A metrizable semitopological semilattice with non-closed partial order

Abstract

We construct a metrizable Lawson semitopological semilattice X whose partial order ≤ X = {(x, y) ∈ X × X : xy = x} is not closed in X × X. This resolves a problem posed earlier by the authors.1991 Mathematics Subject Classification. 54A20, 06A12, 22A26, 37B05.

The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices

“…Remark 1. The completeness of X is essential in Theorem 5: by [6], there exists a metrizable semitopological semilattice X whose partial order is not closed in X × X, and for every x ∈ X the upper set ↑x is finite.…”

A metrizable semitopological semilattice with non-closed partial order

“…This proposition does not generalize to semitopological semilattices as shown by the following example constructed by the authors in [1]. Example 1.…”

“…In this paper we shall construct an example of a metrizable Lawson semitopological semilattice with non-closed partial order, thus answering a problem posed by the authors in [1].…”

“…At the end of the paper [1] the authors asked whether there exsists a Lawson Hausdorff semitopological semilattice X with non-closed partial order. In this paper we shall give an affirmative answer to this question.…”

“…x, y ∈ X 1 . In this case y ∈ X1 \ V α (x) implies y↾[1, α) = x↾[1, α) or y([α, λ)) ⊂ {0, 2}. If y↾[1, α) = x↾[1, α), then V α (x) ∩ V β (y) ⊂ V α (x) ∩ V α (y) = ∅.…”