A note on semitopological properties

Abstract

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

g*b-Homeomorphisms and Contra-g*b-continuous Maps in Topological Spaces

“…Biswas [1], Mashour [9], Sundaram [13], Crossley and Hildebrand [2], and Devi [3] Several topologists have generalized homeomorphisms in topological spaces. Biswas [1], Crossley and Hilde-brand [2], Sundaram [13] have introduced and studied semi-homeomorphism and some what homeomorphism and generalized homeomorphism and gc-homeomorphism respectively.…”

“…Biswas [1], Crossley and Hilde-brand [2], Sundaram [13] have introduced and studied semi-homeomorphism and some what homeomorphism and generalized homeomorphism and gc-homeomorphism respectively.…”