A group \(G\) is a topological group if it is endowed with a topology such that the inversion map \(i(g)=g^{-1}\) and the product map \(p(g_1,g_2)=g_1\cdot g_2\) are continuous (the latter regarding the product topology ).
Examples Link to heading
- \((\mathbb{R}, +)\) or \((\mathbb{C}, +)\)
- \(\mathbb{R}^n=\mathbb{R}\times \cdots \times \mathbb{R}\)
- \((\mathbb{R}\setminus \{0\},\cdot )\) or \((\mathbb{C}\setminus \{0\},\cdot )\)
- \(\mathbb{S}^1 \subseteq \mathbb{C}\)
- \(\mathbb{R}^{>0}\subseteq \mathbb{R}\setminus \{0\}\)
- discrete group
- \((\GL(n, \mathbb{R}), \cdot )\) or \((\GL(n, \mathbb{C}), \cdot)\)
Remarks