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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Marc van Leeuwen
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1Another example, which is connected but not locally connected, is https://en.wikipedia.org/wiki/Solenoid_(mathematics) – Matt Samuel Dec 26 '15 at 04:27
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3Does anyone have an example of a smooth manifold that can be made into a topological group but not a Lie group? – Ishan Banerjee Dec 26 '15 at 06:45
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2@IshanBanerjee, https://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem. – Martín-Blas Pérez Pinilla Dec 26 '15 at 18:13
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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.
Cheerful Parsnip
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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.
Matt Samuel
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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.
Max
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(But of course, the $p$-adic numbers provide an example of a $p$-adic Lie group). – Watson Nov 26 '18 at 13:30