The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices

TL;DR

This paper proves that complete subsemilattices in functionally Hausdorff semitopological semilattices are closed, ensuring the closedness of their images under continuous homomorphisms, with results extendable to weaker separation axioms.

Contribution

It establishes the closedness of complete subsemilattices and their images in functionally Hausdorff semitopological semilattices, extending to the weaker $oldsymbol{ ext{T}_{2 ext{ extdelta}}}$ separation axiom.

Findings

Complete subsemilattices are closed in the ambient semilattice.

Images of complete semilattices under continuous homomorphisms are closed.

Results hold under the weaker $oldsymbol{ ext{T}_{2 ext{ extdelta}}}$ separation axiom.

Abstract

A topologized semilattice $X$ is complete if each non-empty chain $C\subset X$ has $\inf C\in\bar C$ and $\sup C\in\bar C$. It is proved that for any complete subsemilattice $X$ of a functionally Hausdorff semitopological semilattice $Y$ the partial order $P=\{(x,y)\in X\times X:xy=x\}$ of $X$ is closed in $Y\times Y$ and hence $X$ is closed in $Y$. This implies that for any continuous homomorphism $h:X\to Y$ from a compete topologized semilattice $X$ to a functionally Hausdorff semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$. The functional Hausdorffness of $Y$ in these two results can be replaced by the weaker separation axiom $\vec T_{2\delta}$, defined in this paper.