Left-Separating Order Types

Abstract: A well ordering < of a topological space X is "left-separating" if ${x&#39;\in X: x&#39;&lt; x}$ 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 ${\alpha}&lt;{\kappa}<sup>+$ there is a $T_2$ space $X$ with $ord_l(X)={\kappa}\cdot {\alpha}$; (2) if ${\kappa}={\lambda}<sup>+$ and $cf({\lambda})={\lambda}&gt;{\omega}$, then for each ordinal ${\alpha}&lt;{\kappa}<sup>+$ there is a 0-dimensional space $X$ with $ord_l( X)={\kappa}\cdot {\alpha}$; (3) if ${\kappa}=2<sup>{\omega}$ or ${\kappa}=\beth_{{\beta}+1}$, where $cf({\beta})={\omega}$, then for each ordinal ${\alpha}&lt;{\kappa}<sup>+$ 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, $X=Y\cup Z, ord_l(Y)$, $ord_l(Z)&lt;\omega_1 \cdot \omega$, 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 $V<sup>Q$ 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 $V<sup>Q$ we have $ord_l(X)&lt;\omega_1 \cdot \omega$ then X is left-separated even in $V$.