. If G is locally countably S-closed and km-perfect, then G is extremally disconnected.

Proof. 

Let $R\subseteq G$ be a regular open set and fix $x\in G\setminus R$. By local countable S-closedness, there exists a regular open neighborhood U of x that is countably S-closed relative to G. If $U\cap R=\varnothing$, then $x\notin Cl\left(R\right)$ and the result follows. Otherwise, set $A=U\cap R$, which is regular open and, by the nature of relative S-closedness under regular open intersections, countably S-closed relative to G. Since G is km-perfect and $x\notin A$, there exists a sequence ${\left\{G_n\right\}}_{n\in \omega }$ of open sets such that $A\subseteq {\bigcup }_{n\in \omega }{G}_n$ and $x\notin {\bigcup }_{n\in \omega }Cl\left(G_n\right)$. The family $\{Cl\left(G_n\right):n\in \omega \}$ constitutes a countable rc-cover of A, and by the relative countable S-closedness of A, there exists $m\in \omega$ with $A\subseteq {\bigcup }_{n=1}^{m}Cl\left(G_n\right)$. Define $V=G\setminus {\bigcup }_{n=1}^{m}Cl\left(G_n\right)$, which is an open neighborhood of x. Then $U\cap V$ is an open neighborhood of x disjoint from R, implying $x\notin Cl\left(R\right)$. Consequently, $Cl\left(R\right)\subseteq R$, so R is closed. Since every regular open set is closed, G is extremally disconnected. □

Corollary 3.

Let G be an almost Moscow topological group. Then G is extremally disconnected if and only if G is locally countably S-closed and km-perfect.

Proof. 

Assume G is locally countably S-closed and km-perfect. Let $R\in RO\left(G\right)$ and $x\in G\setminus R$. By locally countably S-closedness, there exists $U\in RO\left(G\right)$ such that $x\in U$ and U is countably S-closed relative to G. If $U\cap R=⌀$, then $x\notin Cl\left(R\right)$. Otherwise, set $A=U\cap R$. Then, A is countably S-closed relative to G. Since G is km-perfect and $x\notin A$, there exists a sequence ${\left\{G_n\right\}}_{n\in \omega }$ of open sets such that $A\subseteq {\bigcup }_{n\in \omega }{G}_n$ and $x\notin {\bigcup }_{n\in \omega }{G}_n$. The family $\{Cl\left(G_n\right):n\in \omega \}$ is a countable rc-cover of A. Countable S-closedness relative to G yields $m\in \omega$ with $A\subseteq {\bigcup }_{n=1}^{m}Cl\left(G_n\right)$. Let $V=G\setminus {\bigcup }_{n=1}^{m}Cl\left(G_n\right)$. Then V is open, $x\in V$, and $V\cap R=⌀$, so $x\notin Cl\left(R\right)$. Hence R is closed, proving extremal disconnectedness. The converse follows since extremally disconnected spaces are trivially km-perfect and satisfy local countable S-closedness under regular covers. □

Theorem 25.

Let G be an almost Moscow topological group. If G is locally rc-Lindelöf and mildly Hausdorff, then G is δ-extremally disconnected.

Proof. 

Let $R\in RO\left(G\right)$ and $y\in G\setminus R$. Mildly Hausdorffness implies $R={\bigcup }_{z\in R}{D}_z$ where each $D_z$ is $\delta$-closed. Each $\delta$-closed set is an intersection of regularly closed sets; thus for each $z\in R$, there exists $E_z\in RC\left(G\right)$ with $z\in E_z$ and $y\notin E_z$. Locally rc-Lindelöfness provides $U\in \tau$ such that $y\in U$ and U is rc-Lindelöf relative to G. Set $A=U\cap R$. Then A is rc-Lindelöf relative to G, and $\{E_z:z\in A\}$ is an rc-cover of A. There exists a countable $C\subseteq A$ such that $A\subseteq {\bigcup }_{z\in C}{E}_z$. Define $B_y={\bigcap }_{z\in C}(G\setminus E_z)$. Then $B_y$ is a $G_{\delta }$-set, $y\in B_y$, and $B_y\cap R=⌀$. Hence $G\setminus R$ is a union of $G_{\delta }$-sets, establishing $\delta$-extremally disconnectedness. □

Lemma 2.

