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Results tagged with topological-groups
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user 4354
A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).
20
votes
Accepted
Is a left topological group which is a manifold a topological group?
Here is a counter-example with $G$ homeomorphic to $\mathbb R^2$. Let $f:\mathbb R\to\mathbb R$ be a discontinous additive homomorphism (constructed using a Hamel basis of $\mathbb R$ over $\mathbb Q$ …
- 32.4k
44
votes
Accepted
Why are free groups residually finite?
Here is a direct proof for free groups.
Let $x_1,\dots,x_m$ be the generators of our group. Consider a word $x_{i_n}^{e_n}\dots x_{i_2}^{e_2}x_{i_1}^{e_1}$ where $e_i\in\{\pm 1\}$ and there are no ca …
- 32.4k
5
votes
Sequential topological vector spaces
The space of all functions $\mathbb R\to\mathbb R$ with the topology of pointwise convergence is obviously sequential but not first countable.
Indeed, for every $x\in\mathbb R$, the set $U_x:=\{f:|f( …
- 32.4k