Books and sources concerning $G$-spaces

Question

A $G$-space is (generally) a topological space $X$ equipped with a continuous action by a topological group $G$. I mean generally because, I've never studied before $G$-spaces and after I read a couple of papers involving them, I've been looking for the exact definition on the internet but there are different conditions like:

• $G$ is just a group;
• the action is not necessarily continuous;
• $X$ is Hausdorff (I think this is because the author of the document was working with Hausdorff spaces).

So my request is regarding for books concerning the definition of a $G$-space in the most generally sense and, why not, the precedence of its nature.

Answer 1

See "Topological Groups And Related Structures" by Arhangel'skii and Tkachenko (the definition is right after Example $10.2.6$).

See also "Topological Groups" by Dikranjan, Prodanov and Stoyanov (after Lemma $7.6.6$).