Related papers: Spaces with a Finite Family of Basic Functions

A family $\bfam$ of continuous real-valued functions on a space $X$ is said to be {\sl basic} if every $f \in C(X)$ can be represented $f = \sum_{i=1}^n g_i \circ \phi_i$ for some $\phi_i \in \bfam$ and $g_i \in C(\R)$ ($i=1, ..., n$).…

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…

Any unital separable continuous C(X)-algebra with properly infinite fibres is properly infinite as soon as the compact Hausdorff space X has finite topolog-ical dimension. We study conditions under which this is still the case if the…

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…

A regular topological space $X$ is defined to be a $\mathfrak P_0$-space if it has countable Pytkeev network. A network $\mathcal N$ for $X$ is called a Pytkeev network if for any point $x\in X$, neighborhood $O_x\subset X$ of $x$ and…

Let $f \colon X \rightarrow Y$ be a resolvable-measurable mapping of a metrizable space $X$ to a regular space $Y$. Then $f$ is piecewise continuous. Additionally, for a metrizable completely Baire space $X$, it is proved that $f$ is…

In this paper we introduce and study so-called $k^*$-metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory. By…

For separable metrizable spaces $X,Y$ and a metrizable topological group $Z$ by $S(X\times Y,Z)$ we denote the space of all separately continuous functions $f:X\times Y\to Z$ endowed with the topology of layer-wise uniform convergence,…

If $\mathcal P$ is a family of filters over some set $I$, a topological space $X$ is \emph{sequencewise $\mathcal P$-\brfrt compact} if, for every $I$-indexed sequence of elements of $X$, there is $F \in \mathcal P$ such that the sequence…

A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. In this paper, we mainly give affirmative answers for two questions about $\pi$-metrizable spaces. The main results are that: (1) A space $X$ is…

For a metrizable space $X$ and a finite measure space $(\Omega,\mathfrak{M},\mu)$ let $M_{\mu}(X)$ and $M^f_{\mu}(X)$ be the spaces of all equivalence classes (under the relation of equality almost everywhere mod $\mu$) of…

Any unital separable continuous C(X)-algebra with properly infinite fibres is properly infinite as soon as the compact Hausdorff space X has finite topological dimension. We study conditions under which this is still the case in the…

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…

We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every…

A simple criterion for a functor to be finitary is presented: we call $F$ finitely bounded if for all objects $X$ every finitely generated subobject of $FX$ factorizes through the $F$-image of a finitely generated subobject of $X$. This is…

Let S denote the family of all subspaces of the plane that are graphs of functions from the real line R to itself. We prove that S has two subfamilies G,H of spaces such that the cardinality of G is c (the cardinality of the continuum) and…

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$…

A topological space $X$ is called a topological fractal if $X=\bigcup_{f\in\mathcal F}f(X)$ for a finite system $\mathcal F$ of continuous self-maps of $X$, which is topologically contracting in the sense that for every open cover $\mathcal…

we prove that if $X$ is a locally compact $\sigma$-compact space then on its quotient, $\gamma(X)$ say, determined by the algebra of all real valued bounded continuous functions on $X$, the quotient topology and the completely regular…

A finite family $\mathcal{F}=\{f_1,\ldots,f_n\}$ of continuous selfmaps of a given metric space $X$ is called an iterated function system (shortly IFS). In a case of contractive selfmaps of a complete metric space is well-known that IFS has…