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Anton Lipin

Publications and source records attributed to Anton Lipin.

6 recordsLinked to original sources
arXiv2026

$\mathbb R^{ω_1}$-Factorizable Spaces and Groups

A topological space $X$ is $\mathbb R^{ω_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{ω_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{ω_1}$-factorizable if and only if $X\times D(ω_1)$, where $D(ω_1)$ is a discrete space of cardinality $ω_1$, is $z$-embedded in the product $βX\times βD(ω_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{ω_1}$-factorizability is hereditary and countably multiplicative, that any $\mathbb R^{ω_1}$-factorizable space is hereditarily Lindelöf and hereditarily separable, and that the existence of nonmetrizable $\mathbb R^{ω_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all $\mathbb R^{ω_1}$-factorizable spaces are second-countable, while under MA + $\lnot$CH, the countable Fréchet--Urysohn fan is $\mathbb R^{ω_1}$-factorizable.

math.GN↗
arXiv2025

On resolvability and tightness in uncountable spaces

We investigate connections between resolvability and different forms of tightness. This study is adjacent to [1,2]. We construct a non-regular refinement $τ^*$ of the natural topology of the real line $\mathbb{R}$ with properties such that the space $(\mathbb{R}, τ^*)$ has a hereditary nowhere dense tightness and it has no $ω_1$-resolvable subspaces, whereas $Δ(\mathbb{R}, τ^*) = \frak{c}$. We also show that the proof of the main result of [1], being slightly modified, leads to the following strengthening: if $L$ is a Hausdorff space of countable character and the space $L^ω$ is c.c.c., then every submaximal dense subspace of $L^κ$ has disjoint tightness. As a corollary, for every $κ\geq ω$ there is a Tychonoff submaximal space $X$ such that $|X|=Δ(X)=κ$ and $X$ has disjoint tightness.

math.GN↗
arXiv2025

Resolvability in products and squares

Suppose $X$ and $Y$ are topological spaces, $|X| = Δ(X)$ and $|Y| = Δ(Y)$. We investigate resolvability of the product $X \times Y$. We prove that: I. If $|X| = |Y| = ω$ and $X,Y$ are Hausdorff, then $X \times Y$ is maximally resolvable; II. If $2^κ= κ^+$, $\{|X|, \mathrm{cf}|X|\} \cap \{κ, κ^+\} \ne \emptyset$ and $\mathrm{cf}|Y| = κ^+$, then the space $X \times Y$ is $κ^+$-resolvable. In particular, under GCH the space $X^2$ is $\mathrm{cf}|X|$-resolvable whenever $\mathrm{cf}|X|$ is an isolated cardinal; III. ($\frak{r} = \frak{c}$) If $\mathrm{cf}|X| = ω$ and $\mathrm{cf}|Y| = \mathrm{cf}(\frak{c})$, then the space $X \times Y$ is $ω$-resolvable. If, moreover, $\mathrm{cf}(\frak{c}) = ω_1$, then the space $X \times Y$ is $ω_1$-resolvable.

math.GN↗
arXiv2024

Relatively functionally countable subsets of products

A subset $A$ of a topological space $X$ is called relatively functionally countable (RFC) in $X$, if for each continuous function $f : X \to \mathbb{R}$ the set $f[A]$ is countable. We prove that all RFC subsets of a product $\prod\limits_{n\inω}X_n$ are countable, assuming that spaces $X_n$ are Tychonoff and all RFC subsets of every $X_n$ are countable. In particular, in a metrizable space every RFC subset is countable. The main tool in the proof is the following result: for every Tychonoff space $X$ and any countable set $Q \subseteq X$ there is a continuous function $f : X^ω\to \mathbb{R}^2$ such that the restriction of $f$ to $Q^ω$ is injective.

math.GN↗
arXiv2023

On resolvability, connectedness and pseudocompactness

We prove that: I. If $L$ is a $T_1$ space, $|L|>1$ and $d(L) \leq κ\geq ω$, then there is a submaximal dense subspace $X$ of $L^{2^κ}$ such that $|X|=Δ(X)=κ$; II. If $\frak{c}\leqκ=κ^ω<λ$ and $2^κ=2^λ$, then there is a Tychonoff pseudocompact globally and locally connected space $X$ such that $|X|=Δ(X)=λ$ and $X$ is not $κ^+$-resolvable; III. If $ω_1\leqκ<λ$ and $2^κ=2^λ$, then there is a regular space $X$ such that $|X|=Δ(X)=λ$, all continuous real-valued functions on $X$ are constant (so $X$ is pseudocompact and connected) and $X$ is not $κ^+$-resolvable.

math.GN↗