On the Basics of Ideal Moscow Spaces

1. Introduction

The study of special classes of topological spaces often depends on the possibility of representing closures, neighborhoods, or other topologically significant sets by simpler classes of sets. One important example is the class of Moscow spaces, introduced in the study of general topology and subsequently investigated in connection with topological groups, products, and C-embeddings; see Arhangel’skii [1].

Recall that a subset of a topological space is called a $G_{\delta }$ set if it is a countable intersection of open sets. A Hausdorff space X is called a Moscow space if, for every open set $U\subseteq X$ and every point $x\in {cl}_X\left(U\right)$, there is a $G_{\delta }$ set P such that

$$x\in P\subseteq {cl}_X\left(U\right).$$

Equivalently,

$${cl}_X\left(U\right)=\bigcup \{P:P\mathrm{is}\mathrm{a}{G}_{\delta }\mathrm{subset}\mathrm{of}X,P\subseteq {cl}_X\left(U\right)\}.$$

This formulation is consistent with the standard definition of Moscow spaces; see [1].

Ideals provide another method for organizing subsets of a topological space. An ideal consists of sets regarded as small relative to a particular application. The ideal may be the family of finite sets, the family of countable sets, or another family satisfying the axioms of an ideal. The systematic use of ideals in topology, including the associated local functions and ideal-induced topologies, is developed in works such as Janković and Hamlett [2].

The purpose of this paper is to combine these two ideas. We introduce a notion of an $\mathcal{I}$-$G_{\delta }$ set, in which a set is allowed to differ from a $G_{\delta }$ set by a member of an ideal $\mathcal{I}$. We then define an ideal Moscow space by requiring the classical local $G_{\delta }$ condition to hold with $\mathcal{I}$-$G_{\delta }$ sets in place of ordinary $G_{\delta }$ sets.

The terminology “ideal Moscow space” is used here for this specific proposed definition. The results below are elementary consequences of the definition unless otherwise indicated. They are intended to establish a rigorous foundation for further investigations rather than to suggest that every possible ideal extension of Moscow spaces has been covered.

The paper is organized as follows. Section 2 recalls the necessary notions. Section 3 introduces ideal-$G_{\delta }$ sets and ideal Moscow spaces. Section 4 establishes basic results. Section 5 presents examples and counterexamples. Section 6 discusses subspaces, products, and mappings. Section 7 lists questions for further research.

2. Preliminaries

Throughout the paper, $(X,\mathcal{T})$ denotes a topological space. When a separation axiom is needed, it will be stated explicitly. For $A\subseteq X$, the closure and interior of A are denoted by ${cl}_X\left(A\right)$ and ${int}_X\left(A\right)$, respectively.

Definition 1.

A nonempty family $\mathcal{I}\subseteq \mathcal{P}\left(X\right)$ is called anideal on Xif it satisfies the following conditions:

• If $A\in \mathcal{I}$ and $B\subseteq A$, then $B\in \mathcal{I}$.
• If $A,B\in \mathcal{I}$, then $A\cup B\in \mathcal{I}$.

Thus an ideal is closed under taking subsets and finite unions. Unless stated otherwise, we do not assume that $\mathcal{I}$ is closed under countable unions.

Definition 2.

A triple $(X,\mathcal{T},\mathcal{I})$ consisting of a topological space $(X,\mathcal{T})$ and an ideal $\mathcal{I}$ on X is called anideal topological space.

Example 1.

Let X be any nonempty set. The families

$${\mathcal{I}}_{\mathrm{fin}}={\left[X\right]}^{<\omega }\mathrm{and}{\mathcal{I}}_{\mathrm{ctbl}}={\left[X\right]}^{\le {\aleph }_0}$$

are ideals on X. Here ${\mathcal{I}}_{\mathrm{fin}}$ consists of all finite subsets of X, while ${\mathcal{I}}_{\mathrm{ctbl}}$ consists of all countable subsets of X. The second family is a countably complete ideal: a countable union of countable sets is countable.

Remark 1.

The family $\mathcal{P}\left(X\right)$ is also an ideal on X, called the improper ideal. Some authors reserve the word “ideal” for proper ideals, in which case $\mathcal{P}\left(X\right)$ is excluded. We allow the improper ideal in order to make explicit a degeneracy that must be avoided in applications.

