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The Alexandroff property and the preservation of strong uniform continuity. (English) Zbl 1252.54004
It is well known that the pointwise limit of a sequence of continuous functions need not be continuous. What precisely must be added to pointwise convergence to yield continuity of the limit was given by P. S. Alexandroff in [Einf├╝hrung in die Mengenlehre und die Theorie der reellen Funktionen. Berlin: VEB Deutscher Verlag der Wissenschaften (1956; Zbl 0070.04704)].
Recall that if \((X,d)\) and \((Y,\rho)\) are metric spaces and \((f_{n})_{n\in\mathbb{N}}\) is a sequence of functions from \(X\) to \(Y\), then the sequence \( (f_{n})_{n\in\mathbb{N}}\) has the Alexandroff property with respect to \(f\) if for each \( \varepsilon >0\) and \(n_{0}\in \mathbb{N}\) there exists a strictly increasing sequence \((n_{k})_{k\in\mathbb{N}}\) of integers such that \(n_{1}>n_{0}\) and a countable open cover \(\left\{ V_{k}:k\in\mathbb{N}\right\} \) of \(X\) such that \(\forall k\in \mathbb{N}\), \(\forall x\in V_{k}\) we have \(\rho \left( f(x),f_{n_{k}}(x)\right) <\varepsilon \).
Alexandroff showed that if \((X,d)\) and \((Y,\rho)\) are metric spaces and \((f_{n})_{n\in\mathbb{N}}\) is a sequence of functions from \(X\) to \(Y\) that pointwise converges to \(f\), then the Alexandroff property is equivalent to continuity of \(f\). In fact this result is true without metrizability in the domain.
In the paper under review some modifications of this classical property of Alexandroff are developed for nets of continuous functions that combined with a certain convergence yield continuity of the limits of continuous functions. For instance, in Theorem 4.11, which is the last theorem of this paper, the following equivalences are proved:
Let \((X,\mathcal{T)}\) be a Hausdorff space and \((Y,\mathbf{T})\) be a Hausdorff uniform space. Suppose that \(\mathcal{{B}}\) is a bornology on \( X\) with compact base, and let \(\left( f_{\lambda }\right) _{\lambda \in \Lambda }\) be a net in \(C(X,Y)\) \(\mathcal{T}_{\mathcal{{B}}}\)-convergent to \(f:X\rightarrow Y\). The following conditions are equivalent:
1.
\(f\in C(X,Y)\);
2.
For each nonempty compact subset \(C\) of \(X\), \(T_{0}\in \mathbf{T}\) and \( \lambda _{0}\in \Lambda \), there exists a finite set of indices \(\left\{ \lambda _{1},\lambda _{2},\dots ,\lambda _{n}\right\} \) such that \(\lambda _{j}\geqslant \lambda _{0}\), for \(1\leqslant j\leqslant n\), and a neighborhood \(U\) of \(C\) such that \(\forall x\in U\), \(\exists j\in \left\{ 1,2,\dots ,n\right\} \) such that \(\left( f(x),f_{\lambda _{j}}(x)\right) \in T\);
3.
\(\left( f_{\lambda }\right) _{\lambda \in \Lambda }\) has the classical Alexandroff property with respect to \(f\);
4.
\(\left( f_{\lambda }\right) _{\lambda \in \Lambda }\) is \(\mathcal{T}_{ \mathcal{{B}}}^{\square }\)-convergent to \(f\), where \(\mathcal{T}_{ \mathcal{{B}}}^{\square }\) is the topology corresponding to the uniformity in \(Y^{X}\) whose entourages basis consists of subsets of \( Y^{X}\times Y^{X}\) of the form \[ \left[ B,T\right] ^{\square }:=\left\{ (f,g):\exists U\in \mathcal{T},\;B\subset U;\left( f(x),g(x)\right) \in T,x\in U\right\} \] where \(B\) runs over the bornology \(\mathcal{{B}}\) and \(T\) runs over \(\mathbf{T}\).
In this paper the author develops the extension to the uniform space setting of the theory of strong uniform continuity and strong uniform convergence, which had been developed in the setting of metric spaces in the papers [G. Beer and S. Levi, J. Math. Anal. Appl. 350, No. 2, 568–589 (2009; Zbl 1161.54003)] and [G. Beer and S. Levi, Set-Valued Var. Anal. 18, No. 3–4, 251–275 (2010; Zbl 1236.54012)].

MSC:
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54C05 Continuous maps
54C35 Function spaces in general topology
40A30 Convergence and divergence of series and sequences of functions
54E15 Uniform structures and generalizations
54C08 Weak and generalized continuity
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