A special kind of topology on $C(X)$ lying between the point-open topology and the topology of uniform convergence

Abstract

Let $X$ be a topological space. For each transfinite cardinal number ${\aleph }_{\alpha }$, we define a topology $C_{{\aleph }_{\alpha }}(X)$ on the ring $C(X)$. With ${\aleph }_{\alpha }={\aleph }_0$, $C_{{\aleph }_{\alpha }}(X)$ reduces to the space $C_p(X)$. We prove that for ${\aleph }_{\alpha }\ge {\aleph }_1$, $C_{{\aleph }_{\alpha }}(X)$ is pathwise connected if and only if it is connected if and only if $X$ is pseudocompact. Here, we define ${\aleph }_{\alpha }$-separable space and furthermore, we show that $X$ is ${\aleph }_{\alpha }$-separable when and only when $C_{{\aleph }_{\alpha }}(X)$ is metrizable when and only when it is a sequential space. Later we introduce two new cardinal functions, namely $c{c}_{{\aleph }_{\alpha }}(X)$ and $a{c}_{{\aleph }_{\alpha }}(X)$ which turned out to be the character and pseudocharacter of $C_{{\aleph }_{\alpha }}(X)$ respectively. At the end of this article we show that a number cardinal functions associated with the space $C_{{\aleph }_{\alpha }}(X)$ are equal.