Properties of the space of group-valued continuous functions

Abstract

In this paper, we find necessary and sufficient conditions for countable fan tightness and countable strong fan tightness of the space (briefly, $C_{p}(X,G)$) of all group-valued continuous functions endowed with the topology of pointwise convergence in term of Menger property and Rothberger property respectively. Furthermore, we establish a relationship between countable fan tightness, the Reznichenko property and the Hurewicz property for the space $C_{p}(X,G)$. In addition to this we prove that the Menger property is preserve during $G$-equivalence of topological spaces. Through this paper, we establish a general result regarding fan tightness of $C_{p}(X,G)$ and Hurewicz number of the space $X^{n}$ for every natural number $n$. Finally, we study the monolithicity of the space $C_{p}(X,G)$.