Definition 3.

A subset $G\subseteq X$ is called a $G_{\delta }$ set if

$$G={\displaystyle \bigcap _{n=1}^{\infty }}{U}_n$$

for some sequence ${\left(U_n\right)}_{n\in \mathbb{N}}$ of open subsets of X.

Every open set is a $G_{\delta }$ set, since it can be expressed as a countable intersection of copies of itself. In particular, X and ⌀ are $G_{\delta }$ sets.

Definition 4.

Thepseudocharacterof a point $x\in X$, denoted by $\psi (x,X)$, is the least cardinality of a family $\mathcal{U}$ of open neighborhoods of x such that

$$\left\{x\right\}=\bigcap \mathcal{U}.$$

A space has countable pseudocharacter if $\psi (x,X)\le {\aleph }_0$ for every $x\in X$.

The relation between pseudocharacter and Moscow spaces is particularly useful: if every point is a $G_{\delta }$ set, then the Moscow property follows immediately; (see [1], Section 1).

Definition 5.

A topological space X is called aMoscow spaceif, for every open set $U\subseteq X$ and every $x\in {cl}_X\left(U\right)$, there exists a $G_{\delta }$ set $P\subseteq X$ such that

$$x\in P\subseteq {cl}_X\left(U\right).$$

The local formulation above is equivalent to the usual union-of-$G_{\delta }$-sets formulation. Indeed, the existence of such a set P for each $x\in {cl}_X\left(U\right)$ implies that the union of all eligible $G_{\delta }$ sets equals ${cl}_X\left(U\right)$.

3. Ideal-${\mathbf{G}}_{\mathbf{\delta }}$ Sets and Ideal Moscow Spaces

We now introduce the principal definition of the paper.

Definition 6.

Let $(X,\mathcal{T},\mathcal{I})$ be an ideal topological space. A subset $A\subseteq X$ is called an$\mathcal{I}$-$G_{\delta }$ setif there exists a $G_{\delta }$ set $G\subseteq X$ such that

$$A▵G\in \mathcal{I},$$

where

$$A▵G=(A\setminus G)\cup (G\setminus A)$$

is the symmetric difference of A and G.

Thus A is an $\mathcal{I}$-$G_{\delta }$ set precisely when it agrees with some $G_{\delta }$ set outside an ideal-small set.

Example 2.

Let $X=\mathbb{R}$ with its usual topology and let $\mathcal{I}={\mathcal{I}}_{\mathrm{fin}}$ be the ideal of finite subsets. The set

$$A=(0,1)\cup \left\{2\right\}$$

is an $\mathcal{I}$-$G_{\delta }$ set. Indeed, $(0,1)$ is a $G_{\delta }$ set and

$$A▵(0,1)=\left\{2\right\}\in {\mathcal{I}}_{\mathrm{fin}}.$$

Example 3.

Let X be any topological space and let $\mathcal{I}=\{⌀\}$ be the trivial ideal. Then

$$A\mathrm{is}\mathrm{an}\mathcal{I}-G_{\delta }\mathrm{set}⟺A\mathrm{is}\mathrm{a}{G}_{\delta }\mathrm{set}.$$

This follows because $A▵G=⌀$ is equivalent to $A=G$.

Definition 7.

Let $(X,\mathcal{T},\mathcal{I})$ be an ideal topological space. We call X an$\mathcal{I}$-Moscow space, or anideal Moscow space relative to $\mathcal{I}$, if for every open set $U\subseteq X$ and every point $x\in {cl}_X\left(U\right)$, there exists an $\mathcal{I}$-$G_{\delta }$ set P such that

$$x\in P\subseteq {cl}_X\left(U\right).$$

Equivalently, for every open $U\subseteq X$,

$${cl}_X\left(U\right)=\bigcup \left\{P\subseteq {cl}_X\left(U\right):P\mathrm{is}\mathrm{an}\mathcal{I}-G_{\delta }\mathrm{set}\right\}.$$

Remark 2.

The requirement $P\subseteq {cl}_X\left(U\right)$ is important. Merely requiring a $G_{\delta }$ set to approximate ${cl}_X\left(U\right)$ modulo $\mathcal{I}$ would give a different definition. The present definition preserves the local structure of the classical Moscow property while allowing the witnessing set to differ from a $G_{\delta }$ set by an ideal-small set.

