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I know the definitions of Lie group and topological group are different. Can you give me an example of topological group which is not a Lie group.

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Another example is the rationals under addition with the subspace topology induced from inclusion in $\mathbb R$. Since the rationals are countable, they can't be a manifold of dimension exceeding $0$, so the only possibility would be a $0$ manifold. However, the topology on the rationals is not discrete, so they do not form a $0$-manifold either.

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The Cantor set is a topological group. It is homeomorphic to $\{0,1\}^{\omega}$ in the product topology, which is a topological vector space over $\mathbb{Z}_2$. As the set is totally disconnected and not discrete it is easy to see that it cannot be a manifold.

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  • $\begingroup$ what topology of {0,1}? discrete ? $\endgroup$ Jan 23 at 5:34
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    $\begingroup$ @SamaelManasseh Yes but the product topology is not discrete. $\endgroup$ Jan 23 at 11:35
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The $p$-adic numbers are a topological group, but not a Lie group.

There are many profinite groups which are not Lie groups, for example the profinite group completion of a knot group.

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  • $\begingroup$ (But of course, the $p$-adic numbers provide an example of a $p$-adic Lie group). $\endgroup$
    – Watson
    Nov 26, 2018 at 13:30

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