arXiv subjects
Publications and source records attributed to Kacper Kucharski.
In this note, we prove that an arbitrary product of $W$-spaces is a $κ$-Fréchet-Urysohn space. We also show that the boundary tightness of a product $X \times Y$ is at most $κ$ provided that $X$ is compact with tightness $t(X) \leq κ$ and $Y$ has boundary tightness $tb(Y) \leq κ$. The first result fully resolves, and the second partially addresses, two questions recently raised by Tkachuk.
We prove that if $X$ and $Y$ are first-countable perfect spaces such that the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is a hereditarily Baire space if and only if $Y$ is hereditarily Baire as well. We also establish that for any Tychonoff space, if there exists a continuous linear surjection of the space $C_p(X)$ onto the space $C_p(Y)$ and the space $X$ is either strongly $σ$-scattered or has property $(κ)$, then $Y$ also satisfies those properties. Additionally, we obtain the following result: if $X$ and $Y$ are Tychonoff spaces in which every closed set has a $W$-point, and if the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is scattered if and only if $Y$ is scattered.
In this paper, we provide a complete isomorphism classification of the spaces $C_p(K)$ of real-valued continuous functions endowed with the topology of pointwise convergence for separable compact lines $K$ of weight $ω_1$, under the assumption of Baumgartner's Axiom $\mathsf{BA}$. More specifically, we show that, up to linear homeomorphism, there are exactly two function spaces $C_p(K)$ for such $K$. We also construct an example of a separable compact line $K$ of weight $2^ω$ neither of whose spaces of continuous functions, $C_p(K)$ and $C_w(K)$, is homeomorphic to its square.
A topological space $Y$ has the property (B) of Banakh if there is a countable family $\{A_n:n\in \mathbb{N}\}$ of closed nowhere dense subsets of $Y$ absorbing all compact subsets of $Y$. In this note we show that the space $C_p(X)$ of continuous real-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space $X$ has the following property $(κ)$: every sequence of disjoint finite subsets of $X$ has a subsequence with point--finite open expansion. Additionally, we provide an analogous characterization for the compact--open topology on $C(X)$. Finally, we give examples of Tychonoff spaces $X$ whose all bounded subsets are finite, yet $X$ fails to have the property $(κ)$. This answers a question of Tkachuk.
It is known that both the Menger and Hurewicz property of a Tychonoff space $X$ can be described by the way $X$ is placed in its Čech-Stone compactification $βX$. We provide analogous characterizations for the projective versions of the properties of Menger and Hurewicz.