On Almost Normality

Abstract

Abstract.A topological space X is called almost normal if for any two disjoint closed subsets A and B of X one of which is regularly closed, there exist two open disjoint subsets U and V of X such that A C U and B C V. We will present an example of a Tychonoff almost normal space which is not normal. Almost normality is not productive. We will present some conditions to assure that the product of two spaces will be almost normal.We investigate in this paper a weaker version of normality called almost normality. We will prove that almost normality is a property which lies between mild normality and normality. We will present an example of a Tychonoff almost normal space which is not normal. We will show that almost normality is not productive and we will present some conditions to assure that the product of two spaces will be almost normal.We will denote an order pair by (x,y), the set of positive integers by N and the set of real numbers by R. A T4 space is a T\ normal space. And a Tychonoff space is a T\ completely regular space. The interior of a set A will be denoted by intyl, and the closure of a set A will be denoted by A.

$CC$-normal topological spaces

“…Thus, ω 1 × I ω1 , where I = [0, 1] is the closed unit interval with its usual metric topology and I ω1 is an uncountable product of I , is not CC -normal because it is a countably compact nonnormal space [15], but ω 1 × I ω1 is C -normal being locally compact [2]. The space ω 1 × (ω 1 + 1) is an example of an L -normal space, see [9], which is not CC -normal because it is a countably compact nonnormal space. Here is an example of a CC -normal space that is not L -normal.…”

“…Since any first countable space is Fréchet, the statements are true if X is first countable. In fact, for C -normality the statement is true if X is a k -space, see [9]. For a function that bears L-normality, the following is true: "If X is L -normal and of countable tightness and f : X −→ Y bears the L -normality of X , then f is continuous.…”

“…For a function that bears L-normality, the following is true: "If X is L -normal and of countable tightness and f : X −→ Y bears the L -normality of X , then f is continuous. ", see [9].…”

“…Any L -normal regular separable space of countable tightness is normal, see [9]. This is not true for CCnormality.…”

“…is a homeomorphism for each Lindelöf subspace A ⊆ X [9]. Any normal space is CC -normal, just by taking X = Y and f to be the identity function.…”