We say that a collection $𝒜$ of subsets of $X$ has property $\left(CC\right)$ if there is a set $D$ and point-countable collections $𝒞$ of closed subsets of $X$ such that for any $A\in 𝒜$ there is a finite subcollection $ℱ$ of $𝒞$ such that $A=D\setminus \bigcup ℱ$. Then we prove that any compact space is Corson if and only if it has a point-$\sigma$-$\left(CC\right)$ base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis” (2014) by...

We define and investigate a generalization of the notion of convex compacta. Namely, for semiconvex combination in a semiconvex compactum we allow the existence of non-trivial loops connecting a point with itself. It is proved that any semiconvex compactum contains two non-empty convex compacta, the center and the weak center. The center is the largest compactum such that semiconvex combination induces a convex structure on it. The convex structure on the weak center does not necessarily coincide...

The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal 𝔥 is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindelöf degree in most published models. Similar results, some definitive,...

We present short and elementary proofs of the following two known theorems in General Topology: (i) [H. Wicke and J. Worrell] A $T_1$ weakly $\delta \theta$-refinable countably compact space is compact. (ii) [A. Ostaszewski] A compact Hausdorff space which is a countable union of metrizable spaces is sequential.

Given a subbase $𝒮$ of a space $X$, the game $PO(𝒮,X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$-th move, a point $x_n\in X$ and a set $U_n\in 𝒮$ such that $x_n\in U_n$. The game stops after the moves $\{x_n,U_n:n\in ø\}$ have been made and the player $P$ wins if ${\bigcup }_{n\in ø}{U}_n=X$; otherwise $O$ is the winner. Since $PO(𝒮,X)$ is an evident modification of the well-known point-open game $PO\left(X\right)$, the primary line of research is to describe the relationship between $PO\left(X\right)$ and $PO(𝒮,X)$ for a given subbase $𝒮$. It turns out that, for any subbase $𝒮$, the player $P$ has a winning strategy...

In this paper, we study some properties of relatively strong pseudocompactness and mainly show that if a Tychonoff space $X$ and a subspace $Y$ satisfy that $Y\subset \overline{\mathrm{Int}Y}$ and $Y$ is strongly pseudocompact and metacompact in $X$, then $Y$ is compact in $X$. We also give an example to demonstrate that the condition $Y\subset \overline{\mathrm{Int}Y}$ can not be omitted.

For a cardinal $\alpha$, we say that a subset $B$ of a space $X$ is $C_{\alpha }$-compact in $X$ if for every continuous function $fX\to {ℝ}^{\alpha }$, $f\left[B\right]$ is a compact subset of ${ℝ}^{\alpha }$. If $B$ is a $C$-compact subset of a space $X$, then $\rho (B,X)$ denotes the degree of $C_{\alpha }$-compactness of $B$ in $X$. A space $X$ is called $\alpha$-pseudocompact if $X$ is $C_{\alpha }$-compact into itself. For each cardinal $\alpha$, we give an example of an $\alpha$-pseudocompact space $X$ such that $X\times X$ is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...

A space $X$ is truly weakly pseudocompact if $X$ is either weakly pseudocompact or Lindelöf locally compact. We prove: (1) every locally weakly pseudocompact space is truly weakly pseudocompact if it is either a generalized linearly ordered space, or a proto-metrizable zero-dimensional space with $\chi (x,X)>\omega$ for every $x\in X$; (2) every locally bounded space is truly weakly pseudocompact; (3) for $\omega <\kappa <\alpha$, the $\kappa$-Lindelöfication of a discrete space of cardinality $\alpha$ is weakly pseudocompact if $\kappa ={\kappa }^{\omega }$.

Arhangel’skii [Sci. Math. Jpn. 55 (2002), 153–201] defined notions of relative paracompactness in terms of locally finite open partial refinement and asked if one can generalize the notions above to the well known Michael’s criteria of paracompactness in [17] and [18]. In this paper, we consider some versions of relative paracompactness defined by locally finite (not necessarily open) partial refinement or locally finite closed partial refinement, and also consider closure-preserving cases, such...