Related papers: Baire classification of separately continuous func…

Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions which…

We prove that for every $T_0$ space $X$, there is a well-filtered space $W(X)$ and a continuous mapping $\eta_X: X\lra W(X)$ such that for any well-filtered space $Y$ and any continuous mapping $f: X\lra Y$ there is a unique continuous…

We prove that: I. For every regular Lindel\"of space $X$ if $|X|=\Delta(X)$ and $\mathrm{cf}|X|\ne\omega$, then $X$ is maximally resolvable; II. For every regular countably compact space $X$ if $|X|=\Delta(X)$ and $\mathrm{cf}|X|=\omega$,…

The aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if $X\hat{\otimes}_\pi Y$ is strongly subdifferentiable and either $X$ or $Y$ has the metric…

Let $X$ and $Y$ be separable Banach spaces. Suppose $Y$ either has a shrinking basis or $Y$ is isomorphic to $C(2^\mathbb{N})$ and $A$ is a subset of weakly compact operators from $X$ to $Y$ which is analytic in the strong operator…

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space,…

Let $X$ and $Y$ be Banach or normed linear spaces and $F\subset X$ a closed set. We apply our recent extension theorem for vector-valued Baire one functions arXiv:1512.03717 to obtain an extension theorem for vector-valued functions…

In this paper we introduce a class of functions contained in the disc algebra $\mathcal{A}(D)$. We study functions $f \in \mathcal{A}(D)$, which have the property that the continuous periodic function $u = Ref|_{\mathbb{T}}$, where…

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…

In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition…

We define and study the free topological vector space $\mathbb{V}(X)$ over a Tychonoff space $X$. We prove that $\mathbb{V}(X)$ is a $k_\omega$-space if and only if $X$ is a $k_\omega$-space. If $X$ is infinite, then $\mathbb{V}(X)$…

This paper introduces the ring of all real valued Baire one functions, denoted by $B_1(X)$ and also the ring of all real valued bounded Baire one functions, denoted by $B_1^*(X)$. Though the resemblance between $C(X)$ and $B_1(X)$ is the…

For a Tychonoff space $X$, let $C_k(X)$ and $C_p(X)$ be the spaces of real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open topology and the pointwise topology, respectively. If $X$ is compact, the classic result of…

Let $\I$ be an ideal on $\N$ which is either analytic or coanalytic. Assume that $(f_n)$ is a sequence of functions with the Baire property from a Polish space $X$ into a complete metric space $Z$, which is divergent on a comeager set. We…

It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are…

A function $f:X\to \mathbb R$ defined on a topological space $X$ is called returning if for any point $x\in X$ there exists a positive real number $M_x$ such that for every path-connected subset $C_x\subset X$ containing the point $x$ and…

We prove that: 1. If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable. 2. A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally…

This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic…

Completeness for a (topological) space is often based on the existence of special structures (such as metrics, uniformities, proximities, convergences, etc) that explicitly induce the topology, making the completeness induction-dependent.…

We develop a systematic algorithmic framework that unites global and local classification problems using index sets. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear,…