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Proximal set-open topologies on partial maps. (English) Zbl 0958.54029
Summary: Let \(X\), \(Y\) be \(T_1\) topological spaces. A partial map from \(X\) to \(Y\) is a continuous function \(f\) whose domain is a subspace \(D\) of \(X\) and whose codomain is \(Y\). Let \(P(X,Y)\) be the set of partial maps with domains in a fixed class \(\mathcal D\). In analogy with the global case, we introduce on \(P(X, Y)\), whatever be the nature of the domain class \(\mathcal D\), new function space topologies, the proximal set-open topologies, briefly PSOTs, deriving from general netwoks on \(X\) and proximity on \(Y\) by replacing inclusion with strong inclusion. The PSOTs include the already known generalized compact-open topology on partial maps with closed domains. When domains are supposed closed, the network \(\alpha\) closed and hereditarily closed and the proximity \(\delta\) on \(Y\) Efremovich, then the PSOT attached to \(\alpha\) and \(\delta\) is uniformizable iff \(\alpha\) is an Uryson family in \(X\).

MSC:
54E05 Proximity structures and generalizations
54E15 Uniform structures and generalizations
Keywords:
partial map
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