I am learning about topological groups, and keep getting confused when I try to be precise about exactly what the structure is. I have also seen that most textbooks omit a specific topology on a group altogether, simply stating that it is a topological group.

For example, in Munkres' 'Topology', there are supplementary exercises asking you to prove that $(\mathbb{R}, +)$ is a topological group, but no topology is specified.

Do we always assume that the topology on a group is the most "obvious" one? Do certain groups have a property where the group operation map and inversion map are always continuous, regardless of the topology?

I hope my question is clear enough. Thanks

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    $\begingroup$ In most situations the topology is clear from the context, as you say for the standard topology on $\mathbb R$. If $G$ is a group and you pick any topology on $G$, then you should not expect to obtain a topological group. For example a topological group is a homogeneous topological space, i.e. its homeomorphism group acts transitively. $\endgroup$ Dec 2, 2020 at 22:26
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    $\begingroup$ Okay, thanks for clarifying. $\endgroup$
    – Natasha
    Dec 2, 2020 at 22:39
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    $\begingroup$ Only if your group $G$ is trivial, every topology on $G$ makes $G$ a topological group. $\endgroup$ Dec 2, 2020 at 23:56
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    $\begingroup$ @JulianQuast Why not an official answer? $\endgroup$
    – Paul Frost
    Dec 3, 2020 at 10:34
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    $\begingroup$ If $G$ is a topological group and $\mathrm{Homeo}(G)$ is the group of homeomorphisms $G \to G$, then the action of $\mathrm{Homeo}(G)$ on $G$ by $\varphi \cdot g = \varphi(g)$ is transitive, because if $g,h \in G$ then the left translation $\Lambda_{gh^{-1}}(x) = gh^{-1}x$ is a homeomorphism, that maps $h$ to $g$. This is the definition of "homogeneous space" and means roughly, that the space looks the same at each point. $\endgroup$ Dec 10, 2020 at 5:08


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