Remark 3.

The definition depends on both $\mathcal{T}$ and $\mathcal{I}$. In particular, the same topological space can be an $\mathcal{I}$-Moscow space for one ideal and fail to be an $\mathcal{I}$-Moscow space for another. We will establish a precise monotonicity result below.

4. Basic Theorems

We begin with a simple observation concerning the relation between the classical and ideal versions.

Lemma 1.

Let $(X,\mathcal{T},\mathcal{I})$ be an ideal topological space. Every $G_{\delta }$ subset of X is an $\mathcal{I}$-$G_{\delta }$ set.

Proof. 

If G is a $G_{\delta }$ set, then

$$G▵G=⌀\in \mathcal{I}.$$

Thus G is an $\mathcal{I}$-$G_{\delta }$ set by definition. □

Theorem 1.

Every Moscow space is an $\mathcal{I}$-Moscow space for every ideal $\mathcal{I}$ on its underlying set.

Proof. 

Let X be a Moscow space, and let $U\subseteq X$ be open. For any $x\in {cl}_X\left(U\right)$, the Moscow property provides a $G_{\delta }$ set P satisfying

$$x\in P\subseteq {cl}_X\left(U\right).$$

By the preceding lemma, P is an $\mathcal{I}$-$G_{\delta }$ set. Consequently, X is an $\mathcal{I}$-Moscow space. □

Corollary 1.

If $\mathcal{I}=\{⌀\}$, then X is an $\mathcal{I}$-Moscow space if and only if X is a Moscow space.

Proof. 

The forward implication follows because the $\mathcal{I}$-$G_{\delta }$ sets are exactly the ordinary $G_{\delta }$ sets when $\mathcal{I}=\{⌀\}$. The reverse implication is the preceding theorem. □

The next result describes how the property changes when the ideal is enlarged.

Theorem 2.

(Monotonicity in the ideal) Let I and J be ideals on X such that

$$I\subseteq J.$$

If X is an $\mathcal{I}$-Moscow space, then X is a J-Moscow space.

Proof. 

Suppose that X is an $\mathcal{I}$-Moscow space. Let U be open and let $x\in {cl}_X\left(U\right)$. There exists an $\mathcal{I}$-$G_{\delta }$ set P such that

$$x\in P\subseteq {cl}_X\left(U\right).$$

By definition, some $G_{\delta }$ set G satisfies

$$P▵G\in \mathcal{I}.$$

Since $I\subseteq J$, we also have

$$P▵G\in J.$$

Therefore P is a J-$G_{\delta }$ set, and the defining condition for a J-Moscow space holds. □

Remark 4.

The monotonicity theorem means that enlarging the ideal weakens the requirement. It is therefore important to specify the ideal when studying ideal Moscow spaces. For a nontrivial theory, one should generally choose a proper ideal that reflects the intended meaning of “small” subsets.

Proposition 1.

The property of being an $\mathcal{I}$-Moscow space is invariant under homeomorphisms, provided that the ideal is transported along the homeomorphism.

Proof. 

Let $h:X\to Y$ be a homeomorphism, and define an ideal on Y by

$$J=\left\{h\right(A):A\in I\}.$$

Because h is bijective, J is an ideal on Y. Homeomorphisms preserve open sets, closures, and countable intersections of open sets. Moreover,

$$h(A▵B)=h\left(A\right)▵h\left(B\right).$$

It follows that A is an I-$G_{\delta }$ set if and only if $h\left(A\right)$ is a J-$G_{\delta }$ set.

Suppose that X is I-Moscow. Let $V\subseteq Y$ be open and $y\in {cl}_Y\left(V\right)$. Put $U=h^{-1}\left(V\right)$ and $x=h^{-1}\left(y\right)$. Then U is open and $x\in {cl}_X\left(U\right)$. Choose an $\mathcal{I}$-$G_{\delta }$ set P with

$$x\in P\subseteq {cl}_X\left(U\right).$$

The set $h\left(P\right)$ is a J-$G_{\delta }$ set and

$$y\in h\left(P\right)\subseteq {cl}_Y\left(V\right).$$

Thus Y is J-Moscow. Applying the same argument to $h^{-1}$ gives the converse. □

Theorem 3.

