Answers to Six Questions Concerning Ideal Topological Groups

1. Notations and a Few Rudimentary Terminology

The theory of ideal topological spaces extends ordinary topology by introducing an ideal $\mathcal{I}$ of “small” subsets. The local function associated with an ideal was introduced in the classical theory of ideal topological spaces, and I-open sets and I-continuous mappings were subsequently developed in a systematic form [1,8,9,10].

Let $(X,\tau ,\mathcal{I})$ be an ideal topological space. For $A\subseteq X$, its local function is $A^{★}=\{x\in X:U\cap A\notin \mathcal{I}\mathrm{for}\mathrm{every}U\in \tau \left(x\right)\}$, where $\tau \left(x\right)=U\in \tau \in U$. The associated ★-closure is ${Cl}^{★}\left(A\right)=A\cup A^{★}$. A subset $A\subseteq X$ is called I-open if $A\subseteq Int\left(A^{★}\right)$.

We denote the family of all I-open subsets by $IO\left(X\right)$.

These notions were used by Jafari and Rajesh to introduce ideal topological groups [6]. Their definition is equivalent to requiring the multiplication and inversion to have the appropriate I-continuity properties. In particular, if $G=(G,★,\tau ,\mathcal{I})$ is an ideal topological group, then for every $x,y\in G$ and every ordinary neighborhood W of $x{y}^{-1}$ there exist I-open neighborhoods U of x and V of y such that $U{V}^{-1}\subseteq W$.

The basic translation and inversion properties were established in [6]; see also the subsequent work on p-ideal topological groups [5] and the later systematic treatment in [2].

The paper On Some Questions Concerning Ideal Topological Groups and Related Notions by Jafari and Rajesh, published in Questions and Answers in General Topology 44 (2026), 17–19, poses six questions concerning minimality, sequential completeness, precompactness, $\tau$-precompactness, separately I-continuous multiplication, and jointly ideal-continuous multiplication [7].

The purpose of the present paper is to give detailed answers and, where a completely general assertion does not follow from the definitions, to isolate precise sufficient conditions.

Throughout the paper, all groups are written multiplicatively.

Definition 1.

Let $(X,\tau ,\mathcal{I})$ be an ideal topological space. A subset $A\subseteq X$ is calledI-openif $A\subseteq Int\left(A^{★}\right)$. The family of all I-open sets is denoted by $IO\left(X\right)$.

Definition 2.

Let $f:(X,\tau ,\mathcal{I})⟶(Y,\sigma ,\mathcal{J})$ be a mapping between ideal topological spaces. We say that f is I-J-continuous if $f^{-1}\left(V\right)\in IO\left(X\right)$ for every $V\in \sigma$.

Definition 3.

A bijection $f:(X,\tau ,\mathcal{I})⟶(Y,\sigma ,\mathcal{J})$ is an I-J-homeomorphism if it is I-J-continuous and I-J-open.

Definition 4.

An ideal topological space $(X,\tau ,\mathcal{I})$ is called I-$T_2$ if for every pair of distinct points $x,y\in X$ there exist disjoint I-open sets $U,V$ such that $x\in U,y\in V$.

The above definition is the one used in the 2026 paper containing the six questions [7]. The earlier theory shows that ideal topological groups have strong translation properties. In particular, translations and inversion are I-homeomorphisms [6].

Proposition 1.

Let G be an ideal topological group. Then every left translation $L_g\left(x\right)=gx$ and every right translation $R_g\left(x\right)=xg$ is an I-homeomorphism. The inversion $i\left(x\right)=x^{-1}$ is also an I-homeomorphism.

Proof. 

This is one of the basic structural results for ideal topological groups. The proof follows from the defining condition for an ideal topological group and the invariance of the class of I-open sets under translations and inversion; see [6]. □

Remark 1.

Consequently, every ideal topological group is I-homogeneous. Thus, as in the ordinary theory of topological groups, many questions can be reduced to the identity element.

In what follows, ${\mathbb{T}}^{\tau }={\prod }_{\alpha <\tau }\mathbb{T}$ denotes the product of $\tau$ copies of the circle group $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}$, and $\mathcal{I}={\left[G\right]}^{<\omega }=\{A\subseteq G:|A|<\omega \}$ denotes the ideal of all finite subsets of G, where $\omega ={\aleph }_0$ is the first infinite cardinal.

2. Question 1: Minimal Ideal Topological Groups

Question 1.

We say that an I-$T_2$ ideal topological group $(G,★,\tau ,\mathcal{I})$ is minimal if every I-continuous isomorphism $\phi :(G,★,\tau ,\mathcal{I})⟶(H,★,\sigma ,\mathcal{J})$ is an I-J-homeomorphism. Is there any proper example of such an ideal topological group?

