COMPACTNESS AND CARDINALITY OF THE SPACE OF CONTINUOUS FUNCTIONS UNDER REGULAR TOPOLOGY.

In this paper, we investigate the compactness and cardinality of the space C(X, Y ) of continuous functions from a topological space X to Y equipped with the regular topology. We prove that different forms of compactness, such as sequential compactness, countable compactness, and pseudocompactness, coincide on a subset of C(X, Y ) with regular topology. Moreover, we prove the comparison and coincidence of regular topology with the graph topology on the space C(X, Y ). Furthermore, we examine various cardinal invariants, such as density, character, pseudocharacter, etc., on the space C(X, Y ) equipped with the regular topology. In addition, we define a type of equivalence between X and Y in terms of C(X) and C(Y ) endowed with the regular topology and investigate certain cardinal invariants preserved by this equivalence. [ABSTRACT FROM AUTHOR]