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 $\aleph_\alpha\geq \aleph_1$, $C_{\aleph_\alpha}(X)$ 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 $cc_{\aleph_\alpha}(X)$ and $ac_{\aleph_\alpha}(X)$ 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.