Main Theorems/Techniques for proving Homeomorphism?

General Question: what are the most common Theorems/Methods used to prove Homeomorphism?

I encountered:
- find the map explicitly
- use the Compact-to-Hausdorff Lemma
- find cts maps $f$ and $g$ s.t. $f\circ g=g\circ f= i$ where $i$ is the identity map.

Can anyone deepen/correct/enlarge my list?

Solution 1:

Suppose that $f: X \rightarrow Y$ is a bijective continuous map. In general, from this it does not follow that $f^{-1}$ is continuous, so $f$ is not necessarily a homeomorphism.

However, there are some special cases where $f: X \rightarrow Y$ bijective and continuous implies that $f$ is a homeomorphism. In any introductory course on functional analysis, one comes across to the open mapping theorem. As an easy corollary, you get the bounded inverse theorem:

Bounded Inverse Theorem: Let $X$ and $Y$ be Banach spaces. Suppose that $T: X \rightarrow Y$ is a bijective, linear and continuous map. Then $T$ is a homeomorphism.

This is a very special case, here linearity and the algebraic structure of $X$ and $Y$ is vital for the above theorem to hold. There is also the following theorem, proven in the 1960's.

Theorem: Any two separable infinite-dimensional Banach spaces are homeomorphic.

Solution 2:

Probably the most useful method is to give a full classification of a certain class of spaces. A classification of spaces satisfying some property $P$ amounts to giving a set of criteria for such spaces (hopefully a finite set of criteria) such that two spaces with property $P$ are homeomorphic if and only if both spaces coincide for all criteria. For instance

$2$-dimensional manifolds ($3$-dimensional manifolds are nearly done). These are classified by their genus (or Euler characteristic, or homology, or fundamental group) and their orientability. If you want to consider manifolds with boundary and marked-points too, it amounts to also counting boundary circles and marked points.
Perfect totally-disconnected compact metric spaces. The classification of these spaces is actually really easy - there's only one space up to homeomorphism with these properties and it's the Cantor set. A common technique to show that a space is homeomorphic to the Cantor set is to simply check that it satisfies all of the above properties.
Finite space. This one is an easy one. You just need to check cardinality and then all permutations of the elements and the action of this on the topology. I suppose this amounts to your first point of 'finding a map explicitly', but I mention it because we can do this in an algorithmic way, so finite spaces are particularly suited to this method - even if the computational complexity of such a check isn't feasible to be done in practice.
Knot complements in the sphere. Because a knot complement $S^3\setminus K$ is homeomorphic to $S^3\setminus K'$ if and only if $K$ and $K'$ are isotopic knots, this amounts to then classifying knots. This is a hard problem but (as far as I'm aware) is expected to be solvable and already has a vast array of machinery that has been developed for this purpose.
Denjoy continua. These are spaces which are embedded in the torus $T^2$ which, without getting technical, can be thought of as removing a disk from the torus and then 'stretching' that hole around the torus along some geodesic with irrational slope (we take the limit of this stretching process). If the 'stretching line' of a Denjoy continuum $D$ has slope $\alpha$ with continued fraction expansion $[a_0:a_1,\ldots]$ and another Denjoy continuum $D'$ has stretching line with slope $\beta$ and continued fraction expansion $[b_0:b_1,\ldots]$, then $D$ is homeomorphic to $D'$ if and only if there exist $k$ and $k'$ such that $(a_k,a_{k+1},a_{k+2},\ldots)=(b_{k'},b_{k'+1},b_{k'+2},\ldots)$.
Solenoids. These are inverse limits of maps from the $n$-torus to itself. They can be fully classified by just considering the degrees of the maps appearing in the inverse limit (disregarding order) up to a finite deletion of those maps appearing in the limit. This is another result attributed to Fokkink in his PhD thesis, as with Denjoy continua.