Timeline for Can you give me an example of a totally disconnected subgroup of a topological group that is not a topological group? [closed]
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| May 15, 2020 at 16:36 | history | closed |
YCor ARG Alex M. LeechLattice Stefan Kohl♦ |
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| May 11, 2020 at 6:08 | review | Close votes | |||
| May 15, 2020 at 16:36 | |||||
| May 11, 2020 at 2:02 | answer | added | Todd Trimble | timeline score: 3 | |
| May 11, 2020 at 1:37 | comment | added | Elizeu França | I don't have in mind an example. But, the continuity of $p:G\times G\rightarrow G$ not implies the continuity of $p:H\times H\rightarrow H$. | |
| May 11, 2020 at 1:32 | comment | added | Todd Trimble | Elizeu, that doesn't answer my question. If you have an example, please say what it is. | |
| May 11, 2020 at 1:12 | comment | added | Todd Trimble | You say in general not. I don't think I follow. What example do you have in mind? (FWIW, Wikipedia seems to agree with me, en.wikipedia.org/wiki/Topological_group#Properties, paragraph 5, but it seems to me a straightforward argument.) | |
| May 11, 2020 at 1:06 | comment | added | Elizeu França | Yes, the subgroup $H$ has the subspace topology. The group operations $p:H\times H \rightarrow H$ and $i:H\rightarrow H$, $p(g,h)=gh, i(g)=g^{-1}$ are continuous? This is true when $H$ is closed subgroup. In general not. Therefore, a subgroup is not, in general, a topological group with the subspace topology. | |
| May 11, 2020 at 0:36 | comment | added | Todd Trimble | Where you say more generally, I assume you mean a subgroup equipped with the subspace topology? Because it seems to me the answer is yes, and it has nothing to do with total disconnectedness: a subgroup of a topological group, when given the subspace topology, is a topological group. Mostly it comes down to checking that if $H \hookrightarrow G$ is a subspace, then the product topology on $H \times H$ coincides with the subspace topology on $H \times H$ inherited from the product space $G \times G$. | |
| May 11, 2020 at 0:01 | history | asked | Elizeu França | CC BY-SA 4.0 |