Faculty Publications

Title

Left-separating order types

Document Type

Article

Keywords

Left separated spaces, Left separating order type, Preservation

Journal/Book/Conference Title

Topology and its Applications

Volume

283

Abstract

A well ordering ≺ of a topological space X is left-separating if {x′∈X:x′≺x} is closed in X for any x∈X. A space is left-separated if it has a left-separating well-ordering. The left-separating type ordℓ(X) of a left-separated space X is the minimum of the order types of the left-separating well-orderings of X. We prove that (1) if κ is a regular cardinal, then for each ordinal α<κ+ there is a T2 space X with ordℓ(X)=κ⋅α; (2) if κ=λ+ and cf(λ)=λ>ω, then for each ordinal α<κ+ there is a 0-dimensional T2 space X with ordℓ(X)=κ⋅α; (3) if κ=2ω or κ=ℶβ+1, where cf(β)=ω, then for each ordinal α<κ+ there is a locally compact, locally countable, 0-dimensional space X with ordℓ(X)=κ⋅α. The union of two left-separated spaces is not necessarily left-separated. We show, however, that if X is a countably tight space, X=Y∪Z, ordℓ(Y)<ω1⋅ω and ordℓ(Z)<ω1⋅ω, then X is also left-separated and ordℓ(X)≤ordℓ(Y)+ordℓ(Z). We prove that it is consistent that there is a first countable, 0-dimensional space X, which is not left-separated, but there is a c.c.c poset Q such that VQ⊨ordℓ(X)=ω1⋅ω. However, if X is a topological space and Q is a c.c.c poset such that VQ⊨ordℓ(X)<ω1⋅ω, then X is left-separated even in V.

Original Publication Date

9-1-2020

DOI of published version

10.1016/j.topol.2020.107338

Repository

UNI ScholarWorks, Rod Library, University of Northern Iowa

Language

en

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