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Results tagged with topological-groups
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user 75
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).
4
votes
Accepted
Bohr compactification and "discretization"
This is basically immediate from the definition and Pontryagin duality. Pontryagin duality gives a contravariant involution on the category $LCAb$ of locally compact abelian groups which sends the su …
- 31.2k
15
votes
Fundamental groups of topological groups.
Here is another construction, which I think shows that it isn't surprising that such groups exist. First, we note the fact that for any abelian group $G$, there is a model of $BG$ that is an abelian …
- 31.2k
2
votes
Accepted
A Hausdorff atom in lattice of group topologies
Any compact group with no nontrivial normal closed subgroups has this property, since there can be no coarser Hausdorff topology and in any coarser non-Hausdorff topology the closure of the identity w …
- 31.2k
8
votes
Accepted
Exact sequence of groups to exact sequence of sheaves
The following is perhaps more of an extended comment than an answer. The sequence of sheaves is exact iff the quotient map $G\to H$ has a section over a neighborhood of every point (in fact, because …
- 31.2k
4
votes
Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non...
It's not hard to construct a compact connected LOTS with this property. Let $X_0$ be any countably saturated dense linear order, and let $X$ be its bounded Dedekind completion ("bounded" meaning also …
- 31.2k