I was reading the book Introduction to the topological groups by Taqdir Husain, he gives a proof for this theorem, but I had a problem understanding some part. The proof goes like this
It remains to show that the inversion mapping $x\to x^{-1}$ is continuous. So let $U$ be a open neighborhood of $e$. We wish to show that there exists a compact neighborhood $C$ of $e$ such that $C^{-1}\subseteq U$. Suppose this is not possible, i.e. $C^{-1}\setminus U\neq\varnothing$ for all compact neighborhood $C$ of $e$. Now define the following family
$$
\mathscr{F}=\{C^{-1}\setminus U:C\text{ is a compact neighborhood of }e\}
$$
In the last Lemma proves that $C^{-1}$ is compact for $C$ compact. So the family $\mathscr{F}$ is a family of compact sets. Then claims that $\mathscr{F}$ has the finite intersection property.
This last thing it's not clear to me, because I don't see why for example two elements of $\mathscr{F}$ say $A^{-1}\setminus U$ and $B^{-1}\setminus U$ must have elements in common.
The book actually says that the proof is due to Robert Ellis on A note on the continuity of the inverse. And he says that before his proof this was an open problem. I found no more proofs of this theorem, so It would be really helpful if somebody could help me.
By the way, there is another similar question in the forum but it requires the compactness Prove a compact Hausdorff space with a group structure is a topological group
Best Answer
Here is a proof that pairwise intersections are nonempty. The finite intersection property in general follows by induction.
Recall that the standing assumption is that for every compact neighborhood $C$ of $\{1\}$, $C^{-1}\setminus U$ is nonempty. Consider two compact neighborhoods $A, B$ of $1$. Take $C=A\cap B$. This is again a compact neighborhood of $1$. Then $$ (A^{-1}\setminus U)\cap (B^{-1}\setminus U)= (A^{-1} \cap B^{-1}) \setminus U = (A\cap B)^{-1} \setminus U= C^{-1}\setminus U\ne \emptyset, $$ according to the standing assumption. qed
Consider also reading the proof that every locally compact Hausdorff paratopological group is a topological group given by Alex Ravsky here, in the unnumbered Proposition in his answer.