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.