Topological group
= Topological group
{wiki}
A <topological group> is a <group> $G$ equipped with a <topology> for which both the <group operation> and taking an <inverse element> are <continuous functions>:
$$
m:G\times G\longrightarrow G,\quad (x,y)\longmapsto xy,
\qquad \iota:G\longrightarrow G,\quad x\longmapsto x^{-1}.
$$
Here $G\times G$ has the <product topology>: <continuity> of multiplication concerns both inputs together. Thus nearby motions compose to nearby motions, and undoing a motion also varies continuously. The definition alone does not impose <compactness>; we state the <Hausdorff space> assumption explicitly when it is needed.
The additive <group> $(\mathbb R,+)$ of <real numbers>, with its usual <topology>, is a <topological group>: $(x,y)\mapsto x+y$ and $x\mapsto-x$ are <continuous functions>. Its <identity element> is $0$. The <circle group> instead uses <multiplication> of <complex numbers> on the <unit circle>, with the <subspace topology> from the <complex plane>:
$$
\mathbb T=\{z\in\mathbb C:|z|=1\},\qquad
e^{i\theta}e^{i\phi}=e^{i(\theta+\phi)},\qquad
(e^{i\theta})^{-1}=e^{-i\theta}.
$$
Its <identity element> is $1$, and its <inverse element> is the <complex conjugate>. Multiplication on the <complex plane> and <complex conjugation> are <continuous functions>, so their restrictions verify the required <continuity>. Angles add, but points multiply: a quarter-turn followed by another quarter-turn gives $i\cdot i=-1$, and the next two quarter-turns return to $1$. Ordinary <addition> would fail to stay on the <unit circle>, since $1+1=2$.
For any fixed $a\in G$, the translations $x\mapsto ax$ and $x\mapsto xa$ are <homeomorphisms>, with inverses given by translation by $a^{-1}$. Consequently the <neighbourhoods> of the <identity element> determine the entire <topology>. This is why local information near one motion can control a whole family of symmetries. See https://arxiv.org/html/2006.10956v1#S1.SS1[Tornier, Haar Measures, section 1.1] for these basic properties of <topological groups>.