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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

##### MSC:
 54A05 Topological spaces and generalizations (closure spaces, etc.) 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
##### Citations:
Zbl 0113.163; Zbl 0357.54002