Embeddings of Hausdorff semitopological semilattices in hyperspaces

Abstract

UDC 515.12, 512.56

Beer and Ok showed that a locally compact and order-connected Hausdorff topological semilattice $X$ can be embedded in the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion. First, we show that this result can be generalized to the case of Hausdorff semitopological semilattices. Second, we also prove that a locally compact Hausdorff semitopological semilattice is a topological poset. Third, we conclude that a mapping defined by $x\rightarrow{\downarrow}x$ from a locally compact lower semiclosed space $(X, \tau, \leq)$ to the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion is continuous if and only if $X$ is upper open and $\leq $ is closed in $X\times X.$ Finally, we introduce the concept of $H$-closedness for a $T_{0}$ topological space.