Center not is closed in T0 quasitopological group

Statement

It is possible to have a T0 quasitopological group $G$ such that the center $Z(G)$ is not a closed subgroup of $G$.

Related facts

• Center is closed in T0 topological group

Facts used

1. Infinite group with cofinite topology is a quasitopological group

Proof

Let $G$ be an infinite group whose center is a proper infinite subgroup. Equip $G$ with the cofinite topology (so the proper closed subsets are precisely the finite subsets). By Fact (1), $G$ thereby becomes a quasitopological group. By the definition of the cofinite topology, it is $T_{0}$. however, by construction, the center is not closed.

An example of such a group $G$ might be unitriangular matrix group:UT(3,Z).