I am struggling immensely with topology since the start of the course, probably due to its extremity; the explanations are either "very rough" or "very strict and rigid and hard to comprehend." Either 0 or 100, no 40 or 50. As a complete newbie, here's a question I am approaching which I managed 20% (if, at all).
The question is with respect to the following diagram:
Here is the question
For any $f: X \rightarrow S^1$, define $\bar{f}: f^* \mathbb{R} \rightarrow \mathbb{R}$ and $q:f^* \mathbb{R} \rightarrow X$, and $p: \mathbb{R} \rightarrow S^1$ specifically, $t \rightarrow e^{2 \pi i t}$. Prove that for any $x \in X$, the inverse image $q^{-1}(x) \subset f^* \mathbb{R}$ is homeomorphic to $\mathbb{Z}$ with the discrete topology. $f^*\mathbb{R}$ is the pullback of $p$ on $f$, defined $f^*\mathbb{R}= \{(x,t) \in X \times \mathbb{R} : f(x) = p(t) \in S^1\}$
1.$\,$My remark about topology's extremity; homeomorphism. Sure, I get the "vague" idea that it is a type of mapping and just as how say coffee cups can be transformed into a torus but not a sheet of paper, homeomorphic (topological) spaces are "equivalent." That is, a "homeomorphism exists" between them. But specifically, what are they? My notes gives me the example $[0,1)$ and $(0,1]$ are homeomorphic with $x \rightarrow 1-x$.
a) I thought homemorphisms are defined in "topological" spaces, $[0,1)$ and $(0,1]$ seem to me just "sets." What is the topology defined on them?
b)Sure $x \rightarrow 1-x$ is a "map" but I mean, how does this relate to the "intuitive rough explanation" of transforming a torus and a coffee cup? $[0,1)$ is just an interval, a straight line so... a straight line transformed to a straight line? Is that what's going on here?
Then the stuff gets very rigid; homeomorphisms are bijective continuous maps. Well, I can perhaps try to ram the definition down my throat without a great understanding of it, but while people told me "topology is pretty much about drawing stuff out" I can't imagine anything as to what "homeomorphism" is.
Okay, back to the question, what I figured so far
$(\mathbb{Z}, \tau _{disc})$, $\tau_{disc}$ is the power set, so essentially, this topological space has elements which are all open(right?). So, the homeomorphism between the subset $Q=\{q^{-1}(x): x \in X\}$ should have its preimage i.e. Q to be an open set itself(by definition of "continuity" of a map for topological spaces, which a homeomorphism must satisfy).
is all. I get stuck here; I mean, I don't know what $X$ is, I don't know what topology is defined on it(is there? Or do I have the liberty of coming up with whatever that's convenient?) and I have no means of deducing that $Q$ is open. Well, can I define a discrete topology on $Q$ too? That should make every element or subset open in $Q$.
And even if I manage to show "continuity" I don't know how "bijectivity" can be shown.
The issue is, I really have no means of finding a "homeomorphism" in general. How do people show these? A good guess? Intuition? Years of practice? I don't immediately see "hmm, ah, a map like this might work...or if not, probably this one..yeah that works"
Would someone please show me how this can be solved? Thank you so much in advance