Left-separating order types

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 typeordℓ(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 andordℓ(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 thatVQ⊨ordℓ(X)=ω1⋅ω. However, if X is a topological space and Q is a c.c.c poset such thatVQ⊨ordℓ(X)<ω1⋅ω, then X is left-separated even in V.