Hussemoller's definition of a right G-space confuses me at a few key points:
For a topological group G, a right G-space is a space $X$ together with a map $X \times G \xrightarrow{\quad }X$. The image of $(x, s) \in X \times G$ under this map is $xs$. We assume the following axioms.
(1) For each $x \in X, s,t \in G$, the relation $x(st) = (x s)t$ holds
(2) (Not represented in my diagrams) For each $x \in X$ the relation $x e_G = x$ holds, where $e_G$ is the identity of $G$.
I'm not interested or concerned with (2), so we'll ignore it and focus the concern on (1), which confuses me and I want a clarification on. I want to more explicitly label what group operations are happening at each point. Denote the group law for $G$ as $\left( G\times G \xrightarrow{\quad \ddagger \quad } G \right)$ and our other binary operation as $\left( X \times G \xrightarrow{\quad * \quad}G \right)$.
(I)
Is this more explicit restatement of (1) correct?
For each $x, (x * s) \in X $ and $s, t \in G$, it holds that $x * (s \ddagger t) = (x * s) \ddagger t$
That is, did Husemoller use concatenation to denote two separate binary operations that are at play? The right side of the equality has a strange implication if true: it must hold that $x * s$ is also in $G$, since the group law is being applied to it.
Clearly $X$ and $G$ are topological spaces, with $G$ as a group too.
(II)
What of $X \times G$? Does it need to also be a topological space formed from the product topology, or can it simply be a product of the two sets with the binary function $*(X \times G)$ placed on it? No matter what answer you give, please explain and justify the response.
As always, thank you greatly for the help.
(I) Yes, both (distinct) binary operations are denoted by concatenation. In case one wants to be explicit about the operations, conventional notation would be something like denoting the multiplication $G\times G\rightarrow G$ by an infix "$\cdot$" and the operation $X\times G\rightarrow X$ by an infix "$.$". The correct statement of axiom (1) then becomes that $x.(s\cdot t)=(x.s).t$ for $x\in X$ and $s,t\in G$.
(II) The space $X\times G$ is equipped with the product topology. This is general topological convention: If the product of two topological spaces is introduced, it is always assumed to be equipped with the product topology unless explicitly stated otherwise. The operation is given by a map $X\times G\rightarrow X$, i.e. continuous with respect to this topology.