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Semi-closure and related topics in Hashimoto topologies. (English) Zbl 0714.54001
This paper establishes a connection between the two topological artifacts sketched below - semi-topology and Hashimoto’s construction. Take a space X with topology \({\mathcal T}\). Following N. Levine [Am. Math. Mon. 70, 36-41 (1963; Zbl 0113.163)], one calls a subset A of X semi-open (or more precisely, \({\mathcal T}\)-semi-open) if \(U\subset A\subset cl_{{\mathcal T}}(U)\) for some \(U\in {\mathcal T}\), and more generally, its semi-interior is the union of all smaller semi-open sets. Dually, A is semi-closed if \(X\setminus A\) is semi-open, and its semi-closure is the intersection of all larger semi-closed sets.
Hashimoto’s construction involves an ideal \({\mathcal P}\) of subsets of X as well. For \(x\in X\) and \(A\subset X\), let \(x\in D_{{\mathcal P}}(A)\) iff \(U\cap A\not\in {\mathcal P}\), for each neighbourhood U of x. Under the assumption made throughout this paper (that \(A\in {\mathcal P}\) iff \(A\cap D_{{\mathcal P}}(A)=\emptyset\) iff \(D_{\prod}(A)=\emptyset)\), the rule \(A\mapsto A\cup D_{{\mathcal P}}(A)\) defines the closure operator for a topology \({\mathcal T}({\mathcal P})\) on X. Clearly, \({\mathcal T}({\mathcal P})\) is finer than \({\mathcal T}\) and \({\mathcal T}({\mathcal P})={\mathcal T}\) if \({\mathcal P}=\{\emptyset \}\). Further, H. Hashimoto [Fundam. Math. 91, 5-10 (1976; Zbl 0357.54002)] showed that for \(A\subset X\), \(A\in {\mathcal T}({\mathcal P})\) iff \(A=B\setminus H\), for some \(H\in {\mathcal P}\) and \(B\in {\mathcal T}\), and A is closed under \({\mathcal T}({\mathcal P})\) iff \(A=B\cup H\), for some \(H\in {\mathcal P}\) and some \({\mathcal T}\)-closed B.
This paper extends Hashimoto’s construction to semi-topologies (indeed, the construction seems to make sense even more generally, for pre- topologies). The author compares the semi-topologies defined by \({\mathcal T}({\mathcal P})\) and \({\mathcal T}\), under the meagerness assumption (M) that \({\mathcal T}\cap {\mathcal P}=\{\emptyset \}\). Typical results include: Theorem 2. Assume (M). Then the subset A of X is \({\mathcal T}({\mathcal P})\)-semi-open [or \({\mathcal T}({\mathcal P})\)-semi-closed] iff \(A=B\setminus H\), for some \(H\in {\mathcal P}\) and some \({\mathcal T}\)-semi-open set B [or \(A=B\cup H\), for some \(H\in {\mathcal P}\) and some \({\mathcal T}\)-semi-closed set B]. - Theorem 7. Assume (M). If W is \({\mathcal T}({\mathcal P})\)-semi-open, then its \({\mathcal T}\)- and \({\mathcal T}({\mathcal P})\)-semi-closures coincide. - Theorem 9. Assume (M). The topological space (X,\({\mathcal T})\) is extremally disconnected iff the space (X,\({\mathcal T}({\mathcal P}))\) is extremally disconnected.
Reviewer: M.Schroder

54A05 Topological spaces and generalizations (closure spaces, etc.)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces