A note on convergence of sequences of functions (Q1985650)

The authors introduce and investigate the following types of covergence of sequences of functions between metric spaces \(X\) and \(Y\). (1) \textit{Semi-\(\alpha\) convergence.} The sequence \((f_n)_n\) semi-\(\alpha\) converges to \(f\) at a point \(x\in X\) iff \(f_n(x)\to f(x)\), and for every \(\varepsilon>0\) there is \(\delta>0\) such that \[(\forall n\in\mathbb{N})\; (\exists m\ge n)\; (\forall t\in B(x,\delta))\; \varrho_Y(f_m(t),f(x))<\varepsilon\] (2) \textit{Semi exhaustiveness.} The sequence \((f_n)_n\) is semi-exhaustive at \(x\in X\) iff for every \(\varepsilon>0\) there is \(\delta>0\) such that \[(\forall n\in\mathbb{N})\; (\exists m\ge n)\; (\forall t\in B(x,\delta))\; \varrho_Y(f_m(t),f_m(t))<\varepsilon\] (3) \textit{Semi uniform convergence.} The sequence \((f_n)_n\) is semi-uniformly convergent to \(f:X\to Y\) at \(x\in X\) iff \(f_n(x)\to f(x)\), and for every \(\varepsilon>0\) there is \(\delta>0\) such that \[(\forall n\in\mathbb{N})\; (\exists m\ge n)\; (\forall t\in B(x,\delta))\; \varrho_Y(f_m(t),f(t))<\varepsilon\] The sequence \((f_n)_n\) is semi-\(\alpha\) convergent (respectively, semi-exhaustive, or semi-uniformly convergent) iff it has this property at each \(x\in X\). In the final part of the article the authors study relationships between \(\alpha\) convergence, semi-\(\alpha\) convergence and equal and uniformly equal convergence.