In this article we define the $m$-topology on some rings of quotients of $C\left(X\right)$. Using this, we equip the classical ring of quotients $q\left(X\right)$ of $C\left(X\right)$ with the $m$-topology and we show that $C\left(X\right)$ with the $r$-topology is in fact a subspace of $q\left(X\right)$ with the $m$-topology. Characterization of the components of rings of quotients of $C\left(X\right)$ is given and using this, it turns out that $q\left(X\right)$ with the $m$-topology is connected if and only if $X$ is a pseudocompact almost $P$-space, if and only if $C\left(X\right)$ with $r$-topology is connected. We also observe that...

In this paper, we show that it is possible to extend the Ellis theorem, establishing the relations between axioms of a topological group on a new class $𝒩$ of spaces containing all countably compact spaces in the case of Abelian group structure. We extend statements of the Ellis theorem concerning separate and joint continuity of group inverse on the class of spaces $𝒩$ that gives some new examples and statements for the $C_p$-theory and theory of topologically homogeneous spaces.

For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and...

We prove that if K is a compact space and the space P(K × K) of regular probability measures on K × K has countable tightness in its weak* topology, then L₁(μ) is separable for every μ ∈ P(K). It has been known that such a result is a consequence of Martin's axiom MA(ω₁). Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.

The aim of this paper is to show that every Hausdorff continuous interval-valued function on a completely regular topological space X corresponds to a Dedekind cut in C(X) and conversely.

Jachymski showed that the set $$\left\{(x,y)\in {𝐜}_0\times {𝐜}_0:{\left(\sum _{i=1}^n\alpha \left(i\right)x\left(i\right)y\left(i\right)\right)}_{n=1}^{\infty }\text{is}\text{bounded}\right\}$$ is either a meager subset of ${𝐜}_0\times {𝐜}_0$ or is equal to ${𝐜}_0\times {𝐜}_0$. In the paper we generalize this result by considering more general spaces than ${𝐜}_0$, namely ${𝐂}_0\left(X\right)$, the space of all continuous functions which vanish at infinity, and ${𝐂}_b\left(X\right)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma$-porosity.

Let $C_p\left(X\right)$ be the space of continuous real-valued functions on X, with the topology of pointwise convergence. We consider the following three properties of a space X: (a) $C_p\left(X\right)$ is Scott-domain representable; (b) $C_p\left(X\right)$ is domain representable; (c) X is discrete. We show that those three properties are mutually equivalent in any normal T₁-space, and that properties (a) and (c) are equivalent in any completely regular pseudo-normal space. For normal spaces, this generalizes the recent result of Tkachuk that $C_p\left(X\right)$ is...