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Bornological convergence and shields. (English) Zbl 1275.54003

Starting with an ideal \(\mathcal{S}\), that is, a family \(\mathcal{S}\) of nonempty subsets of a metric space \(\langle X,d\rangle\) that is closed with respect to taking finite unions and taking nonempty subsets and considering the \(\epsilon\)-enlargement of a set \(E\subseteq X\) (for \(\epsilon>0\)), defined by \(E^\epsilon:=\bigcup_{a\in E}S_\epsilon(a)\), where \(S_\epsilon(a)\) is the open \(\epsilon\)-ball with center \(a\), a net \(\langle A_i\rangle_{i\in I}\) of subsets of \(X\) is called \(\mathcal{S}^-\)-convergent to a subset \(A\) of \(X\) if for each \(S\in\mathcal{S}\) and each \(\epsilon>0\), we have eventually \(A\cap S\subseteq A_i^\epsilon\). Dually, the net is called \(\mathcal S^+\)-convergent to \(A\) if for each \(S\in\mathcal S\) and each \(\epsilon>0\), we have eventually \(A_i\cap S\subseteq A^\epsilon\). All of these convergences are now collectively called bornological convergences, as in the most important applications, the ideal \(\mathcal S\) is also a cover, making it a bornology. In the present paper, the authors discuss necessary and sufficient conditions for these bornological convergences to be topological, i.e. that there is a hyperspace topology such that the convergence is compatible with the topology. They consider the hyperspaces \(\mathcal P(X)\), \(\mathcal P_0(X)\), \(\mathcal C(X)\), \(\mathcal C_0(X)\); \(\mathcal P(X)\) \big(\(\mathcal C(X)\)\big) is the collection of all subsets (closed subsets) of \(X\) whereas, \(\mathcal P_0(X)\) \big(\(\mathcal C_0(X)\)\big) is the collection of all nonempty subsets (nonempty closed subsets) of \(X\).
First of all, the authors find a number of necessary and sufficient conditions on an ideal \(\mathcal S\) so that \(\mathcal S^-\)-convergence is topological for the aforesaid hyperspaces. These conditions involve total boundedness and stability of an ideal under small enlargements. A set \(A\in\mathcal P_0(X)\) is called “totally bounded with respect to \(\mathcal S\)” provided \(\forall\,\epsilon>0\), \(\exists\,S\in\mathcal S\) with \(S\subseteq A\subseteq S^\epsilon\). An ideal \(\mathcal S\) is called “stable under small enlargements” if \(\forall\,S\in\mathcal S\), \(\exists\,\epsilon>0\) with \(S^\epsilon\in\mathcal S\). Using this stability of the ideal \(\mathcal S\) under small enlargements, the authors obtain some necessary and sufficient conditions for \(\mathcal S^+\)-convergence to be topological on \(\mathcal P(X)\) and \(\mathcal P_0(X)\). To study similar situations on \(\mathcal C(X)\) and \(\mathcal C_0(X)\), the authors first discuss the concept of “shield” and using this concept the authors describe thoroughly the necessary and sufficient conditions for \(\mathcal S^-\)-convergence, \(\mathcal S^+\)-convergence and \(\mathcal S\)-convergence (which is a conglomeration of both \(\mathcal S^+\) and \(\mathcal S^-\)-convergence) of nets of subsets of \(X\) to be topological on \(\mathcal C(X)\) and \(\mathcal C_0(X)\). A nonempty subset \(S\) of \(X\) is said to be “shielded from closed sets” by a superset \(S_1\) provided whenever \(A\in\mathcal C_0(X)\) and \(A\cap S_1=\emptyset\), then \(A\) is far from \(S\) i.e. \(A^\epsilon\cap S=\emptyset\) for some \(\epsilon>0\). Here \(S_1\) is called a “shield for \(S\)”.
The authors complete their study with a closer look at the notion of shield. For any two sets \(T,S\in\mathcal P_0(X)\), the authors define the set \(K_T(S):=cl(S)\cap cl(X\smallsetminus T)\) and using this set they prove the upper semicontinuity of some special multifunctions, when \(T\) is a shield for \(S\). The paper is concluded with some nice results regarding shields when instead of a metric space \(X\) some metrizable space \(X\) is considered with various compatible metrics. It is also noted that \(\mathcal S\)-convergence is compatible with a metrizable topology on \(\mathcal C(X)\) if and only if \(\mathcal S\) is a bornology closed under small enlargements that has a countable base; this result then culminates into the remarkable fact: there must exist an equivalent remetrization of a metrizable space \(X\) such that \(\mathcal S\)-convergence becomes Attouch-Wets convergence in this new metric.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54B20 Hyperspaces in general topology
46A17 Bornologies and related structures; Mackey convergence, etc.
54C60 Set-valued maps in general topology
54E35 Metric spaces, metrizability
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