Connectedness of some rings of~quotients of $C(X)$ with the $m$-topology

Abstract

In this article we define the $m$-topology on some rings of quotients of $C(X)$. Using this, we equip the classical ring of quotients $q(X)$ of $C(X)$ with the $m$-topology and we show that $C(X)$ with the $r$-topology is in fact a subspace of $q(X)$ with the $m$-topology. Characterization of the components of rings of quotients of $C(X)$ is given and using this, it turns out that $q(X)$ with the $m$-topology is connected if and only if $X$ is a pseudocompact almost $P$-space, if and only if $C(X)$ with $r$-topology is connected. We also observe that the maximal ring of quotients $Q(X)$ of $C(X)$ with the $m$-topology is connected if and only if $X$ is finite. Finally for each point $x$, we introduce a natural ring of quotients of $C(X)/O_x$ which is connected with the $m$-topology.