Every space of countable pseudocharacter is an $\mathcal{I}$-Moscow space for every ideal $\mathcal{I}$.

Proof. 

Let X have countable pseudocharacter. For every $x\in X$, there is a countable family of open neighborhoods ${\left(U_n\right)}_{n\in \mathbb{N}}$ such that

$$\left\{x\right\}={\displaystyle \bigcap _{n=1}^{\infty }}{U}_n.$$

Hence $\left\{x\right\}$ is a $G_{\delta }$ set. If U is open and $x\in {cl}_X\left(U\right)$, take $P=\left\{x\right\}$. Then

$$x\in P\subseteq {cl}_X\left(U\right).$$

The set P is a $G_{\delta }$ set and therefore an $\mathcal{I}$-$G_{\delta }$ set. Thus X is $\mathcal{I}$-Moscow. □

Corollary 2.

Every first-countable $T_1$ space is an $\mathcal{I}$-Moscow space for every ideal $\mathcal{I}$.

Proof. 

In a first-countable $T_1$ space, each singleton is a $G_{\delta }$ set. Indeed, if $\left(U_n\right)$ is a countable local base at x, then

$${\displaystyle \bigcap _{n=1}^{\infty }}{U}_n=\left\{x\right\}.$$

The preceding theorem applies. □

Remark 5.

The countable-pseudocharacter theorem is a sufficient condition, not a characterization. Moscow spaces can have points that are not $G_{\delta }$ points; see the discussion in [1].

5. Countable Ideals and Intersections

The definition of an $\mathcal{I}$-$G_{\delta }$ set does not, for an arbitrary ideal, guarantee that countable intersections of $\mathcal{I}$-$G_{\delta }$ sets are again $\mathcal{I}$-$G_{\delta }$. The reason is that an ideal is generally closed only under finite unions. The following proposition identifies a useful sufficient condition.

Proposition 2.

Let $\mathcal{I}$ be a σ-ideal on X, meaning that it is closed under countable unions. If ${\left(A_n\right)}_{n\in \mathbb{N}}$ is a sequence of $\mathcal{I}$-$G_{\delta }$ sets, then

$${\displaystyle \bigcap _{n=1}^{\infty }}{A}_n$$

is an $\mathcal{I}$-$G_{\delta }$ set.

Proof. 

For each n, choose a $G_{\delta }$ set $G_n$ such that

$$A_n▵G_n\in \mathcal{I}.$$

Put

$$A={\displaystyle \bigcap _{n=1}^{\infty }}{A}_n,G={\displaystyle \bigcap _{n=1}^{\infty }}{G}_n.$$

Since a countable intersection of $G_{\delta }$ sets is again a $G_{\delta }$ set, G is a $G_{\delta }$ set.

We claim that

$$A▵G\subseteq {\displaystyle \bigcup _{n=1}^{\infty }}(A_n▵G_n).$$

Indeed, if a point belongs to exactly one of A and G, then it must belong to exactly one of $A_n$ and $G_n$ for at least one index n. The right-hand side belongs to $\mathcal{I}$ because $\mathcal{I}$ is a $\sigma$-ideal. By heredity,

$$A▵G\in \mathcal{I}.$$

Thus A is an $\mathcal{I}$-$G_{\delta }$ set. □

Remark 6.

The preceding proposition concerns the auxiliary class of $\mathcal{I}$-$G_{\delta }$ sets. It does not by itself establish a new characterization of ideal Moscow spaces. Its purpose is to clarify which closure properties are available when the ideal is countably complete.

Example 4.

The ideal of countable subsets of an arbitrary set is a σ-ideal. Hence, for this ideal, countable intersections of ideal-$G_{\delta }$ sets are ideal-$G_{\delta }$. By contrast, the ideal of finite subsets is not generally a σ-ideal: a countable union of finite sets may be countably infinite.

6. Examples and Counterexamples

Example 5

(Discrete spaces). Let X be a discrete space and let $\mathcal{I}$ be any ideal on X. Every subset of X is open and hence is a $G_{\delta }$ set. In particular, for every open $U\subseteq X$,

$${cl}_X\left(U\right)=U.$$

For any $x\in U$, choose $P=\left\{x\right\}$. Then

$$x\in P\subseteq U={cl}_X\left(U\right),$$

and P is a $G_{\delta }$ set. Thus every discrete space is Moscow and therefore $\mathcal{I}$-Moscow for every ideal.

