# topological group

A topological group is a group (algebra) that is also a topological space.

So, the group operation \(\cdot : G \times G \rightarrow G, (x,y) \rightarrow xy\) and the inversion operation \(G \rightarrow G, x \rightarrow x^-1\) are continuous

The group operation is continuous if for any neighborhood \(W\) that contains \(xy\), there is a neighborhood \(U\) that contains \(x\) and a neighborhood \(V\) that contains \(y\) such that \(UV = \{uv \mid u\in U, v\in V\}\) such that \(UV\subseteq W\).

Recall epsilon-delta proofs. For every \(\epsilon\) neighborhood around \(f(x)\), you can always find a \(\delta\) neighborhood around \(x\) such that the image of that neighborhood is completely within \(\epsilon\) of \(f(x)\)