2.1. The Relation with Classical Minimality

For ordinary Hausdorff topological groups, a group is called minimal when every continuous isomorphism from it onto a Hausdorff topological group is a topological isomorphism. Compact Hausdorff groups are classical examples of minimal topological groups [3,4].

The ideal version is more delicate. An I-continuous mapping is generally weaker than an ordinary continuous mapping. Therefore the classical compact-to-Hausdorff argument cannot automatically be applied to an arbitrary ideal topological group.

The correct general statement is therefore a conditional one.

Definition 5.

We call an I-compact I-$T_2$ ideal topological group strongly I-compact if every I-closed subset is I-compact and every I-compact subset of an I-$T_2$ ideal topological group is I-closed.

This definition isolates precisely the compactness–Hausdorff principle needed in the minimality argument.

Theorem 2.

Let G be a strongly I-compact I-$T_2$ ideal topological group. Suppose $\phi ⟶H$ is a bijective I-J-continuous homomorphism onto a J-$T_2$ ideal topological group H, and suppose that $J = \phi \left(\mathcal{I}\right)$. Then φ is an I-J-homeomorphism.

Proof. 

We prove that $\phi$ is I-J-open. Let U be an I-open subset of G. Put $F=G\setminus U$. Then F is I-closed. By strong I-compactness, F is I-compact. Since $\phi$ is I-J-continuous and surjective, the image $\phi \left(F\right)$ is J-compact. Since H is J-$T_2$, the strong compactness assumption implies that $\phi \left(F\right)$ is J-closed. Observe that $\phi \left(U\right)$ is J-open. Hence $\phi$ is I-J-open. Since $\phi$ is already I-J-continuous, it follows that $\phi$ is an I-J-homeomorphism. □

Corollary 1.

Every strongly I-compact I-$T_2$ ideal topological group is minimal with respect to I-continuous isomorphisms satisfying $J = \phi \left(\mathcal{I}\right)$.

2.2. A Concrete Proper Example

A particularly simple nontrivial source of examples is obtained from finite groups.

Let $G={\mathbb{Z}}_3=\{0,1,2\}$ under addition modulo 3, and let $\tau =\{⌀,1,2,G\},\mathcal{I}=\{⌀,\{0\left\}\right\}$. Then the I-open subsets are $IO\left(G\right)=\mathcal{P}\left(G\right)\setminus \left\{0\right\}$.

This example is explicitly recorded in the later literature on ideal topological groups [2].

Although this particular finite example is not sufficient by itself to establish the unrestricted universal minimality statement in Question 1, it is a genuine proper ideal topological group and illustrates the type of finite rigid structures that should be used when studying minimality.

Remark 2.

Thus the answer to Question 1 should be stated carefully:

There are natural sufficient conditions for minimality, especially strong ideal compactness together with an ideal Hausdorff condition, but the six definitions in the 2026 paper alone do not justify the unqualified assertion that every compact-looking ideal topological group is minimal.

The main obstruction is that I-continuity is weaker than ordinary continuity and I-closedness need not behave exactly like ordinary closedness. Consequently, the ideal analogue of the classical compact-Hausdorff proof requires an additional compactness–closedness principle.

3. Question 2: Sequentially Complete Ideal Topological Groups

Question 3.

We say that an I-$T_2$ ideal topological group G is sequentially I-complete if all I-continuous homomorphic images of G are sequentially complete. Is there any proper example?

3.1. Sequential Completeness

For an ordinary topological group, a sequence $\left(x_n\right)$ is Cauchy if, for every neighborhood U of the identity, there exists N such that $x_m^{-1}{x}_n\in U$, where $m,n\ge N$. The group is sequentially complete if every Cauchy sequence converges. This is the standard notion of sequential completeness in topological groups [3].

In an ideal setting one may replace ordinary neighborhoods by appropriate I-neighborhoods. The resulting notion depends on the chosen ideal convergence structure, so it is useful to distinguish this generalized notion from ordinary sequential completeness.

Definition 6.

An I-Cauchy sequence in an ideal topological group G is a sequence $\left(x_n\right)$ such that for every I-neighborhood U of the identity there exists N satisfying $x_m^{-1}{x}_n\in U(m,n\ge N)$.

The group is sequentially I-complete if every I-Cauchy sequence I-converges to an element of G.

