On SI2-continuous Spaces

Luo, Shunlong; Xu, Xiaoquan

doi:10.1016/j.entcs.2019.07.021

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Definition 31 ([16]2

Preliminaries1

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2023

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(3 citation statements)

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“…Definition 2.4 ( [16,20]) Let (X, τ ) be a T 0 space and x, y ∈ X. Define x ≪ r y if for every irreducible set E, y ∈ E δ implies there exists e ∈ E such that x ≤ e. We denote the set {y ∈ X : y ≪ r x} by ⇓ r x and the set {y ∈ X : x ≪ r y} by ⇑ r x.…”

Section: Preliminariesmentioning

confidence: 99%

“…(3) ⇑ r x ∈ τ for all x ∈ X. 16,20]) Let (X, τ ) be a T 0 space. A subset U ⊆ X is called weakly irreducibly open if the following conditions are satisfied:…”

Section: Definition 25 ([20]mentioning

confidence: 99%

“…Lemma 3.8 ( [16]) Let F be a directed family of nonempty finite sets in a T 0 space X. If G ≪ r x with F ⊆↑ x then there exists some F ∈ F such that F ⊆↑ G.…”

Section: Definition 31 ([16]mentioning

confidence: 99%

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$SI_2$-quasicontinuous spaces

Ruan¹,

Xu²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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In this paper, as a common generalization of $SI_{2}$-continuous spaces and $s_{2}$-quasicontinuous posets, we introduce the concepts of $SI_{2}$-quasicontinuous spaces and $\mathcal{GD}$-convergence of nets for arbitrary topological spaces by the cuts. Some characterizations of $SI_{2}$-quasicontinuity of spaces are given. The main results are: (1) a space is $SI_{2}$-quasicontinuous if and only if its weakly irreducible topology is hypercontinuous under inclusion order; (2) A $T_{0}$ space $X$ is $SI_{2}$-quasicontinuous if and only if the $\mathcal{GD}$-convergence in $X$ is topological.

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Section: Preliminariesmentioning

confidence: 99%

“…(3) ⇑ r x ∈ τ for all x ∈ X. 16,20]) Let (X, τ ) be a T 0 space. A subset U ⊆ X is called weakly irreducibly open if the following conditions are satisfied:…”

Section: Definition 25 ([20]mentioning

confidence: 99%