Related papers: Continuous extension of functions from countable s…

Let $X$ be a topological vector space of complex-valued sequences and $Y$ be a subset of $X$. We provide conditions for $X \setminus Y \cup \{0\}$ to contain uncountably infinitely many linearly independent dense vector subspaces of $X$. We…

Assume that $\mathcal{P}$ is a topological property of a space $X$, then we say that $X$ is {\it dense-$\mathcal{P}$} if each dense subset of $X$ has the property $\mathcal{P}$. In this paper, we mainly discuss dense subsets of a space $X$,…

We present applications of $C_p$-theory, the branch of general topology concerned with spaces of real-valued continuous functions, to model theory, mostly in the context of continuous logics. We include $C_p$-theoretic results and proofs in…

Being motivated by the study of the space $C_c(X)$ of all continuous real-valued functions on a Tychonoff space $X$ with the compact-open topology, we introduced in [15] the concepts of a $cp$-network and a $cn$-network (at a point $x$) in…

Some new classes of compacta $K$ are considered for which $C(K)$ endowed with the pointwise topology has a countable cover by sets of small local norm--diameter.

We give an abstract approach to approximations with a wide range of regularity classes $X$ in spaces of pseudocontinuable functions $K^p_\vartheta$, where $\vartheta$ is an inner function and $p>0$. More precisely, we demonstrate a general…

We propose the notion of countable decomposability of maps on C*-algebras: a bounded linear map $\varphi : \mathscr{A}\to B(\mathcal{H})$, where $\mathscr{A}$ is a C*-algebra and $\mathcal{H}$ a Hilbert space, will be called countably…

An open (resp., closed) subset A of a topological space (X, T ) is called C-open (resp., C-closed) set if cl(A) \ A (resp., A \ int(A)) is a countable set. This paper aims to present the concept of C-open and C-closed sets. We first…

We consider *-linear maps into a commutative C*-algebra C (X) of continuous functions on a locally compact Hausdorff space X with certain specified properties and prove two results: (1) an extension result for a class of *-linear maps Y -->…

We survey discrete and continuous model-theoretic notions which have important connections to general topology. We present a self-contained exposition of several interactions between continuous logic and $C_p$-theory which have applications…

In this paper, we study some properties of $*-$open and $*-$closed subsets of a space. The collection of all $*-$open subsets of a space $X$ form a topology on $X$ which is denoted by $^{*}O(X)$. We investigate the relations between…

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…

The concept of the strong Pytkeev property, recently introduced by Tsaban and Zdomskyy in [32], was successfully applied to the study of the space $C_c(X)$ of all continuous real-valued functions with the compact-open topology on some…

We study definably complete locally o-minimal expansions of ordered groups in this paper. A definable continuous function defined on a closed, bounded and definable set behave like a continuous function on a compact set. We demonstrate…

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 show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of…

The separately continuity topology is considered and some its properties are investigated. With help of these properties a generalization of Sierpinski theorem on determination of real separately continuous function by its values on an…

Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial…

Let $X$ be a zero-dimensional space and $C_c(X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K(X)$ (resp., $C_{c}^{\psi}(X)$) the set of all functions in $C_c(X)$ with…

We present some sufficient conditions for continuity of the mapping $f:\langle X,\tau_X^*\rangle \to \langle Y,\tau_Y^*\rangle$, where $\tau_X^*$ and $\tau_Y^*$ are topologies induced by the local function on $X$ and $Y$, resp. under the…