Example 6

(The real line). The real line with its usual topology is first-countable and $T_1$. Therefore it is Moscow and is $\mathcal{I}$-Moscow for every ideal $\mathcal{I}$ on $\mathbb{R}$. In particular, this holds for the finite-set ideal and the countable-set ideal.

Example 7

(The trivial ideal). If $\mathcal{I}=\{⌀\}$, the ideal Moscow property is exactly the classical Moscow property. Consequently, all classical examples and counterexamples transfer directly to this special case.

Example 8

(The improper ideal). Let $\mathcal{I}=\mathcal{P}\left(X\right)$. Then every subset $A\subseteq X$ is an $\mathcal{I}$-$G_{\delta }$ set, because

$$A▵⌀=A\in \mathcal{I}$$

and ⌀ is a $G_{\delta }$ set. Hence every topological space is $\mathcal{I}$-Moscow for the improper ideal.

This example shows why a nontrivial theory should specify a proper ideal rather than allow the improper ideal without comment.

Example 9

(A classical non-Moscow space). Let D be an uncountable discrete space and let

$$X=\alpha D=D\cup \{\infty \}$$

be its one-point compactification. A neighborhood of ∞ contains all but finitely many points of D. Choose a countably infinite subset $A\subseteq D$ and put $U=A$. Then U is open and

$${cl}_X\left(U\right)=A\cup \{\infty \}.$$

Every $G_{\delta }$ neighborhood of ∞ contains all but countably many points of D. To see this, write such a neighborhood as ${\bigcap }_{n=1}^{\infty }{V}_n$, where each $V_n$ is a neighborhood of ∞. Each complement $D\setminus V_n$ is finite, so

$$D\setminus {\displaystyle \bigcap _{n=1}^{\infty }}{V}_n={\displaystyle \bigcup _{n=1}^{\infty }}(D\setminus V_n)$$

is countable. Since D is uncountable, the intersection contains uncountably many points of D.

It follows that no $G_{\delta }$ set containing ∞ can be contained in $A\cup \{\infty \}$. Thus $\alpha D$ is not a Moscow space. This is a standard example; see [1].

Remark 7.

The preceding example is a non-Moscow space for the trivial ideal. It does not automatically provide a counterexample for every nontrivial ideal. The monotonicity theorem shows that the answer may change when the ideal is enlarged.

Proposition 3.

Let X be any topological space and let $\mathcal{I}$ be an ideal on X. If every point of X is an $\mathcal{I}$-$G_{\delta }$ set, then X is $\mathcal{I}$-Moscow.

Proof. 

Let U be open and $x\in {cl}_X\left(U\right)$. By hypothesis, $\left\{x\right\}$ is an $\mathcal{I}$-$G_{\delta }$ set. Taking $P=\left\{x\right\}$ gives

$$x\in P\subseteq {cl}_X\left(U\right).$$

Thus X is $\mathcal{I}$-Moscow. □

Remark 8.

This proposition is the ideal analogue of the elementary sufficient condition based on points being $G_{\delta }$ sets. It is not necessary for an ideal Moscow space that every singleton be an $\mathcal{I}$-$G_{\delta }$ set.

7. Subspaces, Products, and Mappings

The classical theory of Moscow spaces has a rich relation to subspaces, products, and continuous mappings. For example, every dense subspace of a Moscow space is Moscow, whereas closed subspaces need not preserve the property; see [1]. For ideal Moscow spaces, additional care is required because the ideal must also be transferred to the relevant subspace or image.

7.1. Subspaces

Let $Y\subseteq X$. The trace ideal of $\mathcal{I}$ on Y is defined by

$${\mathcal{I}|}_Y=\{A\subseteq Y:A\in \mathcal{I}\}.$$

This is an ideal on Y.

Proposition 4.

Let $(X,\mathcal{T},\mathcal{I})$ be an ideal topological space and let $Y\subseteq X$. Suppose that Y is an open subspace of X and that X is $\mathcal{I}$-Moscow. Then Y is $\left(\mathcal{I}{|}_Y\right)$-Moscow.

Proof. 

