On images of complete topologized subsemilattices in sequential semitopological semilattices

TL;DR

The paper proves that the image of a complete topologized semilattice under a continuous homomorphism into a sequential Hausdorff semitopological semilattice is always closed.

Contribution

It establishes a new closure property for images of complete topologized semilattices under continuous homomorphisms in sequential settings.

Findings

Images are closed under continuous homomorphisms

Completeness ensures closure in sequential semitopological semilattices

Advances understanding of topological semilattice homomorphisms

Abstract

A topologized semilattice $X$ is called complete if each non-empty chain $C\subset X$ has $\inf C\in\bar C$ and $\sup C\in\bar C$. We prove that for any continuous homomorphism $h:X\to Y$ from a complete topologized semilattice $X$ to a sequential Hausdorff semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$.