4
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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
5
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.