Almost uniform convergence (Q1314914)

A net \((f_ n)\) of functions on a topological space \(X\) to a uniform space \((Y,{\mathcal U})\) converges almost uniformly to a function \(f\) at \(x_ 0\in X\) if for each \(U\in{\mathcal U}\) there exists a neighborhood \(W\) of \(x_ 0\) such that eventually \((f_ n(x),f(x))\in U\) for each \(x\in W\). Clearly almost uniform convergence lies between uniform convergence and quasi-uniform convergence. The author shows that if \(X\) is Hausdorff, then almost uniform convergence coincides with quasi-uniform convergence if and only if \(X\) is discrete. A paracompact Tykhonov space \(X\) is compact if and only if uniform convergence and almost convergence coincide. A Tykhonov space \(X\) is locally compact if and only if uniform convergence on compacta equals almost uniform convergence. The author also provides suitable examples. (Reviewer's remarks: In Corollary 2-3 read paracompact for compact. For related work see [\textit{A. Di Concilio} and the reviewer, Monatsh. Math. 103, 93-102 (1987; Zbl 0607.54013)] and [the reviewer, Czech. Math. J. 37(112), 608-612 (1987; Zbl 0647.54003)].