arXiv subjects
Publications and source records attributed to Volodymyr Mykhaylyuk.
We prove that every continuous function $f:E\to Y$ depends on countably many coordinates, if $E$ is an $(\aleph_1,\aleph_0)$-invariant pseudo-$\aleph_1$-compact subspace of a product of topological spaces and $Y$ is a space with a regular $G_δ$-diagonal. Using this fact for any $α<ω_1$ we construct an $(α+1)$-embedded subspace of a completely regular space which is not $α$-embedded.
A function $f:X\to Y$ between topological spaces is said to be a {\it weakly Gibson function} if $f(\overline{G})\subseteq \overline{f(G)}$ for any open connected set \mbox{$G\subseteq X$}. We call a function $f:X\to Y$ {\it segmentary connected} if $X$ is topological vector space and $f([a,b])$ is connected for every segment $[a,b]\subseteq X$. We show that if $X$ is a hereditarily Baire space, $Y$ is a metric space, \mbox{$f:X\to Y$} is a Baire-one function and one of the following conditions holds: (i) $X$ is a connected and locally connected space and $f$ is a weakly Gibson function, (ii) $X$ is an arcwise connected space and $f$ is a Darboux function, (iii) $X$ is a topological vector space and $f$ is a segmentary connected function, then $f$ has a connected graph.
A function $f:X\to Y$ between topological spaces is said to be a {\it weakly Gibson function} if $f(\overline{U})\subseteq \overline{f(U)}$ for any open connected set \mbox{$U\subseteq X$}. We prove that if $X$ is a locally connected hereditarily Baire space and $Y$ is a $T_1$-space then an $F_σ$-measurable mapping $f:X\to Y$ is weakly Gibson if and only if for any connected set $C\subseteq X$ with the dense connected interior the image $f(C)$ is connected. Moreover, we show that each weakly Gibson $F_σ$-measurable mapping $f:\mathbb R^n\to Y$, where $Y$ is a $T_1$-space, has a connected graph.
We investigate the Baire classification of mappings $f:X\times Y\to Z$, where $X$ belongs to a wide class of spaces, which includes all metrizable spaces, $Y$ is a topological space, $Z$ is an equiconnected space, which are continuous in the first variable and for a dense set in $X$ these mappings are functions of a Baire class $α$ in the second variable.
We prove that for a topological space $X$, an equiconnected space $Z$ and a Baire-one mapping $g:X\to Z$ there exists a separately continuous mapping $f:X^2\to Z$ with the diagonal $g$, i.e. $g(x)=f(x,x)$ for every $x\in X$. Under a mild assumptions on $X$ and $Z$ we obtain that diagonals of separately continuous mappings $f:X^2\to Z$ are exactly Baire-one functions, and diagonals of mappings $f:X^2\to Z$ which are continuous on the first variable and Lipschitz (differentiable) on the second one, are exactly the functions of stable first Baire class.
We prove that for every separately twice differentiable solution $f$ of the PDE $f"_{xx}=f"_{yy}$ is of the form $f(x,y)=ϕ(x+y)+ψ(x-y)$ for some twice differentiable functions $ϕ, ψ$.
For a non-isolated point $x$ of a topological space $X$ the network character $nw_χ(x)$ is the smallest cardinality of a family of infinite subsets of $X$ such that each neighborhood $O(x)$ of $x$ contains a set from the family. We prove that (1) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with $nw_χ(x)=\aleph_0$; (2) for each point $x\in X$ with countable character there is an injective sequence in $X$ that $\F$-converges to $x$ for some meager filter $\F$ on $ω$; (3) if a functionally Hausdorff space $X$ contains an $\F$-convergent injective sequence for some meager filter $\F$, then for every $T_1$-space $Y$ that contains two non-empty open sets with disjoint closures, the function space $C_p(X,Y)$ is meager. Also we investigate properties of filters $\F$ admitting an injective $\F$-convergent sequence in $βω$.