Left-Separating Order Types

TL;DR

This paper investigates the properties and construction of left-separating well-orderings in topological spaces, establishing existence results for various cardinalities and analyzing how these properties behave under unions and forcing extensions.

Contribution

It provides new constructions of left-separated spaces with prescribed left-separating types and analyzes their behavior under unions and forcing extensions.

Findings

Existence of spaces with specific left-separating types for various cardinals.

Union of two left-separated spaces with certain properties remains left-separated.

Forcing can change the left-separating type of a space, but some properties are absolute.

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 ${\kappa}$ is a regular cardinal, then for each ordinal ${\alpha}<{\kappa}^+$ there is a $T_2$ space $X$ with $ord_l(X)={\kappa}\cdot {\alpha}$; (2) if ${\kappa}={\lambda}^+$ and $cf({\lambda})={\lambda}>{\omega}$, then for each ordinal ${\alpha}<{\kappa}^+$ there is a 0-dimensional space $X$ with $ord_l( X)={\kappa}\cdot {\alpha}$; (3) if ${\kappa}=2^{\omega}$ or ${\kappa}=\beth_{{\beta}+1}$, where $cf({\beta})={\omega}$, then for each ordinal ${\alpha}<{\kappa}^+$ there is a locally compact, locally countable, 0-dimensional space $X$ with $ord_l( X)={\kappa}\cdot {\alpha}$. The union of two left-separated spaces is not necessarily left-separated. We show, however, that if X is a countably tight space, $X=Y\cup Z, ord_l(Y)$, $ord_l(Z)<\omega_1 \cdot \omega$, 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 $ord_l(X)=\omega_1 \cdot \omega$. 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 $ord_l(X)<\omega_1 \cdot \omega$ then X is left-separated even in $V$.