# Completeness and absolute $H$-closedness of topological semilattices

## TL;DR

This paper investigates conditions under which continuous homomorphisms between topological semilattices have closed images, focusing on completeness and absolute $H$-closedness.

## Contribution

It introduces new completeness conditions that ensure the closedness of images of continuous homomorphisms in topological semilattices.

## Key findings

- Identifies specific completeness conditions guaranteeing closed images.
- Establishes criteria for absolute $H$-closedness in topological semilattices.
- Provides theoretical results linking completeness and closedness properties.

## Abstract

We find (completeness type) conditions on topological semilattices $X,Y$ guaranteeing that each continuous homomorphism $h:X\to Y$ has closed image $h(X)$ in $Y$.