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Probabilistic approach spaces. (English) Zbl 1463.54017

The author studies a generalisation of Lowen’s approach spaces. Given a point \(p\in S\) and a subset \(A\subseteq S\), a distribution function \(\delta(p,A)\) is assigned: its value at \(x\), \(\delta(p,A)(x)\), is interpreted as the probability that the distance of \(p\) from \(A\) is less than \(x\). The author introduces suitable axioms and shows that the resulting category is isomorphic to the category of left-continuous probabilistic topological convergence spaces and, as a consequence, is a topological category. The category of Lowen’s approach spaces is isomorphic to a simultaneously bireflexive and bicoreflexive subcategory; he also shows that the category of quasi-metric spaces is isomorphic to a bicoreflexive subcategory of the category of probabilistic aproach spaces.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54E70 Probabilistic metric spaces
54E99 Topological spaces with richer structures
54B30 Categorical methods in general topology
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