# A metrizable semitopological semilattice with non-closed partial order

## TL;DR

This paper constructs a metrizable semitopological semilattice with a non-closed dense partial order and explores conditions for topologies with prescribed convergent sequences.

## Contribution

It introduces a novel example of a metrizable semitopological semilattice with a non-closed partial order and characterizes conditions for specific convergent sequences in topologies.

## Key findings

- Existence of a metrizable semitopological semilattice with non-closed dense partial order
- Necessary and sufficient conditions for topologies with prescribed convergent sequences
- Characterization of topologies on sets, acts, semigroups, or semilattices

## Abstract

We construct a metrizable semitopological semilattice $X$ whose partial order $P=\{(x,y)\in X\times X:xy=x\}$ is a non-closed dense subset of $X\times X$. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.