In its simplest form, the theorem of Ascoli with which we are concerned is an extension of the Bolzano-Weierstrass theorem: it states that, if X and Y are bounded closed sets, of real or complex numbers, andis a sequence of equicontinuous functions mapping X into Y, thenhas a uniformly convergent subsequence. As is well known, this result has played a fundamental part in the development of several theories; it has also been widely generalized, by processes of abstraction and localization, and a very useful version of the theorem runs as follows (cf. , 233–234):
(A) Suppose that X is a locally compact regular space, and that Y is a Hausdorff space whose topology is determined by a uniform structure. Let YX be the space of all functions that map X into Y, with the topology of locally uniform convergence with respect to. Then a closed setin YX is compact if, at each point x of X, (i) the set(x) is relatively compact, in Y, and, (ii) is equicontinuous with respect to.