3.2. A Finite Proper Example

The finite group $G={\mathbb{Z}}_3$ with $\tau =\{⌀,1,2,G\},\mathcal{I}=\{⌀,0\}$ is a proper example in the following precise sense. Every homomorphic image of ${\mathbb{Z}}_3$ is finite. Indeed, if $\phi :{\mathbb{Z}}_3\to H$ is a group homomorphism, then $\phi \left({\mathbb{Z}}_3\right)$ is either the trivial group or a cyclic group of order 3.

Hence every homomorphic image is finite. Under any Hausdorff group structure compatible with the corresponding ideal convergence, every Cauchy sequence is eventually confined to a finite group and therefore has a convergent subsequence. In the finite $T_1$ situation this implies eventual stabilization. Thus, every such image is sequentially complete.

Proposition 2.

Let G be a finite ideal topological group whose admissible I-Hausdorff images are finite $T_1$ spaces. Then every I-continuous homomorphic image of G is sequentially complete.

Proof. 

Let $f:G\to H$ be an I-continuous homomorphism. Since G is finite, $f\left(G\right)$ is finite. Let $\left(y_n\right)$ be a Cauchy sequence in $f\left(G\right)$. Because $f\left(G\right)$ is finite, some point y occurs infinitely many times. In a finite Hausdorff topological group, the Cauchy property forces the sequence to eventually remain in every neighborhood of y. Hence $y_n⟶y$. Thus $f\left(G\right)$ is sequentially complete. □

Remark 3.

Question 2 therefore has a positive answer for finite ideal topological groups under the natural finite-image hypothesis. For infinite groups, the situation is substantially more delicate. One should not infer sequential completeness merely from I-$T_2$ or from the existence of many I-open sets. A useful direction is to investigate ideal analogues of complete and precompact topological groups.

4. Question 3: Precompact Ideal Topological Groups

Question 4.

We say that an I-$T_2$ ideal topological group G is precompact if, for every I-neighborhood U of the identity, there exists a finite subset $F\subseteq G$ such that $G=UF$. Is there any example of a precompact ideal topological group?

The answer is affirmative.

Theorem 5.

Let G be a compact Hausdorff topological group and let $\mathcal{I}$ be the ideal of finite subsets of G. Then G is a precompact ideal topological group.

Proof. 

Let $\mathcal{I}={\left[G\right]}^{<\omega }$ be the ideal of finite subsets of G. Every ordinary open subset of G is I-open. Indeed, if U is ordinary open and $x\in U$, then one can choose an open neighborhood V of x with $V\subseteq U$. Since V is infinite whenever G has no isolated points, its intersection with U is not finite. Hence $x\in U^{★}$, and it follows that $U\subseteq Int\left(U^{★}\right)$.

Now let W be an I-neighborhood of the identity e. By definition, there exists an I-open set V such that $e\in V\subseteq W$. In particular, V contains an ordinary open neighborhood O of e. Since G is compact, the family $\{gO:g\in G\}$ is an open cover of G. Hence there exist $g_1,\dots ,g_n\in G$ such that $G=g_{1}O\cup \dots \cup g_{n}O$. Let $F=\{g_1,\dots ,g_n\}$. Then $G=FO$. Using the symmetry of neighborhoods, or replacing O by a suitable symmetric neighborhood, we obtain $G=OF$. Since $O\subseteq W$, we have $G=OF\subseteq WF\subseteq G$. Therefore, $G=WF$. Thus G is precompact in the ideal sense. □

Example 1.

The circle group $\mathbb{T}=z\in \mathbb{C}:\left|z\right|=1$ with multiplication, its usual compact topology, and the ideal $\mathcal{I}={\left[\mathbb{T}\right]}^{<\omega }$ of finite subsets is a precompact ideal topological group.

This is a particularly transparent answer to Question 3.

5. Question 4: $\mathbf{\tau }$-Precompact Ideal Topological Groups

Question 6.

Let τ be an infinite cardinal. An I-$T_2$ ideal topological group G is called τ-precompact if for every I-neighborhood U of the identity there exists $F\subseteq G$ such that $G=UF\mathrm{and}\left|F\right|\le \tau .$ Is there any example of a τ-precompact ideal topological group?

The answer is again affirmative, and in fact compact groups give a stronger result.

Theorem 7.

Let τ be any infinite cardinal. Every compact Hausdorff topological group, equipped with the ideal of finite subsets, is τ-precompact.

Proof. 

By the proof of Theorem 3.1, for every I-neighborhood U of the identity there exists a finite subset $F\subseteq G$ such that $G=UF$. Since F is finite and $\tau$ is an infinite cardinal, we have $\left|F\right|<\omega \le \tau$, where $\omega$ means the first infinite cardinal, e.i., the cardinality of the natural numbers: $\omega =\left|\mathbb{N}\right|={\aleph }_0$. It follows that, $\left|F\right|\le \tau$. Therefore, G is $\tau$-precompact. □

In fact, we can produce examples with arbitrarily large weight.

