Functional equicontinuity and uniformities in topological groups (Q1886711)

Let \(X\) be a topological space and let \(\mathcal M\) be a collection of subspaces of \(X\); \(X\) is said to be \textit{strongly functionally generated} by \(\mathcal M\) if for every real-valued discontinuous mapping \(f:X\rightarrow \mathbb{R}\) there is a set \(A\in\mathcal M\) such that the restriction \(f_{| A}:A\rightarrow \mathbb{R}\) is discontinuous. A subspace \(Y\) of \(X\) is said to be relatively pseudocompact in \(X\) at \(y\in Y\) if there is a sequence \((V_n)_{n\in \mathbb{N}}\) of neighborhoods of \(y\) in \(X\) such that for every continuous function \(f:X\rightarrow \mathbb{R}\), there is \(n\in \mathbb{N}\) such that \(f\) is bounded on \(V_n\cap Y\). If \(Y\) is relatively pseudocompact in \(Y\) at each of its points, then \(Y\) is said to be \textit{pointwise relatively pseudocompact} in \(X\). The main result of the paper states: Let \(X\) be a topological space that is strongly functionally generated by the set of all its pointwise relatively pseudocompact subspaces, let \((Y,\mathcal U)\) be a uniform space and let \(\mathcal H\) be a subset of \({\mathcal C}(X,Y)\). Then the following are equivalent: (1) \(\mathcal H\) is equicontinuous. (2) Every countable subset of \(\mathcal H\) is functionally equicontinuous. (3) For each \(x\in X\), (a) every countable subset of \(\mathcal H\) is evenly equidistant at \(x\), and (b) every countable subset of \(\mathcal H\) which is uniformly discrete at \(x\) is functionally equicontinuous at \(x\). This statement implies, for example, that for every \(X\) belonging to a surprisingly wide class of topological spaces (including all quasi-\(k_{\mathbb{R}}\)-spaces), any set of continuous mappings from \(X\) to any uniform space \(Y\) which is functionally equicontinuous is in fact equicontinuous.