For an almost Moscow topological group G, the following are equivalent:

(i) 

G is rc-Lindelöf.

(ii) 

Every semi-open cover of G admits a countable subfamily whose closures cover G.

Proof. 

Assume (i). Let ${\left\{S_{\alpha }\right\}}_{\alpha \in I}$ be a semi-open cover. For each $\alpha$, choose $U_{\alpha }\in RO\left(G\right)$ such that $U_{\alpha }\subseteq S_{\alpha }$ and $Cl\left(U_{\alpha }\right)=Cl\left(S_{\alpha }\right)$. The family $\left\{U_{\alpha }\right\}$ is an rc-cover of G, so there exists a countable $J\subseteq I$ with $G={\bigcup }_{\alpha \in J}Cl\left(U_{\alpha }\right)={\bigcup }_{\alpha \in J}Cl\left(S_{\alpha }\right)$. Conversely, assume (ii) and let ${\left\{R_{\beta }\right\}}_{\beta \in \Lambda }$ be an rc-cover. Regular closed sets are semi-open, so there exists a countable $K\subseteq \Lambda$ such that $G={\bigcup }_{\beta \in K}Cl\left(R_{\beta }\right)={\bigcup }_{\beta \in K}{R}_{\beta }$, proving (i). □

Proposition 3.

Let G and H be almost Moscow topological groups that are weak P-spaces, and let $f:G\to H$ be an irresolute surjective homomorphism. If G is rc-Lindelöf, then H is rc-Lindelöf.

Proof. 

Let ${\left\{V_i\right\}}_{i\in I}$ be a semi-open cover of H. Irresoluteness implies ${\left\{f^{-1}\left(V_i\right)\right\}}_{i\in I}$ is a semi-open cover of G. By Lemma 2, there exists a countable $J\subseteq I$ such that $G={\bigcup }_{i\in J}Cl\left(f^{-1}\left(V_i\right)\right)$. Set $U={\bigcup }_{i\in J}{f}^{-1}\left(V_i\right)$. The weak P-space property yields $Cl\left(U\right)={\bigcup }_{i\in J}Cl\left(f^{-1}\left(V_i\right)\right)=G$. Irresoluteness and surjectivity imply $H=f\left(G\right)\subseteq f(Cl\left(U\right))\subseteq Cl\left(f\left(U\right)\right)=Cl\left({\bigcup }_{i\in J}{V}_i\right)$. Since H is a weak P-space, $Cl\left({\bigcup }_{i\in J}{V}_i\right)={\bigcup }_{i\in J}Cl\left(V_i\right)$, so H is covered by the closures of a countable subfamily. Lemma 2 gives rc-Lindelöfness of H. □

Lemma 3.

In any almost Moscow topological group, a locally finite union of regularly closed subsets is regularly closed.

Proof. 

Let ${\left\{F_{\alpha }\right\}}_{\alpha \in A}$ be a locally finite family in $RC\left(G\right)$. Regular closed sets are semi-open, and arbitrary unions of semi-open sets are semi-open, so $F={\bigcup }_{\alpha \in A}{F}_{\alpha }$ is semi-open. Local finiteness implies $Cl\left(F\right)={\bigcup }_{\alpha \in A}Cl\left(F_{\alpha }\right)={\bigcup }_{\alpha \in A}{F}_{\alpha }=F$. Thus F is closed and semi-open, hence regularly closed. □

Theorem 26.

Let G be an almost Moscow topological group that is locally countably rc-paracompact. Then the semi-regularization $G_s=(G,{\tau }_s)$ is an ω-extremally disconnected almost Moscow topological group.

Proof. 

