Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?

The question is inspired by and related to this one.

Surely, globally non-topologizable infinite groups do exist.

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    $\begingroup$ I don't know, but the class of groups admitting no such topology is closed under taking subgroups, because if $H\subset G$ we can extend the topology on $H$ so that $G\smallsetminus H$ is open and discrete. In particular, a group as in your question will be torsion, and the question boils down to finitely generated groups. $\endgroup$ – YCor Aug 23 '14 at 14:44

Having slept on it, I realise that the answer is No (even if we omit the condition on the inverse). The first example of infinite countable non-topologizable group due to Olshanskii (1980) is actually locally non-topologizable.

Indeed, Olshanskii's group $G$ has exponent $p^2$ and the cyclic centre of order $p$. In addition, $g^p\ne1$ for any non-central element of the group. So, if we have a topology on $G$ and the multiplication is continuous at $1$, then the function $f\colon x\mapsto x^p$ is also continuous at $1$. But $f$ takes only finitely many values ($p$ values). Thus, this continuous at $1$ function to a discrete (finite) set must be constant on a neighbourhood of $1$ and, hence, this neighbourhood must be finite (because the $f(x)=1$ only for central $x$).

Details of Olshanskii's construction can be found if his book.

  • $\begingroup$ which page‌‌‌‌‌‌‌‌‌‌? $\endgroup$ – H. Khas Aug 26 '14 at 0:09
  • $\begingroup$ I have a Russian edition. This is Chapter 10, Section 31, Subsection 3. $\endgroup$ – Anton Klyachko Aug 26 '14 at 10:25
  • $\begingroup$ So this link may be better: books.google.com/… btw, I have not found it yet. $\endgroup$ – H. Khas Aug 27 '14 at 21:34
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    $\begingroup$ It is on page 339 of English edition. Unfortunately, Google preview rejects to show this page to me (but I may read the table of contents). $\endgroup$ – Anton Klyachko Aug 27 '14 at 21:58

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