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**On the topology generated by preopen sets.**
*(English)*
Zbl 0646.54002

Back in 1965, O. Njastad has shown [Pac. J. Math. 15, 961-970 (1965; Zbl 0137.419)] that the class \(\tau_{\alpha}\) of \(\alpha\)-sets of a topological space (X,\(\tau)\) is a topology on X. An \(\alpha\)-set is a set contained in the interior of the closure of its interior.

The topology \(\tau_{\alpha}\) consists of exactly those sets A for with \(A\cap B\in SO(X)\) for all \(B\in SO(X)\). Here SO(X) is the class of semi- open sets, and a set A is semi-open if it is contained in the closure of its interior. Replacing here SO(X) by PO(X) [see preceding review] the author defines \(\gamma\)-sets. He then proceeds to show that the class \(\tau_{\gamma}\) of \(\gamma\)-sets is a topology on X such that \(\tau_{\alpha}\subset \tau_{\gamma}\). \(\{\) There is a misprint in 2.2 Theorem: \(\tau_{\alpha}\) is a topology... should be replaced by \(\tau_{\gamma}\) is a topology...!\(\}\) This topology is investigated, with particular emphasis on the properties of its closure operator.

The topology \(\tau_{\alpha}\) consists of exactly those sets A for with \(A\cap B\in SO(X)\) for all \(B\in SO(X)\). Here SO(X) is the class of semi- open sets, and a set A is semi-open if it is contained in the closure of its interior. Replacing here SO(X) by PO(X) [see preceding review] the author defines \(\gamma\)-sets. He then proceeds to show that the class \(\tau_{\gamma}\) of \(\gamma\)-sets is a topology on X such that \(\tau_{\alpha}\subset \tau_{\gamma}\). \(\{\) There is a misprint in 2.2 Theorem: \(\tau_{\alpha}\) is a topology... should be replaced by \(\tau_{\gamma}\) is a topology...!\(\}\) This topology is investigated, with particular emphasis on the properties of its closure operator.

Reviewer: S.Gacsályi

### MSC:

54A05 | Topological spaces and generalizations (closure spaces, etc.) |

54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |