S-paracompactness and S2-paracompactness

Abstract

A topological space X is an S-paracompact if there exists a bijective function f from X onto a paracompact space Y such that for every separable subspace A of X the restriction map f | A from A onto f (A) is a homeomorphism. Moreover, if Y is Hausdorff, then X is called S 2-paracompact. We investigate these two properties.

Pre-Existing Traits Associated with Covid-19 Illness Severity

“…40,41 While there remains scant data currently available to explain sex differences for Covid-19, male sex bias was also observed for SARS and MERS. 42,43 Similar to the findings in our study, this increase risk was not attributable to a greater prevalence of smoking among men. Notably, prior murine studies have also demonstrated male versus female bias in susceptibility to SARS-CoV infection, which may be related to the effects of sex-specific steroids and X-linked gene activity on modulation of both the innate and adaptive immune response to viral infection.…”

Results about S2-paracompactness

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We present new results regarding S2-paracompactness, that we established in [1], and its relation with other properties such as S-normality, epinormality and L-paracompactness.

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“…Hence, {[0, α] : α < ω 1 } is an open cover of A that has no countable subcover, which implies that A is not Lindelöf. Since ω 1 satisfies the condition in Theorem 2.6, then ω 1 is L 2 -paracompact because it is S 2 -paracompact, (see [1]). A family {A s } s∈S of subsets of a space X is called point-finite if for each x ∈ X, the set {s ∈ S : x ∈ A s } is finite, (see [3]).…”

“…The closure and the interior of the subset A of a topological space X will be denoted respectively by A and int(A). Throughout this paper, a T 1 normal space is called T 4 and a T 1 completely regular space is called Tychonoff space (T 3 1 2 ). In the definitions of compactness, countable compactness, paracompactness, and local compactness we do not assume T 2 .…”

“…The countable complement topology defined on R, (R, CC) (see [9, Example 20]), is an example of a space that is S 2 -paracompact but not L 2 -paracompact. It is S 2 -paracompact because it is P -space, (see [1]), but not L 2 -paracompact because it is Lindelöf and not paracompact space. In fact, it is not even L-paracompact.…”