If $G$ is a paratopological group, is true that if $K$ is a compact subset of $G$ and $F$ is a closed subset of $G$ then $KF$ is closed in $G$?

Question

A group $G$ endowed with a topology is called a paratopological group if the multiplication $G×G\to G$ is continuous.

It is know that that in a topological group $G$ if $K$ is a compact subset of $G$ and $F$ is a closed subset of $G$ then $KF$ and $FK$ are closed in $G$.

The above is true if $G$ is only a paratopological group?

I think that $KF$ is not closed but I don't a counterexample for this!

Thanks for your help!

Answer 1

This is false for paratopological groups.

Let $\tau = \{(a,\infty) : a\in\mathbb R\}\cup\{\mathbb{R}\}$. Then $(\mathbb R, \tau, +)$ is a paratopological abelian group. Let $K$ be the compact set $[0,1)$ and let $F$ be any closed set $(-\infty,a]$.

We have then $$K+F=F+K=(-\infty,a+1),$$ which is not closed in $\tau$.