Generic left-separated spaces and calibers

Left-Separating Order Types

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…