[[Group theory MOC]]
# Topological group

A **topological group** $(G, \cdot, \mathcal{T})$ is a [[group]] $(G, \cdot)$ that is also a [[topological space]] $(G, \mathcal{T})$,
such that the group operation and taking the inverse is continuous, #m/def/group 
i.e. a topological group is a [[Group object]] in [[Category of topological spaces]].

## Properties

- Every topological group is a [[Homogenous space]].
- [[Connected subgroup|The connected component of the identity is a normal subgroup]].

## Related concepts

- A [[Lie group]] is a topological group on which one can perform calculus. 

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#state/tidy | #lang/en | #SemBr