I'm using this document to learn about the representation theory of compact groups. They define a topological group $G$ to be a group with a topology such that
- The multiplication and inversion operations are continuous.
- $\{e\}$ is closed in $G$, where $e$ is the identity element.
This differs from the definition that I am familiar with (which may be found on wikipedia, for example) in requiring the second condition. I believe the definition above is the same as the one in Munkres, which suggests (Munkres being a highly recommended book) there is a good reason for this requirement. My questions are as follows:
Q1. Does the second requirement somehow automatically follow by requiring the topology to be compatible with multiplication and inversion? i.e is the definition on wikipedia equivalent to the definition above?
I've thought about this for a little bit, trying to consider what the limit points of $\{e\}$ may be, but I wasn't able to convince myself this is / should be true. I do not have a counterexample, however - e.g $\mathbb R$ with addition is a topological group in the sense of the definition above.
Q2. If the answer to Q1. is no, what benefit does the second assumption provide? What do we gain by limiting the scope here?
The first place I see this assumption come into play is Theorem 2.1.9, where it is used to prove the claim that for $H \lhd G$, $G/H$ is also a topological group if and only if $H$ is closed in $G$.
The second condition is equivalent to the condition that $G$ is Hausdorff, as others have said, and it does not follow from the first. It's true in all interesting examples and is a very mild additional hypothesis, for the following reason: if $G_e = \overline{ \{ e \} }$ denotes the closure of the identity, then $G_e$ is a normal subgroup, and we have a canonical short exact sequence
$$1 \to G_e \to G \to G/G_e \to 1$$
where $G/G_e$ is the universal Hausdorffification. Morever the topology of $G_e$ is indiscrete, so it is not very interesting.