Related papers: On topological spaces and topological groups with …

It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says…

When a topological group $G$ acts on a compact space $X$, its enveloping semigroup $E(X)$ is the closure of the set of $g$-translations, $g\in G$, in the compact space $X^X$. Assume that $X$ is metrizable. It has recently been shown by the…

We characterize $\kappa$-Fr\'{e}chet--Urysohn topological groups. Using this characterization we show that: (1) a hemicompact topological group is $\kappa$-Fr\'{e}chet--Urysohn iff it is locally compact, and (2) if $F$ is a closed…

If $I$ is an ideal in the ring $C(X)$ of all real valued continuous functions defined over a Tychonoff space $X$, then $X$ is called $I$-$pseudocompact$ if the set $X\setminus \bigcap Z[I]$ is a bounded subset of $X$. Corresponding to $I$,…

It is proved that any countable topological vector space over a finite field $\mathbb F_p$ or, equivalently, any countable Abelian topological group of prime exponent has a closed discrete basis.

We prove that if $K$ is a compact space and the space $P(K\times K)$ of regular probability measures on $K\times K$ has countable tightness in its $weak^*$ topology, then $L_1(\mu)$ is separable for every $\mu\in P(K)$. It has been known…

Game-theoretic characterizations of selection principles provide a powerful framework for analyzing covering properties through strategic interactions. For a Tychonoff space $X$ and a non-trivial metrizable arc-connected topological group…

Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \in \{\omega D,…

We consider special subclasses of the class of Lindel\"of Sigma-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space $X$ is in the class $L\Sigma(\leq\kappa)$ if it…

We initiate the study of the $p$-local commensurability graph of a group, where $p$ is a prime. This graph has vertices consisting of all finite-index subgroups of a group, where an edge is drawn between $A$ and $B$ if $[A : A\cap B]$ and…

For a Hausdorff zero-dimensional topological space $X$ and a totally ordered field $F$ with interval topology, let $C_c(X,F)$ be the ring of all $F-$valued continuous functions on $X$ with countable range. It is proved that if $F$ is either…

$c$-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41(8), 2018, 1135-1167. We offer a characterization of these spaces $X$ via $c$-stable family of closed sets in $X$ by showing that $X$ is $c$-realcompact if…

In 1957, Lacombe initiated a systematic study of the different possible notions of "computable topological spaces". However, he interrupted this line of research, settling for the idea that "computably open sets should be computable unions…

This paper studies various completeness properties of the open-point and bi-point-open topologies on the space C(X) of all real-valued continuous functions on a Tychonoff space X. The properties range from complete metrizability to the…

A space $X$ is of countable type (resp. subcountable type) if every compact subspace $F$ of $X$ is contained in a compact subspace $K$ that is of countable character (resp. countable pseudocharacter) in $X$. In this paper, we mainly show…

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections.…

This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we…

Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is…

The Grothendieck property has become important in research on the definability of pathological Banach spaces [CI], [HT], and especially [HT20]. We here answer a question of Arhangel'ski\u{\i} by proving it undecidable whether countably…

A topological space $X$ is $\mathbb R^{\omega_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{\omega_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$…