The semi-regularization ${\tau }_s$ preserves $RO\left(G\right)$ and $RC\left(G\right)$, so Definition 32 hold for $G_s$. Let $R\in RO\left(G_s\right)$ and $x\in G_{\delta }-{Cl}_{{\tau }_s}\left(R\right)$. There exists $G\in {\tau }_s$ with $x\in G$, $G\cap R=⌀$, and $G={\bigcap }_{n\in \omega }{G}_n$ for $G_n\in {\tau }_s$. Choose $U_n\in RO(G,\tau )$ such that $x\in U_n\subseteq G_n$. Locally countably rc-paracompactness yields $U\in \tau$ with $x\in U$ and U countably rc-paracompact relative to G. If $U\cap R=⌀$, then $x\notin {Cl}_{{\tau }_s}\left(R\right)$. Otherwise, $A=U\cap R$ is countably rc-paracompact relative to G, and $\mathcal{U}=\{G\setminus U_n:n\in \omega \}$ is a countable rc-cover of A. There exists a locally finite rc-refinement $\mathcal{V}$ of $\mathcal{U}$. By Lemma 3, $V=G\setminus \bigcup \mathcal{V}$ is in $RO(G,\tau )$ and $x\in V$. Then $U\cap V\in \tau$, $x\in U\cap V$, and $(U\cap V)\cap R=⌀$, so $x\notin {Cl}_{{\tau }_s}\left(R\right)$. Thus ${Cl}_{{\tau }_s}\left(R\right)\subseteq G_{\delta }-{Cl}_{{\tau }_s}\left(R\right)$, proving $\omega$-extremally disconnectedness. □

Corollary 4.

Let G be a Hausdorff, semi-regular, first countable almost Moscow topological group that is locally countably rc-paracompact. Then G is discrete.

Proof. 

Theorem 26 implies $G_s$ is $\omega$-extremally disconnected. Semi-regularity gives $\tau ={\tau }_s$, so G is $\omega$-extremally disconnected. In a Hausdorff first countable space, $\omega$-extremally disconnectedness implies extremally disconnectedness. Since, any Hausdorff first countable extremally disconnected topological group is discrete, G is discrete. □

Theorem 27.

Let G be an almost Moscow topological group. If G is locally countably S-closed and km-perfect, then G is a topological group that is extremally disconnected. Consequently, G is a strong PT-group and hence a PT-group.

Proof. 

By Theorem 24, G is extremally disconnected. In extremally disconnected spaces, regularly open sets are clopen, so $Int(Cl(U\left)\right)=Cl\left(U\right)$ for every open U. Consider $\phi :G\times G\to G$, then for every open $W\ni xy$, there exist open $U\ni x$, $V\ni y$ such that $\phi (U\times V)\subseteq Int(Cl(W\left)\right)$. Since G is extremally disconnected, $Int(Cl(W\left)\right)=Cl\left(W\right)$. But in topological groups, the closure of an open set containing the identity has nonempty interior, and homogeneity implies $Cl\left(W\right)=W$ for neighborhoods. Thus $\phi (U\times V)\subseteq W$, establishing full continuity of multiplication. An identical argument applies to inversion. Hence G is a topological group. Since extremally disconnected spaces are Moscow, G forms a Moscow topological group, therefore G is a strong PT-group and hence a PT-group.

□

4. Conclusion

The introduction of almost Moscow topological groups establishes a cohesive framework that unifies almost continuous operations with $G_{\delta }$-regular spatial structure. By proving that every almost Moscow space is $\delta$-extremally disconnected, the class is firmly positioned within Arhangel’skii’s Moscow hierarchy while retaining stronger closure-interior control. This duality enables precise analysis of translation invariance, homomorphic permanence, and quotient stability under weakened continuity, extending classical topological group techniques to non-standard regularity environments. Covering properties yield sharp structural classifications within this framework. Extremal disconnectedness arises precisely under locally countably S-closedness and km-perfectness, while local rc-Lindelöf and locally countably rc-paracompact conditions force $\delta$- and $\omega$-extremally disconnectedness, respectively. Notably, Hausdorff semi-regular first countable instances collapse to the discrete topology, illustrating inherent structural rigidity. Crucially, locally countably S-closed km-perfect groups upgrade to classical topological groups that are strong PT-groups. These findings position almost Moscow topological groups as a robust platform for exploring algebraic operations under relaxed continuity, with direct implications for completion theory, quotient dynamics, and generalized topological group representations. Future research may investigate the interplay between almost continuity and uniform structures, characterize completion-friendly subgroups, and extend equivariant dynamics to almost continuous transformation groups, further solidifying the class at the intersection of topological algebra, covering properties, and generalized regularity.