Properties of the space of group-valued continuous functions

TL;DR

This paper characterizes various tightness properties of the space of group-valued continuous functions with pointwise convergence topology, linking them to classical selection principles and properties of the underlying space.

Contribution

It provides necessary and sufficient conditions for fan tightness and strong fan tightness in terms of Menger and Rothberger properties, and explores their relationships with other topological properties.

Findings

Characterizes countable fan tightness using Menger property.

Establishes relationships between fan tightness, Reznichenko, and Hurewicz properties.

Shows Menger property is preserved under G-equivalence.

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 $C_{p}(X,G)$.