This paper investigates necessary and sufficient conditions for a space to have an H-closed extension with countable remainder. For countable spaces we are able to give two characterizations of those spaces admitting an H-closed extension with countable remainder. The general case is more difficult, however, we arrive at a necessary condition — a generalization of Čech completeness, and several sufficient conditions for a space to have an H-closed extension with countable remainder. In particular,...

There is a locally compact Hausdorff space which is linearly Lindelöf and not Lindelöf. This answers a question of Arhangel'skii and Buzyakova.

In our previous paper, we introduced the notion of a lonely point, due to P. Simon. A point $p\in X$ is lonely if it is a limit point of a countable dense-in-itself set, it is not a limit point of a countable discrete set and all countable sets whose limit point it is form a filter. We use the space ${𝒢}_{\omega }$ from a paper of A. Dow, A.V. Gubbi and A. Szymański [Rigid Stone spaces within ZFC, Proc. Amer. Math. Soc. 102 (1988), no. 3, 745–748] to construct lonely points in ${\omega }^{*}$. This answers the question of P. Simon...

Given idempotents e and f in a semigroup, e ≤ f if and only if e = fe = ef. We show that if G is a countable discrete group, p is a right cancelable element of G* = βG∖G, and λ is a countable ordinal, then there is a strictly decreasing chain $⟨q_{\sigma }{⟩}_{\sigma <\lambda }$ of idempotents in $C_p$, the smallest compact subsemigroup of G* with p as a member. We also show that if S is any infinite subsemigroup of a countable group, then any nonminimal idempotent in S* is the largest element of such a strictly decreasing chain of idempotents....

J. Terasawa in "$\beta X-\left\{p\right\}$ are non-normal for non-discrete spaces $X$" (2007) and the author in “On non-normality points and metrizable crowded spaces” (2007), independently showed for any metrizable crowded space $X$ that each point $p$ of its Čech–Stone remainder $X^{*}$ is a non-normality point of $\beta X$. We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.

Let X be a crowded metric space of weight κ that is either ${\kappa }^{\omega }$-like or locally compact. Let y ∈ βX∖X and assume GCH. Then a space of nonuniform ultrafilters embeds as a closed subspace of (βX∖X)∖y with y as the unique limit point. If, in addition, y is a regular z-ultrafilter, then the space of nonuniform ultrafilters is not normal, and hence (βX∖X)∖y is not normal.

We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2n}$ such that $C\left(K_{2n}\right)$ has no uncountable (semi)biorthogonal sequence ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of at most 2n-1 points of $K_{2n}$, but has biorthogonal systems ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves that its is...

Let $X$ be the Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff non-single point spaces $X_{\alpha }$. Let $p\in X^{*}$ be a point in the closure of some $G\subset X$ whose weak Lindelöf number is strictly less than the cofinality of $\tau$. Then we show that $\beta X\setminus \left\{p\right\}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $\beta X$. In particular, this is true if either $X={\omega }^{\tau }$ or $X=R^{\tau }$ and $\tau$ is infinite and not countably cofinal.

We show that ${\omega }^{*}\setminus \left\{p\right\}$ is not normal, if $p$ is a limit point of some countable subset of ${\omega }^{*}$, consisting of points of character ${\omega }_1$. Moreover, such a point $p$ is a Kunen point and a super Kunen point.

An $\alpha$-space is a topological space in which the topology is generated by the family of all $\alpha$-sets (see [N]). In this paper, minimal-$\alpha 𝒫$-spaces (where $𝒫$ denotes several separation axioms) are investigated. Some new characterizations of $\alpha$-spaces are also obtained.

Let a space $X$ be Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff nonsingle point spaces $X_{\alpha }$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau$. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau }$ or ${\omega }^{\tau }$ and a cardinal $\tau$ is infinite and not countably cofinal.

Given a topological space $X$, let $𝒰X$ and ${\eta }_X:X\to 𝒰X$ denote, respectively, the Salbany compactification of $X$ and the compactification map called the Salbany map of $X$. For every continuous function $f:X\to Y$, there is a continuous function $𝒰f:𝒰X\to 𝒰Y$, called the Salbany lift of $f$, satisfying $\left(𝒰f\right)\circ {\eta }_X={\eta }_Y\circ f$. If a continuous function $f:X\to Y$ has a stably compact codomain $Y$, then there is a Salbany extension $F:𝒰X\to Y$ of $f$, not necessarily unique, such that $F\circ {\eta }_X=f$. In this paper, we give a condition on a space such that its Salbany map is open. In particular,...

We discuss the following result of A. Szymański in “Retracts and non-normality points" (2012), Corollary 3.5.: If $F$ is a closed subspace of ${\omega }^{*}$ and the $\pi$-weight of $F$ is countable, then every nonisolated point of $F$ is a non-normality point of ${\omega }^{*}$. We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in “Some non-normal subspaces of the Čech–Stone compactification of a discrete space" (1980) by A. Błaszczyk and A. Szymański. Perhaps our proofs...

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a\in X^{\text{'}}$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive dynamical...

We shall show that there is an ultrafilter on singular $\kappa$ with countable cofinality, which cannot be reached from the set of all subuniform ultrafilters by iterating the closure of sets of size $<\kappa$.

Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain $T_{\alpha }:\alpha <\lambda$ of infinite subsets of ω, there exists $ℳ\subset {\left[\omega \right]}^{\omega }$, an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain $T_{\alpha }:\alpha <\lambda$, hence a ψ-space with Stone-Čech remainder...