A countably compact topological group with the non-countably pracompact square

On powers of countably pracompact groups

In 1990, Comfort asked: is there, for every cardinal number $α\leq 2^{\mathfrak{c}}$, a topological group $G$ such that $G^γ$ is countably compact for all cardinals $γ<α$, but $G^α$ is not countably compact? A similar question can also be asked for countably pracompact groups: for which cardinals $α$ is there a topological group $G$ such that $G^γ$ is countably pracompact for all cardinals $γ< α$, but $G^α$ is not countably pracompact? In this paper we construct such group in the case $α= ω$, assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, and in the case $α= κ^{+…

Some pseudocompact-like properties in certain topological groups

We construct in ZFC a countably compact group without non-trivial convergent sequences of size $2^{\mathfrak{c}}$, answering a question of Bellini, Rodrigues and Tomita. We also construct in ZFC a selectively pseudocompact group which is not countably pracompact, showing that these two properties are not equivalent in the class of topological groups. Using the same technique, we construct a group which has all powers selectively pseudocompact but is not countably pracompact, assuming the existence of a selective ultrafilter.

Closed subsets of compact-like topological spaces

We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of $ω{\timesω}$-matrix units cannot be embedded into a topological semigroup which is a weakly H-closed topological space. We show that each Hausdorff topological space is a closed subspace of some $ω$-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. Also, we construct a pseudocomp…

On feebly compact topologies on the semilattice $\exp_nλ$

We study feebly compact topologies $τ$ on the semilattice $\left(\exp_nλ,\cap\right)$ such that $\left(\exp_nλ,τ\right)$ is a semitopological semilattice. All compact semilattice $T_1$-topologies on $\exp_nλ$ are described. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ for a $T_1$-topology $τ$ on $\exp_nλ$ the following conditions are equivalent: $(i)$ $\left(\exp_nλ,τ\right)$ is a compact topological semilattice; $(ii)$ $\left(\exp_nλ,τ\right)$ is a countably compact topological semilattice; $(iii)$ $\left(\exp_nλ,τ\right)$ is a feebly compact…

拓扑群幂中的可数紧性与紧性的拉姆齐理论变体

Countable compactness in powers of topological groups and Ramsey theoretic variations of compactness

该研究针对拓扑群引入级联可数紧性，证明其与有限乘积、可数紧性的关联，构造了多个满足特定级联紧性但不满足可数紧性的拓扑空间与布尔群。

Covering an uncountable square by countably many continuous functions

We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form $X\times X$, where $X$ is an uncountable subset of the real line. This extends Sierpiński's theorem from 1919, saying that $S\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if the size of $S$ does not exceed $\aleph_1$. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles.