On topological Rudin's lemma, well-filtered spaces and sober spaces

Abstract

Based on topological Rudin's Lemma, we investigate two new kinds of sets -Rudin sets and well-filtered determined sets in T 0 topological spaces. Using such sets, we formulate and prove some new characterizations for well-filtered spaces and sober spaces. Part of the work was inspired by Xi and Lawson's work on wellfiltered spaces. Our study also lead to a new class of spaces -strong d-spaces and some problems whose solutions will strengthen our understanding of the related structures.

Some open problems on well-filtered spaces and sober spaces

“…( [51]) For any T 0 space X, S c (X) ⊆ D c (X) ⊆ RD(X) ⊆ WD(X) ⊆ Irr c (X). In [36,Example 4.15], Liu, Li and Wu constructed a T 0 space X in which some well-filtered determined sets are not Rudin sets, and hence gave a negative answer to a queston posed by Xu and Zhao in [55]: Does RD(X) = WD(X) hold for every T 0 space X? It is not difficult to check that the space X is a WD space but not a Rudin space.…”

“…Proposition 8.5. ( [55]) If X is a d-space and ↓(↑x ∩ A) ∈ Γ(X) for all x ∈ X and A ∈ Γ(X), then X is a strong d-space. Lemma 8.6.…”

“…([37]) Is KBSob reflective in Top k ?8. Strong d-spacesIn order to more finer links between d-spaces and well-filtered spaces, the notion of strong d-spaces has been introduced in[55].Definition 8.1. A T 0 space X is called a strong d-space if for any D ∈ D(X), x ∈ X and U ∈ O(X), d∈D ↑d ∩ ↑x ⊆ U implies ↑d ∩ ↑x ⊆ U for some d ∈ D. The category of all strong d-spaces with continuous mappings is denoted by S-Top d .…”

The reflectivity of some categories of T0 spaces in domain theory

“…The strong d-spaces were introduced by Xu and Zhao [19], which lie between the classes of T 1 spaces and that of d-spaces.…”

On some open problems concerning strong d-spaces and super H-sober spaces

“…Sobriety, monotone convergence and well-filteredness are three of the most important and useful properties in non-Hausdorff topological spaces and domain theory (see [4], [5], [7] and [10]). In recent years, sober spaces, monotone convergence spaces (shortly called d-spaces), well-filtered spaces and their related structures have been introduced and investigated.…”