A number of properties enjoyed by two specially constructed topologies on $C(X)$

Abstract

If $I$ is an ideal in the ring $C(X)$ of all real valued continuous functions defined over a Tychonoff space $X$, then $X$ is called $I$-$pseudocompact$ if the set $X\setminus \bigcap Z[I]$ is a bounded subset of $X$. Corresponding to $I$, the $m^I$-topology and $u^I$-topology on $C(X)$, generalizing the well-known $m$-topology and $u$-topology in $C(X)$ respectively are already there in the literature. It is proved amongst others that the $m^I$-topology is first countable if and only if the $u^I$-topology= $m^I$-topology on $C(X)$ if and only if $X$ is $I$-$pseudocompact$. A special case of this result on choosing $I=C(X)$ reads: the $u$-topology and $m$-topology on $C(X)$ coincide if and only if $X$ is pseudocompact. It is established that the $m^I$-topology on $C(X)$ is second countable if and only if it is ${\aleph }_0$-$bounded$ if and only if $X$ is compact, metrizable and $I=C(X)$. Furthermore it is realized that the $m^I$ topology on $C(X)$ is hemicompact if and only if it is $\sigma$-compact if and only if this topology is $H$-$bounded$ if and only if $X$ is finite and $I=C(X)$.