Learning topology has been hard. I just cannot see how some people can come up with complex functions that link one space to another, in a homeomorphic sense.
The explanations are always "Well if you define this kind of function it satisfies continuity, bijectivity so there" Which is irrefutable simply because that person is (mathematically) right.
I've always been wanting the "how did that function pop in your head."
What motivated you to think of such a function in the first place? A wild and lucky guess? Well that's not mathematical or logical, it can't be.
My conclusion is that these people "know" some of the well-known, most seen, most common homeomorphisms by heart and tweaks it here and there to build a new one to adapt. So I guess that's why they cannot answer to my "but how did you find that function?" I also often see "$X$ and $Y$ are clearly homeomorphic so..." in a discussion, where, to those practiced minds it is indeed crystal clear but to me, I cannot say "hmm, yes because well...uh huh, here's an example of a homeomorphism between $X,Y$ so yes, go on..."
I'm like, "but what's the homeomoprhism? I don't know where to start from scratch! What is the train of thought??"
Words like obvious bijection, or natural way of mapping only makes sense once one has seen these specifically and gotten comfortable with it. It's hard to explain why it's obvious because it is obvious but that's an issue when the less experienced is not comfortable with the notion of just obvious.
This is the problem with a lot of my professors too. They've been doing it for decades. Most may be born with topolo-genes (as I call it) who gets most of the abstract concepts in the speed of light. Perhaps that's why they don't understand why students like me say "it's obvious to you and you indeed sound convincing and I cannot argue back, but I simply can't feel comfortable"
In a nuthsell, at this level, things just seem to come out of nowhere(but still, correct and rigorous).
So my response now is to at least familiarize myself with these common homeomorphisms. Just remember them for now, use them like an idiot and gradually gain comfort.
So my request here is this: Would people provide me as much examples as possible(proof not necessary but appreciated of course) of most basic, common homeomorphisms between spaces?
For instance, the homeomorphism between a disk in a plane and a square? or something in $\mathbb{R}^n$?
It would be great if I can get "a homeomorphism between $X$ and $Y$ is given by $f$" for the spaces you think are standard and know by heart. Even without thinking maybe.
I am thinking this should help me build on intuition and topolo-sense (take it as a topology version of spider-sense) which I lack severely.
