A note on k-c-semistratifiable spaces and strong β-spaces

Title:

A note on k-c-semistratifiable spaces and strong β-spaces

Creator:

Wang, Li-Xia and Peng, Liang-Xue

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:31279ff0-921b-4ef5-93fa-163336d58233
uuid:31279ff0-921b-4ef5-93fa-163336d58233
doi:10.21136/MB.2011.141650

Subject:

c-semistratifiable space, k-c-semistratifiable space, submesocompact space, g function, and strong β-space

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are Gδ-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T2-space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x ∈ X, {x} = ∩ {g(x, n): n ∈ ℕ} and g(x, n + 1) ⊆ g(x, n) for each n ∈ N. (2) If a sequence {xn}n∈N of X converges to a point x ∈ X and yn ∈ g(xn, n) for each n ∈ N, then for any convergent subsequence {ynk }k∈N of {yn}n∈N we have that {ynk }k∈N converges to x. By the above characterization, we show that if X is a submesocompact locally k-csemistratifiable space, then X is a k-c-semistratifible space, and the countable product of k-c-semistratifiable spaces is a k-c-semistratifiable space. If X = ∪ {Int(Xn): n ∈ N} and Xn is a closed k-c-semistratifiable space for each n, then X is a k-c-semistratifiable space. In the last part of this note, we show that if X = ∪ {Xn : n ∈ N} and Xn is a closed strong β-space for each n ∈ ℕ, then X is a strong β-space.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

287-299

Source:

Mathematica bohemica | 2011 Volume:136 | Number:3

Harvested from:

CDK

Metadata only:

false