semi topological and quasi topological group

Question

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Why is every infinite group $G$ with co-finite topology $\tau$, a semi-topological and a quasi-topological group, but is not a para-topological group?

Answer 1

To verify that $G$ is a quasitopological group it suffices to remark that co-finite topology is invariant with respect to the shifts and the inversion. The group $G$ is not paratopological because the multiplication is discontinuous at the unit. Indeed, pick any not-unit element $g\in G$. Then $U=G\setminus\{g\}$ is a neighborhood of the unit, but there is no neighborhood $V$ of the unit such that $VV\subset U$. Indeed, the set $gV^{-1}$ is co-finite, so it intersects a cofinite set $V$. Pick any $h\in gV^{-1}\cap V$. Then $g\in hV\subset VV$, so $VV\not\subset U$.