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

Hereditary properties of semi-separation axioms and their applications

“…For instance, consider the simple K-path (P := (p i ) i∈ [1,5] Z , κ 2 P ) in Figure 1(a). Then (P, κ 2 P ) is a semi-T 1 -space.…”

“…Next, whereas the singleton {q}, q ∈ {p 1 , p 3 , p 5 }, is open in (P, κ 2 P ) and it is obviously semi-closed in (P, κ 2 P ). (2) Under the hypothesis, to prove that the given simple K-path P := (p i ) [1,5] Z is not a semi-T 1 -space, we may assume that p 1 belongs to (Z n ) e . Let us now take the point p 2 ∈ P. Then the singleton {p 2 } is not semi-closed so that P is not a semi-T 1 -space.…”

“…Let us now take the point p 2 ∈ P. Then the singleton {p 2 } is not semi-closed so that P is not a semi-T 1 -space. For instance, consider the simple K-path (P := (p i ) i∈ [1,5] Z , κ 2 P ) in Figure 1(b). Then the singleton {p 2 } is not semi-closed in (P, κ 2 P ) because…”

“…The separation axioms semi-T i , where i = 0, 1 2 , 1, 2, etc (see [3,5,25,27]), are obtained from the definitions of the usual separation axioms T i after replacing open sets by semi-open ones. Hence the axiom T i obviously implies the axiom semi-T i [7] but the converse does not hold.…”

On Strong Form of Irresolute Functions

“…In 1972, Crossley and Hildebrand [2] introduced the notion of irresoluteness. Various types of irresolute functions have been introduced over the course of years.…”