Results about κ-normality

Results about the Alexandroff duplicate space

“…It is clear from the definitions that every quasi-normal space is mildly normal. Mild normality is not preserved by the Alexandroff Duplicate space [7]. Quasi-normality is not preserved by the Alexandroff Duplicate space and here is a counterexample.…”

“…A finite intersection of closed domains is called π-closed [9]. A topological space X is called mildly normal [7] if for any two disjoint closed domains A and B of X, there exist two open disjoint subsets U and V of X such that A ⊆ U and B ⊆ V . A topological space X is called quasi-normal [9] if for any two disjoint π-closed subsets A and B of X, there exist two open disjoint subsets U and V of X such that A ⊆ U and B ⊆ V .…”

$CC$-normal topological spaces

“…A space X is called mildly normal [14] (also called κ -normal [12]) if for any two disjoint closed domains A and B of X there exist two disjoint open subsets U and V of X such that A ⊆ U and B ⊆ V , see also [6,7]. A space X is called almost normal [13] if for any two disjoint closed subsets A and B of X , one of which is a closed domain, there exist two disjoint open subsets U and V of X such that A ⊆ U and B ⊆ V , see also [8].…”

On Almost Normality

“…The converse is not always true. The space u\ x ui\ + 1 is mildly normal, see [2] and [3], but not almost normal because the closed subset A = u>i x {wi} is disjoint from the regularly closed subset B = {(A, A) : a < LO\} and they cannot be separated by two disjoint open subsets, see [1].…”

“…The product space M x P is not almost normal. It is still unknown if the Michael product M x P is mildly normal or not, [3]. Also, whether the Dowker theorem version for almost normality is true or not, which is the following problem: If X is almost normal countably paracompact and Y is compact second countable, is then X x Y almost normal?…”