I'm reading the following paper https://arxiv.org/pdf/1804.10306.pdf in particular proposition (2.1) and trying to understand some facts I'm missing about compact groups theory,
If I have a compact group $G$ (a group that is also a compact topological space) and a set $X$ and define the left action of $G$ on $X$ as
\begin{align} &G \times X \rightarrow X\\ &(g,x) \mapsto g \cdot x \end{align}
Does this mapping turn out to be continuous with respect to the product topology on $G \times X$? And if yes, why? What are the general assumptions on $X$? From what I've understood it is assumed to be a vector space which provides a linear representation of the group $G$, i.e. there exists a homomorphism (taking $X = \mathbb{R}^n$ for example)
$$\varphi: G \rightarrow GL_n(\mathbb{R})$$
But I would like to understand more about when $G$ is a topological space how can I show that the action is continuous with respect to the product topology.
If you have an action of a group on a set and the group happens to also be a topological group and the set happens to have a topology, there is no reason to expect the action to be continuous. However, normally when people talk about an action of a topological group on a topological space, they are referring to an action which is continuous. This is similar to how people often just say "map" to mean "continuous map" when talking about maps between two topological spaces.