Abstract.This paper shows that semitopological classes are subsemilattices of the lattice of topologies, and gives a new characterization for the finest topology in the semitopological class.Introduction.In [4] Levine defined a set, A, to be semiopen if there is some open set U so that i/<= A<= c(U), where c( ) denotes closure in the topological space. In [1] it was shown that if (X, r) is a topological space, there is a finest topology [we shall call it F(t)] so that the semiopen sets are the same as for t. If X is a set of points, let T(X) be the lattice of topologies on X. If t e TiX), let [t] denote the equivalence class of all topologies which have the same semiopen sets as r.[t] is called a semitopological class of topologies on X. The object of this note is to show that if t e TiX), [t] is a subsemilattice of TiX) with respect to the usual join operation on topologies, and to give a new characterization for F(t).
Abstract. The strongly Hausdorff and Urysohn properties of a topological space are shown to be semitopological properties.I. Introduction. Levine [6] defined a set A to be semiopen in a topological space if and only if there is an open set U so that U c A c c(U) where c( ) denotes the closure in the topological space.In [2], semiclosed sets, semi-interior, and semiclosure were defined in a manner analogous to the corresponding concepts of closed, interior, and closure. Then in [3] a property of topological spaces was defined to be a semitopological property if it was preserved by semihomeomorphisms (bijections so that the images of semiopen sets are semiopen and inverses of semiopen sets are semiopen). In [3] the first category, Hausdorff, separable, and connected properties of topological spaces were shown to be semitopological properties.The new separation axioms (semi-T0, semi-Tx, and semi-T2) defined by Maheshwari and Prasad [7] are also semitopological properties, and Hamlett showed [5] that the property of a topological space being a Baire space is semitopological. In this note two additional separation axioms closely related to the Hausdorff separation axiom are shown to be semitopological
Abstract. The strongly Hausdorff and Urysohn properties of a topological space are shown to be semitopological properties.I. Introduction. Levine [6] defined a set A to be semiopen in a topological space if and only if there is an open set U so that U c A c c(U) where c( ) denotes the closure in the topological space.In [2], semiclosed sets, semi-interior, and semiclosure were defined in a manner analogous to the corresponding concepts of closed, interior, and closure. Then in [3] a property of topological spaces was defined to be a semitopological property if it was preserved by semihomeomorphisms (bijections so that the images of semiopen sets are semiopen and inverses of semiopen sets are semiopen). In [3] the first category, Hausdorff, separable, and connected properties of topological spaces were shown to be semitopological properties.The new separation axioms (semi-T0, semi-Tx, and semi-T2) defined by Maheshwari and Prasad [7] are also semitopological properties, and Hamlett showed [5] that the property of a topological space being a Baire space is semitopological. In this note two additional separation axioms closely related to the Hausdorff separation axiom are shown to be semitopological
Abstract.This paper shows that semitopological classes are subsemilattices of the lattice of topologies, and gives a new characterization for the finest topology in the semitopological class.Introduction.In [4] Levine defined a set, A, to be semiopen if there is some open set U so that i/<= A<= c(U), where c( ) denotes closure in the topological space. In [1] it was shown that if (X, r) is a topological space, there is a finest topology [we shall call it F(t)] so that the semiopen sets are the same as for t. If X is a set of points, let T(X) be the lattice of topologies on X. If t e TiX), let [t] denote the equivalence class of all topologies which have the same semiopen sets as r.[t] is called a semitopological class of topologies on X. The object of this note is to show that if t e TiX), [t] is a subsemilattice of TiX) with respect to the usual join operation on topologies, and to give a new characterization for F(t).
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