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Associated topologies of generalized $$\alpha$$-closed sets and $$\alpha$$- generalized closed sets. (English) Zbl 0821.54002
O. Njåstad [Pac. J. Math. 15, 961-970 (1965; Zbl 0137.419)] defined a subset $$A$$ of a topological space $$(X, \tau)$$ to be $$\alpha$$- open if $$A \subset \text{int(cl(int} A))$$, and a subset $$B$$ of $$X$$ to be $$\alpha$$-closed if $$X - B$$ is $$\alpha$$-open. The collection $$\tau^ \alpha$$ of all $$\alpha$$-open subsets of $$(X, \tau)$$ is a topology on $$X$$, and $$\tau \subset \tau^ \alpha$$.
The authors of the paper under review introduce two classes of generalized $$\alpha$$-closed subsets, with the following definitions. A subset $$B$$ of $$(X, \tau)$$ is defined to be $$\alpha$$-generalized closed $$[\alpha^{**}$$-generalized closed] in $$(X, \tau)$$ if $$\tau^ \alpha \text{cl} B \subset U$$ $$[\tau^ \alpha \text{cl} B \subset \text{int(cl} U)]$$ whenever $$B \subset U$$ and $$U$$ is open in $$(X, \tau)$$. The paper considers the basic properties of these two classes of subsets, and their associated topologies.

##### MSC:
 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54E55 Bitopologies
##### Keywords:
generalized closed sets
##### Citations:
Zbl 0137.41903; Zbl 0137.419