An improvement of the Arzela-Ascoli theorem (Q411801)

Let \(C(X,Y)\) be the set of continuous functions from the topological space \(X\) into the uniform space \(Y\) endowed with the topology of uniform convergence. The classical version of the Arzelà-Ascoli theorem asserts that if \(X\) is a compact space and \(Y\) is a metric space, then a subset \(S\) of \(C(X,Y)\) is relatively compact if and only if \(S\) is equicontinuous and \(S(x)= \{f(x); f\in S\}\) is relatively compact for each \(x\in X\).NEWLINENEWLINE In the paper under review the authors prove that, if \(X\) is an almost compact space (a notion weaker than compactness) and \(Y\) is a uniform space, then every equicontinuous subset \(S\) of \(C(X,Y)\) such that \(S(x)\) is relatively compact for each \(x\in X\) is necessarily relatively compact. They also show that if \(X\) is a compact space, \(Y\) is a pseudometric space and \(S\subset C(X,Y)\) is relatively compact, then \(S\) is equicontinuous and \(S(x)\) is relatively compact for each \(x\in X\).