Following Kombarov we say that $X$ is $p$-sequential, for $p\in {\alpha }^{*}$, if for every non-closed subset $A$ of $X$ there is $f\in {}^{\alpha }X$ such that $f\left(\alpha \right)\subseteq A$ and $\overline{f}\left(p\right)\in X\setminus A$. This suggests the following definition due to Comfort and Savchenko, independently: $X$ is a FU($p$)-space if for every $A\subseteq X$ and every $x\in A^{-}$ there is a function $f\in {}^{\alpha }A$ such that $\overline{f}\left(p\right)=x$. It is not hard to see that $p\le {}_{RK}q$ ($\le {}_{RK}$ denotes the Rudin–Keisler order) $\iff$ every $p$-sequential space is $q$-sequential $\iff$ every FU($p$)-space is a FU($q$)-space. We generalize the spaces $S_n$ to construct examples of $p$-sequential...

In this article we introduce the notion of strongly $\mathrm{KC}$-spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space $(X,\tau )$ is maximal countably compact if and only if it is minimal strongly $\mathrm{KC}$, and apply this result to study some properties of minimal strongly $\mathrm{KC}$-spaces, some of which are not possessed by minimal $\mathrm{KC}$-spaces. We also give a positive answer to a question proposed by O. T. Alas and R. G. Wilson, who asked whether every...

It is shown that a space $X$ is $L\left({}^{\mu }p\right)$-Weakly Fréchet-Urysohn for $p\in {\omega }^{*}$ iff it is $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn for arbitrary $\mu ,\nu <{\omega }_1$, where ${}^{\mu }p$ is the $\mu$-th left power of $p$ and $L\left(q\right)=\{{}^{\mu }q:\mu <{\omega }_1\}$ for $q\in {\omega }^{*}$. We also prove that for $p$-compact spaces, $p$-sequentiality and the property of being a $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn space with $\nu <{\omega }_1$, are equivalent; consequently if $X$ is $p$-compact and $\nu <{\omega }_1$, then $X$ is $p$-sequential iff $X$ is ${}^{\nu }p$-sequential (Boldjiev and Malyhin gave, for each $P$-point $p\in {\omega }^{*}$, an example of a compact space $X_p$ which is ${}^{2}p$-Fréchet-Urysohn and it is...

This paper deals with the behavior of $M$-spaces, countably bi-quasi-$k$-spaces and singly bi-quasi-$k$-spaces with point-countable $k$-systems. For example, we show that every $M$-space with a point-countable $k$-system is locally compact paracompact, and every separable singly bi-quasi-$k$-space with a point-countable $k$-system has a countable $k$-system. Also, we consider equivalent relations among spaces entried in Table 1 in Michael’s paper [15] when the spaces have point-countable $k$-systems.

A sufficient condition that the product of two compact spaces has the property of weak approximation by points (briefly WAP) is given. It follows that the product of the unit interval with a compact WAP space is also a WAP space.

A space $X$ is $ℒ$-starcompact if for every open cover $𝒰$ of $X,$ there exists a Lindelöf subset $L$ of $X$ such that $\mathrm{S}t(L,𝒰)=X.$ We clarify the relations between $ℒ$-starcompact spaces and other related spaces and investigate topological properties of $ℒ$-starcompact spaces. A question of Hiremath is answered.

A space $X$ is $𝒞$-starcompact if for every open cover $𝒰$ of $X,$ there exists a countably compact subset $C$ of $X$ such that $\mathrm{St}(C,𝒰)=X.$ In this paper we investigate the relations between $𝒞$-starcompact spaces and other related spaces, and also study topological properties of $𝒞$-starcompact spaces.

Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{\omega }$ is $F_{\sigma \delta }$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{\omega }$. We show that $S_{\omega }$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover,...