A reverse viewpoint on the upper semicontinuity of multivalued maps. (English) Zbl 1289.54069

With \(\mathcal {A}(X)\) the set of all nonempty subsets of \(X\), \(\mathcal {A}(t)\) the upper semifinite topology on \(\mathcal {A}(X)\) for a given topology \(\tau \) on \(X\), \(\phi \: Y\rightarrow X\) a multimap that associates with each point \(y\) of the topological space \(Y\) a nonempty subset \(\phi (y)\) of the topological space \(X\), the notion of upper semicontinuity of the multimap \(\phi \) is defined as follows : for every open \(U\subset X\), the set \(\phi ^{-1}(U)\) is open in \(Y\), where \(\phi ^{-1}(U)= \{y\in Y\: \phi (y)\subset U\}\); \(\phi \) can as well be treated as a single map from \(Y\) into the power set \(\mathcal {A}(X)\) and the usual notation is \(\widetilde {\phi }\: Y\rightarrow \mathcal {A}(X)\). It is well known that the multimap \(\phi \: Y\rightarrow (X,\tau )\) is upper semi continuous iff the (single-valued) map \(\widetilde {\phi }\: Y\rightarrow (\mathcal {A}(X),\mathcal {A}(\tau ))\) is continuous.
On \(\mathcal {A}(X)\), a topology \(\Gamma =\{\emptyset ,\{X\},\mathcal {A}(X)\}\) is defined and there is no topology \(\tau \) on \(X\) such that \(\Gamma =\mathcal {A}(\tau )\).
The function \(\sigma \: \mathcal {A}(\mathcal {A}(X))\rightarrow \mathcal {A}(X)\) is defined by \(\sigma (\xi )=\cup \{E\: E\in \xi \}\); this function can be extended to a function from \(\mathcal {A}(\mathcal {A}(X))\cup \{\emptyset \} \) into \(\mathcal {A}(X)\cup \{\emptyset \}\), defining \(\sigma (\emptyset ) = \emptyset \). With \(\sigma \), termed as the reversal function, because \(\sigma (\mathcal {A}(Z))=Z\) for all \(Z \subseteq X\), the image \(\sigma (\Gamma )\) is considered which forms a subbasis for a topology denoted as \(\mathcal {R}(\Gamma )\) and termed as the reversal topology from \(\Gamma \).
The paper actually tries to describe the constraints under which the upper semi-continuity of a multimap \(\phi \: Y\rightarrow (X,\mathcal {R}(\Gamma ))\) is equivalent to the continuity of the single map \(\widetilde {\phi }\: Y\rightarrow (\mathcal {A}(X),\Gamma )\).


54C60 Set-valued maps in general topology
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
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