Quasi-equicontinuous sets of functions

The well-known theorem of analysis that if F is a family of functions defined, equicontinuous, and uniformly bounded on a bounded closed set E in n-dimensional real cartesian space Rn, then from every sequence {fn } of functions of F it is possible to select a uniformly convergent subsequence, has been recently generalized to various abstract spaces [1; 4; 6]. Consider a set F of continuous functions on one topological space X to another, Y. For any point x of X and any open set W of Y we denote by (x, W) the totality of functions f in F for which f(x) ? W. The topology in F obtained by employing all sets of the (x, W) as a subbase in F is called the p-topology by Arens [2]. The purpose of this note is to find the necessary and sufficient conditions that it be possible to select a subsequence converging pointwise to a continuous function from any given sequence of continuous functions and the necessary and sufficient conditions that a set of continuous functions be compact in the p-topology. DEFINITION. Let {fn } be a sequence of functions from a topological space X to be metric space Y. {fn } is said to be e-related at a point xCX if for every arbitrarily chosen e>0 there is a neighborhood U(x) of x such that, corresponding to each point x' U(x), a positive number N,(x, x') can be determined satisfying the condition: