Left-separating order types

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 αω, then for each ordinal α