Concept of a topological group

Question

I have read the definition multiple times and have skimmed a few posts here, yet I still cannot fully grasp the idea of a topological group. Would anyone care to explain it with a few examples?

Answer 1

If you understand what a group is, and what a topological space is, there's no excuse: you want something that's both at the same time in a reasonable way (i.e., multiplication $G\times G \to G$ and inversion $G \to G$ should be continuous).

1. the circle $\Bbb S^1$ with scalar multiplication, because $\Bbb S^1\times \Bbb S^1 \ni (z,w)\mapsto zw^{-1} \in \Bbb S^1$ is continuous.

2. $\Bbb R$ with addition, because $\Bbb R\times \Bbb R \ni (x,y)\mapsto x-y\in \Bbb R$ is continuous.

3. any finite-dimensional vector space with addition is a topological group (similar as above)

4. matrix groups such as ${\rm GL}(n,\Bbb R)$, ${\rm SL}(n,\Bbb R)$, ${\rm O}(n)$, ${\rm SO}(n)$, ${\rm U}(n)$, ${\rm SU}(n)$, etc., since $(A,B)\mapsto AB^{-1}$ is continuous (it's a rational function of the entries of $A$ and $B$).

5. $\Bbb S^3$ with multiplication of unit quaternions.

6. any group $G$ equipped with the discrete topology becomes a topological group.

7. direct product of topological groups becomes a topological group with the product topology. E.g., $\Bbb R^n$ again, the torus $\Bbb T^n = (\Bbb S^1)^n$.

8. if $G$ is a topological group and $H\lhd G$, then $G/H$ is a topological group with the quotient topology. E.g., torus again $\Bbb T^n = \Bbb R^n/\Bbb Z^n$.

9. if $G$ is a topological group, the connected component $G^0$ of the identity is a topological subgroup which is normal (and $G/G^0$ is discrete).

10. if $\mathcal{A}$ is an associative finite-dimensional unital $\Bbb K$-algebra (where $\Bbb K$ is $\Bbb R$ or $\Bbb C$), then the group $\mathcal{G}(\mathcal{A})$ of invertible elements in $\mathcal{A}$ becomes a topological group.

I'm tired. You can probably cook up another $10$ examples after understanding those...

(Fun fact: if $G$ is a group equipped with a topology such that the multiplication $G\times G \to G$ is continuous, and $G$ is compact Hausdorff, then the inversion $G\to G$ is automatically continuous and thus $G$ is a topological group.)