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Does there exist a group $G$ such that $G$ has no topology on it such that $G$ is a topological group apart from the (in)discrete topology (or other such trivalish topologies)? I am asking as interested in the general methods that one construct a topological group from a group.

I am quite interested in how the problem changes if $G$ is infinite or finite.

Meow
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  • The indiscrete topology also always makes $G$ into a topological group. And to answer your question, take the trivial group. – Najib Idrissi Jan 12 '15 at 19:57
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    @NajibIdrissi, usually one includes $T_1$ness into the definition of topological groups. – Mariano Suárez-Álvarez Jan 12 '15 at 20:05
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    From the other side of the coin: if a topological space $G$ can be made into a topological group, the first homotopy group $\pi_1(G,e)$ is abelian. Hence, if $X$ is a topological space with $\pi_1(X,x_0)$ non abelian, it cannot be made into a topological group compatible with this topology. – Pedro Jan 12 '15 at 20:06
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    @ MO: http://mathoverflow.net/questions/165783/existence-of-infinite-groups-that-are-too-reluctant-to-be-topological/ – Grigory M Jan 12 '15 at 20:18
  • (Re: «I am quite interested in how the problem changes if G is infinite or finite.») That's easy to answer: changes from completely trivial (finite) to very hard problem that stayed open for almost 40 years (infinite). – Grigory M Jan 12 '15 at 20:26
  • @NajibIdrissi Okay, edited. – Meow Jan 13 '15 at 00:05
  • @PedroTamaroff Thanks, these are the sort of things I really appreciate in trying to work out how to intuit the topologisation of groups. – Meow Jan 13 '15 at 00:06
  • Fortunately one does not always assume $T_0/T_1/T_2$ in the definition of topological groups (they are equivalent in this case). Otherwise it's embarrassing when non-Hausdorff groups naturally appear, e.g. as quotients by non-closed normal subgroups. – YCor May 01 '19 at 21:45

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You can find this discussed very nicely in the section about Markov's problems in these notes by Dikran Dikranjan.

In particular, there do exists groups which have no non-discrete compatible topologies; the notes include the examples of Adian groups,which are countable, and noncountable examples due to Shelah. A nice result is that a group with infinite center has some non-discrete Hausdorff topology.

Mariano Suárez-Álvarez
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