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) uses::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).