arXiv subjects

Bogdan Bokalo

Publications and source records attributed to Bogdan Bokalo.

6 recordsLinked to original sources

arXiv2019

Scattered compact sets in continuous images of Čech-complete spaces

Assume hat a functionally Hausdorff space $X$ is a continuous image of a Čech complete space $P$ with Lindelöf number $l(P)<\mathfrak c$. Then the following conditions are equivalent: (i) every compact subset of $X$ is scattered, (ii) for every continuous map $f:X\to Y$ to a functionally Hausdorff space $Y$ the image $f(X)$ has cardinality $|f(X)|\le \max\{l(P),ψ(Y)\}$, (iii) no continuous map $f:X\to[0,1]$ is surjective. Also we prove the equivalence of the conditions: (a) $ω_1<\mathfrak b$, (b) a K-analytic space $X$ (with a unique non-isolated point) is countable if and only if every compact subset of $X$ is countable.

math.GN↗

arXiv2019

On some functional generalizations of the regularity of topological spaces

We introduce and study some generalizations of regular spaces, which were motivated by studying continuity properties of functions between (regular) topological spaces. In particular, we prove that a first-countable Hausdorff topological space is regular if and only if it does not contain a topological copy of the Gutik hedgehog.

math.GN↗

arXiv2017

Weakly discontinuous and resolvable functions between topological spaces

We prove that a function $f:X\to Y$ from a first-countable (more generally, Preiss-Simon) space $X$ to a regular space $Y$ is weakly discontinuous (which means that every subspace $A\subset X$ contains an open dense subset $U\subset A$ such that $f|U$ is continuous) if and only if $f$ is open-resolvable (in the sense that for every open subset $U\subset Y$ the preimage $f^{-1}(U)$ is a resolvable subset of $X$) if and only if $f$ is resolvable (in the sense that for every resolvable subset $R\subset Y$ the preimage $f^{-1}(R)$ is a resolvable subset of $X$). For functions on metrizable spaces this characterization was announced (without proof) by Vinokurov in 1985.

math.GN↗

arXiv2016

Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

A map $f:X\to Y$ between topological spaces is called weakly discontinuous if each subspace $A\subset X$ contains an open dense subspace $U\subset A$ such that the restriction $f|U$ is continuous. A bijective map $f:X\to Y$ between topological spaces is called a weak homeomorphism if $f$ and $f^{-1}$ are weakly discontinuous. We study properties of topological spaces preserved by weakly discontinuous maps and weak homeomorphisms. In particular, we show that weak homeomorphisms preserve network weight, hereditary Lindelöf number, dimension. Also we classify infinite zero-dimensional $σ$-Polish metrizable spaces up to a weak homeomorphism and prove that any such space $X$ is weakly homeomorphic to one of 9 spaces: $ω$, $2^ω$, $\mathbb N^ω$, $\mathbb Q$, $\mathbb Q\oplus 2^ω$, $\mathbb Q\times 2^ω$, $\mathbb Q\oplus\mathbb N^ω$, $(\mathbb Q\times 2^ω)\oplus\mathbb N^ω$, $\mathbb Q\times\mathbb N^ω$.

math.GN↗

arXiv2012

On $\infty$-convex sets in spaces of scatteredly continuous functions

Given a topological space $X$, we study the structure of $\infty$-convex subsets in the space $SC_p(X)$ of scatteredly continuous functions on $X$. Our main result says that for a topological space $X$ with countable strong fan tightness, each potentially bounded $\infty$-convex subset $F\subset SC_p(X)$ is weakly discontinuous in the sense that each non-empty subset $A\subset X$ contains an open dense subset $U\subset A$ such that each function $f|U$, $f\in F$, is continuous. This implies that $F$ has network weight $nw(F)\le nw(X)$.

math.GN↗

arXiv2012

On σ-convex subsets in spaces of scatteredly continuous functions

We prove that for any topological space $X$ of countable tightness, each σ-convex subspace $\F$ of the space $SC_p(X)$ of scatteredly continuous real-valued functions on $X$ has network weight $nw(\F)\le nw(X)$. This implies that for a metrizable separable space $X$, each compact convex subset in the function space $SC_p(X)$ is metrizable. Another corollary says that two Tychonoff spaces $X,Y$ with countable tightness and topologically isomorphic linear topological spaces $SC_p(X)$ and $SC_p(Y)$ have the same network weight $nw(X)=nw(Y)$. Also we prove that each zero-dimensional separable Rosenthal compact space is homeomorphic to a compact subset of the function space $SC_p(ω^ω)$ over the space $ω^ω$ of irrationals.

math.GN↗