Let V be open in Y. Since Y is open in X, the set V is open in X. Let

$$y\in {cl}_Y\left(V\right).$$

Then

$$y\in Y\cap {cl}_X\left(V\right).$$

Because X is $\mathcal{I}$-Moscow, there exists an $\mathcal{I}$-$G_{\delta }$ set $P\subseteq X$ such that

$$y\in P\subseteq {cl}_X\left(V\right).$$

The set $P\cap Y$ is a $G_{\delta }$ set in Y modulo the trace ideal. More precisely, if $P▵G\in \mathcal{I}$ for some $G_{\delta }$ set G in X, then

$${(P\cap Y)▵(G\cap Y)\subseteq (P▵G)\cap Y\in \mathcal{I}|}_Y.$$

The set $G\cap Y$ is $G_{\delta }$ in Y. Also,

$$y\in P\cap Y\subseteq Y\cap {cl}_X\left(V\right)={cl}_Y\left(V\right).$$

Thus Y is $\left(\mathcal{I}{|}_Y\right)$-Moscow. □

Remark 9.

The openness assumption in the preceding proposition ensures that open sets of Y are open in X and that the closure comparison used in the proof is valid. For arbitrary subspaces, a separate argument is required.

7.2. Products

Suppose that $(X,{\mathcal{T}}_X,{\mathcal{I}}_X)$ and $(Y,{\mathcal{T}}_Y,{\mathcal{I}}_Y)$ are ideal topological spaces. A natural product ideal is the ideal generated by sets of the form

$$(A\times Y)\cup (X\times B),A\in {\mathcal{I}}_X,B\in {\mathcal{I}}_Y.$$

This ideal records subsets whose exceptional coordinates are controlled by the ideals on the factors.

Definition 8.

Theproduct ideal${\mathcal{I}}_X⊠{\mathcal{I}}_Y$ on $X\times Y$ is defined by

$${\mathcal{I}}_X⊠{\mathcal{I}}_Y=\left\{C\subseteq X\times Y:\begin{array}{c}{\displaystyle C\subseteq {\displaystyle \bigcup _{j=1}^n}\left((A_j\times Y)\cup (X\times B_j)\right)}\hfill \\ \mathrm{for}\mathrm{some}n\in \mathbb{N},A_j\in {\mathcal{I}}_X,B_j\in {\mathcal{I}}_Y\hfill \end{array}\right\}.$$

Remark 10.

The displayed family is an ideal: it is hereditary, and the union of two members is controlled by concatenating their finite lists of exceptional sets. This definition is one possible choice of product ideal; it is not the only conceivable choice.

Question 4.

Under what additional assumptions on X, Y, ${\mathcal{I}}_X$, and ${\mathcal{I}}_Y$ is $X\times Y$, an $({\mathcal{I}}_X⊠{\mathcal{I}}_Y)$-Moscow space whenever both factors are ideal Moscow spaces?

Remark 11.

Observe that the classical product theory of Moscow spaces is nontrivial [1]; the product of two ideal Moscow spaces cannot be asserted to be ideal Moscow solely from the definition without a proof and suitable hypotheses.

Proposition 5.

Let $h:X\to Y$ be a homeomorphism and let $\mathcal{I}$ be an ideal on X. If $J=\left\{h\right(A):A\in I\}$, then X is I-Moscow if and only if Y is J-Moscow.

Proof. 

This is precisely the homeomorphism-invariance result proved in Section 4. □

8. Conclusions

We introduced an ideal-theoretic extension of the classical notion of a Moscow space. The central idea is to replace ordinary $G_{\delta }$ witnesses by sets that differ from $G_{\delta }$ sets by elements of a prescribed ideal.

The main elementary results establish that every Moscow space is ideal Moscow, that the property is monotone with respect to enlargement of the ideal, and that it is invariant under homeomorphisms when the ideal is transported accordingly. We also established sufficient conditions based on countable pseudocharacter and proved a subspace result for open subspaces with the trace ideal.

The examples show that the choice of ideal is essential. In particular, the improper ideal makes every space ideal Moscow, while the trivial ideal recovers the classical property exactly. These observations motivate the use of proper ideals and the investigation of hypotheses that make the ideal extension both nontrivial and structurally useful.

Further work should focus on product and mapping theorems, the behavior of classical counterexamples under specific ideals, and applications to topological groups and completion theory. Such investigations will help determine the precise scope and mathematical significance of the ideal Moscow property.