Example 2.

For any infinite cardinal τ, consider $G={\mathbb{T}}^{\tau }$, the product of τ copies of the circle group. By the Tychonoff theorem, ${\mathbb{T}}^{\tau }$ is compact. Equip it with the product topology and the ideal $\mathcal{I}={\left[G\right]}^{<\omega }$. Then G is an ideal topological group and is actuallyfinitely precompact, hence τ-precompact.

Remark 4.

The example is stronger than what Question 4 asks. The required set F can always be chosen finite, so $\left|F\right|<\omega \le \tau$. Thus the cardinal parameter τ is irrelevant for compact examples.

6. Question 5: Separately Ideal-Continuous Multiplication

Question 8.

Are there proper examples of separately I-continuous mappings and ideal semitopological groups?

The answer is yes. Recall that a mapping $f:Q\times Y\to Z$ is separately I-continuous if it is I-continuous in each variable when the other variable is fixed.

Definition 7.

An ideal semitopological group is a group G equipped with an ideal topology such that the multiplication $m:G\times G\to G,m(x,y)=xy$, is separately ideal continuous.

Every ideal topological group is an ideal semitopological group.

Proposition 3.

Every ideal topological group is an ideal semitopological group.

Proof. 

Let $m:G\times G\to G,m(x,y)=xy$ be the multiplication mapping. Fix $a\in G$. Let us define $L_a\to G,L_a\left(x\right)=ax$. Since G is an ideal topological group, left translation is I-continuous. Similarly, for fixed $a\in G$, $R_a\left(x\right)=xa$ is I-continuous. Therefore multiplication is separately I-continuous. □

6.1. A Non-Classical Example

We now use the finite example $G={\mathbb{Z}}_3$ with $\tau =\{⌀,1,2,G\},\mathcal{I}=\{⌀,\{0\left\}\right\}$. This example has been recorded as an ideal topological group which is not an ordinary topological group [2]. The family of I-open sets is $IO\left(G\right)=\mathcal{P}\left(G\right)\setminus \left\{0\right\}$. The multiplication and inversion mappings are I-continuous. Therefore multiplication is jointly ideal continuous and hence separately ideal continuous.

Theorem 9.

The above ${\mathbb{Z}}_3$ example is an ideal semitopological group which is not an ordinary semitopological group with respect to the given topology.

Proof. 

The multiplication map is jointly I-continuous by the fact that the structure is an ideal topological group. Therefore it is separately I-continuous.

On the other hand, the topology $\tau =\{⌀,1,2,G\}$ is not a group topology. For example, ordinary continuity of the multiplication fails at suitable points, as observed in the explicit construction in [2]. Hence the structure is genuinely an ideal, rather than an ordinary, semitopological group. □

Thus Question 5 has a positive answer in a particularly strong form.

7. Question 6: Joint Ideal Continuity and Ideal Paratopological Groups

Question 10.

Are there proper examples of jointly ideal continuity and ideal paratopological groups?

Again the answer is affirmative.

Definition 8.

Let $f:X\times Y\to Z$ be a mapping between ideal topological spaces. We say that f is jointly ideal continuous at $(x,y)$ if for every appropriate neighborhood W of $f(x,y)$ there exist I-open neighborhoods U of x and J-open neighborhoods V of y such that $f(U\times V)\subseteq W$.

Definition 9.

An ideal paratopological group is a group equipped with an ideal topology for which the multiplication $m\times G\to G$ is jointly ideal continuous.

Theorem 11.

Every ideal topological group is an ideal paratopological group.

Proof. 

Let $G=(G,★,\tau ,\mathcal{I})$ be an ideal topological group. Let $x,y\in G$ and let W be an ordinary neighborhood of $xy$. Since $xy=x{\left(y^{-1}\right)}^{-1}$, the defining property of an ideal topological group gives I-open neighborhoods U of x and V of $y^{-1}$ such that $U{V}^{-1}\subseteq W$. Using the fact that inversion is an I-homeomorphism, we may replace V by $V^{-1}$ and obtain an I-open neighborhood of y. Consequently there exist I-open neighborhoods U of x and V of y satisfying $UV\subseteq W$. Thus multiplication is jointly ideal continuous at $(x,y)$. Since x and y were arbitrary, multiplication is jointly ideal continuous on $G\times G$. □

7.1. The ${\mathbb{Z}}_3$ Example

The same example $G={\mathbb{Z}}_3,\tau =\{⌀,1,2,G\},\mathcal{I}=\{⌀,\left\{0\right\}\}$ is therefore an ideal paratopological group. The importance of this example is that it is not an ordinary topological group. Hence Question 6 has a genuinely generalized answer rather than merely an ordinary topological-group example.

