arXiv subjects
Publications and source records attributed to Mikołaj Krupski.
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.
We show that every Lindelöf scattered $W$-space is $σ$-compact. This result generalizes a theorem proved recently by Avilés and the author in [Topology Appl. 363 (2025), Paper No. 109234] and answers a question posed by Tkachuk.
We prove that a compact space $K$ embeds into a $σ$-product of compact metrizable spaces ($σ$-product of intervals) if and only if $K$ is (strongly countable-dimensional) hereditarily metalindelöf and every subspace of $K$ has a nonempty relative open second-countable subset. This provides novel characterizations of $ω$-Corson and $NY$ compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the $σ$-product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space $K$ the property of being $NY$ compact is determined by the topological structure of the space $C_p(K)$ of continuous real-valued functions of $K$ equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.
We prove that every Lindelöf scattered subspace of a $Σ$-product of first-countable spaces is $σ$-compact. In particular, we obtain the result stated in the title. This answers some questions of Tkachuk from [Houston J. Math. 48 (2022), no. 1, 171--181].
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.
An old question of A.V. Arhangel'skii asks if the Menger property of a Tychonoff space $X$ is preserved by homeomorphisms of its function space $C_p(X)$. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of fuction spaces $C_p(X)$ by looking at the way $X$ is positioned in its (Čech-Stone) compactification.
The class of $LΣ(\leqω)$-spaces was introduced in 2006 by Kubiś, Okunev and Szeptycki as a natural refinement of the classical and important notion of Lindelöf $Σ$-spaces. Compact $LΣ(\leqω)$-spaces were considered earlier, under different names, in the works of Tkachuk and Tkachenko in relation to metrizably fibered compacta. In this paper we give counterexamples to several open questions about compact $LΣ(\leqω)$-spaces that are scattered in the literature. Among other things, we refute a conjecture of Kubiś, Okunev and Szeptycki by constructing a separable Rosenthal compactum which is not an $LΣ(\leqω)$-space. We also give insight to the structure of first-countable $(K)LΣ(\leqω)$-compacta.
A Tychonoff space $X$ is called $κ$-pseudocompact if for every continuous mapping $f$ of $X$ into $\mathbb{R}^κ$ the image $f(X)$ is compact. This notion generalizes pseudocompactness and gives a stratification of spaces lying between pseudocompact and compact spaces. It is well known that pseudocompactness of $X$ is determined by the uniform structure of the function space $C_p(X)$ of continuous real-valued functions on $X$ endowed with the pointwise topology. In respect of that A.V. Arhangel'skii asked in [Topology Appl., 89 (1998)] if analogous assertion is true for $κ$-pseudocompactness. We provide an affirmative answer to this question.
We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hernández-Gutiérrez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hrušák, Zamora Avilés, and Hernández-Gutiérrez concerning countable dense homogeneous products.
We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter $\mathcal{F}$ and prove that it is equivalent to $K(\mathcal{F})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_r(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$ which is not completely metrizable with $P_r(X)$ hereditarily Baire. As far as we know this is the first example of this kind.
A classical theorem of Malykhin says that if $\{X_α:α\leqκ\}$ is a family of compact spaces such that $t(X_α)\leq κ$, for every $α\leqκ$, then $t\left( \prod_{α\leq κ} X_α\right)\leq κ$, where $t(X)$ is the tightness of a space $X$. In this paper we prove the following counterpart of Malykhin's theorem for functional tightness: Let $\{X_α:α<λ\}$ be a family of compact spaces such that $t_0(X_α)\leq κ$ for every $α<λ$. If $λ\leq 2^κ$ or $λ$ is less than the first measurable cardinal, then $t_0\left( \prod_{α<λ} X_α\right)\leq κ$, where $t_0(X)$ is the functional tightness of a space $X$. In particular, if there are no measurable cardinals, then the functional tightness is preserved by arbitrarily large products of compacta. Our result answers a question posed by Okunev.
For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we discuss the following basic question which seems to be open: Let $K$ and $L$ be infinite compact spaces. Can it happen that $C_w(K)$ and $C_p(L)$ are homeomorphic? M. Krupski proved that the above problem has a negative answer when $K=L$ and $K$ is finite-dimensional and metrizable. We extend this result to the class of finite-dimensional Valdivia compact spaces $K$.
We give an example of an infinite metrizable space $X$ such that the space $C_p(X)$, of continuous real-valued function on $X$ endowed with the pointwise topology, is not homeomorphic to its own square $C_p(X)\times C_p(X)$. The space $X$ is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A.V. Arhangel'skii.
For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that $K$ is an infinite (metrizable) compact space. Is it true that $C_w(K)$ and $C_p(K)$ are homeomorphic? We show that the answer is "no", provided $K$ is an infinite compact metrizable $C$-space. In particular our proof works for any infinite compact metrizable finite-dimemsional space $K$.
This paper has been withdrawn by the author
Modifying a construction of W. Marciszewski we prove (in ZFC) that there exists a subspace of the real line $\mathbb{R}$, such that the realcompact space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology does not admit a continuous bijection onto a $σ$-compact space. This answers a question of Arhangel'skii.