Suppose \(G\) is a topological group and \(X\) is a topological space . A continuous action of \(G\) on \(X\) is an action by homeomorphisms .

The converse is true if \(G\) has the discrete topology . [1, Proposition 3.87]

Proof

Let \(g\in G\). The map \(x\mapsto g\cdot x\) is continuous, since it is the composition of the continuous maps \(x\mapsto (g,x)\) and \((g,x)\mapsto g\cdot x\). The latter is continuous due to the definition of continuous actions. The inverse map is given by the map \(x\mapsto g^{-1}\cdot x\), which is again continuous by the same reasoning.

Consider the reference for the second part of the proof.

References Link to heading

  1. J. Lee, Introduction to Topological Manifolds. New York, NY: Springer New York, 2011. doi:10.1007/978-1-4419-7940-7