How do you proceed when constructing a homeomorphism?

Question

I have maybe quite stupid and quite methodological question: how do you proceed when constructing homeomorphism?

I am asking, because I came across many exercises, proofs etc., where some steps would be to construct a homeomorphism with certain properties. My questions are (regardless of further properties):

1. How do you start when constructing a homeomorphism? How to correctly think about this? Do you start with something canonical, like mostly identity or constant mapping and then look whether it is surjective and injective and continuous with cont. inverse?

2. How do you check bijection? Just 1) arbitrary $y$ has an $x$ s.t. $f(x) = y$ and 2) arbitrary $f(x) = f(y)$ imply $x = y$? Or other methods?

I am sorry if this question is too stupid, but I also apprectiate theoretical answers, however examples would be nice too.

I will possibly summarize all the comments to a comprehensive answer that can help anyone in the future.

Thank you.

Answer 1

Building on SV-97's comment, I often find it's a good idea to do it in smaller steps: If you're looking for a homeomorphism $f : X \to Y$, you instead try to find homeomorphisms $f_1 : X\to X_1$, $f_2 : X_1 \to X_2$ until $f_k : X_{k-1} \to Y$ for some $k \in \mathbb{N}$, and then you simply define $f := f_k \circ \ldots \circ f_1$.

A nice application of this approach is showing that $\mathbb{R}^2 \cong \mathbb{R}^2 \setminus ( [0,\infty) \times \{0\})$. Here you'd do the following: Define the canonical homeomorphism $\varphi : \mathbb{R}^2 \to \mathbb{C}$. Then define $f: \mathbb{C}\to \mathbb{C}$ by $z \mapsto \sqrt{z}$ and $g : \mathbb{R}^2 \to \mathbb{R}^2$ by $(x,y) \mapsto (x,\log(y))$. Then, $$ F := g \circ \varphi^{-1} \circ f \circ \varphi $$ is the desired homeomorphism. Intuitively, you're first "closing a fan" from $360^\circ$ to $180^\circ$ (so you get the upper half plane) and then you're stretching downward to infinity (so you get the entire plane).

Of course, this only works if you know that a homeomorphism can be constructed in the first place, but I still hope this was helpful!

Answer 2

The important step is usually not constructing the homeomorphism in the sense you seem to be talking about here; rather, it's determining whether or not such a homeomorphism exists in the first place. Usually, the construction will be obvious if you have a good intuition for why the spaces are homeomorphic.