Corollary 2.

There exists a group which is an ideal paratopological group but is not a topological group with respect to its underlying topology.

Proof. 

Apply ${\mathbb{Z}}_3$ example. □

8. Relations Among the Six Questions

The six questions are not independent. There is a natural hierarchy.

Proposition 4.

For an ideal topological group G,

Ideal topological group ⟹ Ideal paratopological group ⟹ ideal semitopological group.

Proof. 

The first implication follows from joint ideal continuity of multiplication. Joint ideal continuity implies separate ideal continuity by fixing one variable. □

Thus Questions 5 and 6 are automatically answered positively once one has a proper ideal topological group.

The example of ${\mathbb{Z}}_3$ is particularly interesting and useful because it shows that these implications are meaningful outside the ordinary category of topological groups [2].

For Questions 3 and 4, compactness gives a much stronger conclusion: $\mathrm{compact}\mathrm{ideal}\mathrm{group}⟹\mathrm{finite}-\mathrm{precompact}⟹\tau -\mathrm{precompact}$ for every infinite cardinal $\tau$.

Question 1 is of a different nature. Minimality concerns the behavior of bijective homomorphisms and therefore requires an ideal analogue of the compact-Hausdorff open mapping principle. Question 2 similarly requires an explicit theory of ideal Cauchy sequences and ideal completeness.

9. Summary of the Answers

We summarize the six questions in the following table.

Question	Answer	Main construction/result
1	Minimality	Positive under a strong ideal compactness–closedness hypothesis; the unrestricted statement requires additional structure.
2	Sequential completeness	Positive for finite proper examples and for classes in which all admissible images are finite or otherwise complete.
3	Precompactness	Yes. Every compact Hausdorff group with the ideal of finite subsets is precompact in the ideal sense.
4	$\tau$-precompactness	Yes for every infinite cardinal $\tau$. In fact the compact examples are finitely precompact.
5	Separately I-continuous	Yes. Every ideal topological group is an ideal semitopological group.
6	Jointly ideal continuous	Yes. Every ideal topological group is an ideal paratopological group. The ${\mathbb{Z}}_3$ example is genuinely non-classical.

Remark 5.

Our results and answers indicate several structural principles.

1. Compactness gives ideal precompactness

Let G be a compact Hausdorff group and let $\mathcal{I}$ be the finite-set ideal. Then every I-neighborhood contains an ordinary neighborhood of the identity, and compactness gives a finite right cover. Thus $G=UF$ for some finite F. Hence the ideal version of precompactness is compatible with the classical compact-to-precompact implication.

2. Cardinal bounds

If $G=UF$ with F finite, then it follows automatically that $\left|F\right|\le \tau$ for every infinite cardinal τ. Therefore compact examples solve all τ-precompactness questions simultaneously.

3. An interesting generalized finite group

The group ${\mathbb{Z}}_3$ with $\tau =\{⌀,1,2,{\mathbb{Z}}_3\}$ and $\mathcal{I}=\{⌀,\{0\left\}\right\}$ is especially useful because it is not merely an ordinary topological group equipped with additional notation. Its ideal topology permits the group operations to satisfy ideal continuity even though ordinary continuity fails [2].

10. Conclusions

The six questions posed in the 2026 paper of Jafari and Rajesh can be organized into three levels.

First, Questions 5 and 6 follow structurally from the definition of an ideal topological group:

Ideal topological group ⇒ Ideal paratopological group ⇒ Ideal semitopological group.

Second, Questions 3 and 4 have particularly simple affirmative answers through compact groups. If G is compact Hausdorff and $\mathcal{I}={\left[G\right]}^{<\omega }$, then $G=UF$ for every I-neighborhood U of the identity and some finite $F\subseteq G$. Consequently G is both precompact and $\tau$-precompact for every infinite cardinal $\tau$.

Third, Questions 1 and 2 require genuinely new ideal structure. Minimality and sequential completeness are not automatic consequences of I-$T_2$ or of ideal topological-group axioms. A satisfactory general theory should introduce an appropriate ideal compactness, ideal completeness, and ideal Cauchy theory. Under natural strong compactness and completeness assumptions, affirmative results follow.

The finite ${\mathbb{Z}}_3$ example indicates that the results and insight we obtained is not merely reformulations of classical topological-group questions. Indeed, the ideal structure enlarges the class of admissible group topologies and makes it possible for algebraic operations to satisfy generalized continuity conditions that fail in the ordinary topology.