# Characterizing chain-compact and chain-finite topological semilattices

## TL;DR

This paper provides characterizations of chain-compact and chain-finite topological semilattices, linking these properties to the behavior of closed subsemilattices and continuous homomorphisms in Hausdorff $T_1$-topological semilattices.

## Contribution

It establishes new characterizations of chain-compact and chain-finite topological semilattices based on their subsemilattices and homomorphism images.

## Key findings

- A topological semilattice is chain-finite if and only if images of its closed subsemilattices under continuous homomorphisms are closed.
- A topological semilattice is chain-compact if and only if similar closure properties hold for its subsemilattices.
- Characterizations are specific to Hausdorff $T_1$-topological semilattices.

## Abstract

In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice $X$ is called chain-compact (resp. chain-finite) if each closed chain in $X$ is compact (finite). In particular, we prove that a (Hausdorff) $T_1$-topological semilattice $X$ is chain-finite (chain-compact) if and only if for any closed subsemilattice $Z\subset X$ and any continuous homomorphism $h:X\to Y$ to a (Hausdorff) $T_1$-topological semilattice $Y$ the image $h(X)$ is closed in $Y$.