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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:
 5.4e+06 Proximity structures and generalizations 5.4e+16 Uniform structures and generalizations
partial map
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