Related papers: On the sequential closure of the set of continuous…

We examine conditions on a (compact metrizable) space $X$ such that for any space $Y$ and closed subspace $Z$, the set of continuous functions from $Z$ to $X$ which extend to $Y$ is either open or closed in the set of continuous functions…

It is investigated the existence of a separately continuous function $f:X\times Y\to \mathbb R$ with an onepoint set of discontinuity for topological spaces $X$ and $Y$ which satisfy compactness type conditions. In particular, it is shown…

We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the…

Let $X$ be a locally compact topological space, $(Y,d)$ be a boundedly compact metric space and $LB(X,Y)$ be the space of all locally bounded functions from $X$ to $Y$. We characterize compact sets in $LB(X,Y)$ equipped with the topology of…

Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\mathcal{I}}$-Tychonoff space for some ideal $\mathcal{I}$ of compact sets of $X$. Denote by $C_{\mathcal{I}}(X,Y)$ the space of continuous functions from…

In this paper we continue our research of functions on the boundary of their domain and obtain some results on cluster sets of functions between topological spaces. In particular, we prove that for a metrizable topological space $X$, a…

We prove that every vertically nearly separately continuous function defined on a product of a strong PP-space and a topological space and with values in a strongly $\sigma$-metrizable space with a special stratification, is a pointwise…

We prove that for any topological space $X$ of countable tightness, each \sigma-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…

We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindel\"of subspace…

For a Tychonoff space $X$, we denote by $C_k(X)$ the space of all real-valued continuous functions on X with the compact-open topology. In this paper, we have gave characterization for $C_k(X)$ to satisfy $S_{fin}(S, S)$.

Let $(X_n)_{n}$ be a sequence of uniform spaces such that each space $X_n$ is a closed subspace in $X_{n+1}$. We give an explicit description of the topology and uniformity of the direct limit $u-lim X_n$ of the sequence $(X_n)$ in the…

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology of pointwise convergence into the Tychonoff product of topological spaces, but applied to the function space $C(X,Y)$…

For a topological space X, let (RX)s := (RX,Ts) be the cartesian product of |X| copies of the real line R with the topology of the uniform convergence on separable subsets of X. In this article we analyze the subspace C(X) of (RX)s of all…

It is solved a problem of construction of separately continuous functions on the product of compacts with a given discontinuity points set. We obtaine the following results. 1. For arbitrary \v{C}ech complete spaces $X$, $Y$ and a separable…

It is solved the problem on constructed of separately continuous functions on product of two topological spaces with given restriction. In particular, it is shown that for every topological space $X$ and first Baire class function $g:X\to…

It is solved the problem on construction of separately continuous functions on product of $n$ topological spaces with given restriction. In particular, it is shown that for every topological space $X$ and $n-1$ Baire class function $g:X\to…

Let ${\mathbb T}=({\bf T},\leq)$ and ${\mathbb T}_{1}=({\bf T}_{1},\leq_{1})$ be linearly ordered sets and $\mathscr{X}$ be a topological space. The main result of the paper is the following: If function $\boldsymbol{f}(t,x):{\bf…

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is…

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…

A map $f:X\to Y$ between topological spaces is defined to be {\em scatteredly continuous} if for each subspace $A\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\to Y$ from a perfectly paracompact…