Regarding function syntax of topological spaces In in the Separation Axioms section of Schaum's General Topology, the notion of point separation is mentioned briefly which is then followed by a short proposition. Attached below is the definition and proposition.

I have a concern with this proof. The author uses the notion that $\mathbb{R}$ is a Hausdorff space to show that that the space $X$ is Hausdorff. But I don't believe this is true. $\mathbb{R}$ is merely a set, not a space. It might be argued that $\mathbb{R}$ needs to be endowed with some topology, but no topology was mentioned. Now, had the author said that $\mathbb{R}$ was endowed with the natural open ball topology, then it certainly would be a Hausdorff space.
I suppose my question comes down to an issue of function notation / syntax. In lower level maths, we are brought up with the notion that when we see $f:X \to Y$, we see a function that inputs and outputs values from sets: namely, a domain and a codomain. But these are merely sets with no topology endowed upon them.
So, to make a long story short, in the proof for Proposition $10.9$, does the author need to make a mention of what topology $\mathbb{R}$ is endowed with? Because, like I said, when I see $f:X \to Y$, I see a relationship between two sets at bare minimum, but not necessarily two topological spaces.
 A: Sometimes, when we are being extra careful, we write $f:(X,\tau):\to (Y,\rho),$ where $\tau,\rho$ are topologies on $X,Y,$ respectively.
But usually, we just write $f:X\to Y.$ when there is an implied topology. In particular, if we are given that $X$ is a topological space, there is an implied single topology on $X.$
Unless the conditions are quite specific to say it is not true, when discussing continuity involving $\mathbb R^n,$ we assume the usual topology, derived from the Euclidean metric.
Essentially “$X$ is a topological space” can be treated as

$X$ is an alias for a pair $(X_0,\tau),$   where $X_0$ is a set and $\tau$ is a topology on $X_0.$

The only time this risks confusion is when we talk about multiple topologies on the same set. $(X,\tau_1)$ and $(X,\tau_2)$ are different spaces, with the same underlying set.
It’s a pretty common shorthand. In group theory, a homomorphism is a function having certain properties, and they can be written: $$f:(G,1_G,\cdot_G)\to (H,1_H,\cdot_H)$$ but we usually just shorthand it to $f:G\to H,$ with the implied structures of the groups.
A: The author talks about the "class $C(X,\Bbb R)$ of real-valued continuous functions": this has some implicit assumptions: we mean the functions from $X$ (a given space with a pregiven topology) to $\Bbb R$ (which always has, unless another topology is explicitly mentioned, the natural topology induced from its metric (or equivalently order), that are continuous (w.r.t. those two topologies; we always need $2$ topologies to talk about continuity).
So the proof is perfectly fine one you talke that in account. Always look at context: maths is not a purely formal game where everything is completely spelled out in notation and if things are well-known or standard in a certain context they're no longer mentioned (or expressed in notation).
