Left-Separating Order Types

Abstract

A well ordering < of a topological space X is "left-separating" if $\{x'\in X: x'< x\}$ is closed in X for any x in X. A space is "left-separated" if it has a left-separating well-ordering. The left-separating type, $ord_l(X)$, 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 $T_2$ space $X$ with $ord_l(X)=κ\cdot α$; (2) if $κ=λ^+$ and $cf(λ)=λ>ω$, then for each ordinal $α<κ^+$ there is a 0-dimensional space $X$ with $ord_l( X)=κ\cdot α$; (3) if $κ=2^ω$ or $κ=\beth_{β+1}$, where $cf(β)=ω$, then for each ordinal $α<κ^+$ there is a locally compact, locally countable, 0-dimensional space $X$ with $ord_l( X)=κ\cdot α$. The union of two left-separated spaces is not necessarily left-separated. We show, however, that if X is a countably tight space, $X=Y\cup Z, ord_l(Y)$, $ord_l(Z)<ω_1 \cdot ω$, then $X$ is also left-separated and $ord_l(X)\le ord_l(Y)+ord_l(Z)$. 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 in the generic extension $V^Q$ we have $ord_l(X)=ω_1 \cdot ω$. However, if $X$ is a topological space and $Q$ is a c.c.c poset such that in in the generic extension $V^Q$ we have $ord_l(X)<ω_1 \cdot ω$ then X is left-separated even in $V$.