Results about the Alexandroff duplicate space

Abstract

In this paper, we present some new results about the Alexandroff Duplicate Space. We prove that if a space X has the property P , then its Alexandroff Duplicate space A(X) may not have P , where P is one of the following properties: extremally disconnected, weakly extremally disconnected, quasi-normal, pseudocompact. We prove that if X is α-normal, epinormal, or has property wD, then so is A(X). We prove almost normality is preserved by A(X) under special conditions. 2010 MSC: 54F65; 54D15; 54G20.

C -Paracompactness and C 2 -paracompactness

“…open cover for E that has no finite subcover, which contradicts the compactness of E . Thus, (1) holds. Now, assume E is compact and satisfies (1).…”

“…Similarly, we can show that if E ∩ B i is infinite, then c i ∈ E . Now, assume E is compact and satisfies (1) and (2). Suppose that K 2 is infinite but a ̸ ∈ E .…”

“…Indeed, here is a coarser Hausdorff compact topology on ω 1 . Define a topology V on ω 1 generated by the following neighborhood system: Each nonzero element β < ω 1 will have the same open neighborhood as in the usual ordered topology in ω 1 . Each open neighborhood of 0 is of the form U = (β, ω 1 ) ∪ {0} where β < ω 1 .…”

“…Recall that a topology τ on a nonempty set X is said to be minimal Hausdorff if (X, τ ) is Hausdorff and there is no Hausdorff topology on X strictly coarser than τ ; see [4,5]. In the next theorem we will use the following theorem: "A minimal Hausdorff space is compact if and only if it is completely Hausdorff (T 2 1 2 )" [14, 1.4] (2) ⇒ (3) Since any T 2 locally compact space is Tychonoff, by the minimality, X is compact [14, 1.4].…”

Epinormality

“…A(X) with this topology is called the Alexandroff Duplicate of X. In [2], the following theorem was proved.…”