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Hashimoto topologies and quasi-continuous maps. (English) Zbl 0768.54017
Let \((X,{\mathcal T})\) be a topological space and let \(\mathcal P\) be an ideal of subsets of \(X\). Then the collection \({\mathcal B}({\mathcal P})=\{U-H: U\in{\mathcal T}, H\in{\mathcal P}\}\) is a basis for a topology \({\mathcal T}({\mathcal P})\) on \(X\), which contains \(\mathcal T\). This topology is called by the author Hashimoto topology [see H. Hashimoto, Fundam. Math. 91, 5-10 (1976; Zbl 0357.54002)]. For a detailed study of such types of topologies see also the paper of D. Janković and T. R. Hamlett [Am. Math. Mon. 97, No. 4, 295-310 (1990; Zbl 0723.54005)].
In this paper the author under some assumptions on \(\mathcal P\) (for example \({\mathcal T}\cap{\mathcal P}=\emptyset\) or assuming that ideal \({\mathcal P}\) is a \(\sigma\)-ideal) finds some properties of an upper (lower) quasi- continuous multifunction \(F: (X,{\mathcal T}({\mathcal P}))\to (Y,{\mathcal T}_ 1)\).
54C60 Set-valued maps in general topology
54C08 Weak and generalized continuity
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