Homeomorphism of unit disc onto itself interchanging two points.
117151
Conformal automorphism of unit disk that interchanges two given points
112306
Two manifolds $X$ and $Y$ with two bijective continuous functions from $X$ to $Y$ and $Y$ to $X$, but not homeomorphic
95086
Homeomorphism is bijective
61204
Mapping that takes unit circle to unit circle
34293
Homeomorphism between pairs of letters
65522
The wedge sum of two circles has fixed point property?
32765
$\mathbb{C}\mathbb{P}^1$ is homeomorphic to $S^2$
143269
Existence of area-preserving diffeomorphism between two sets in $\mathbb{R}^2$
152478
Is there a homeomorphism between the $n$-dimensional cube to the $n$-dimensional simplex that doesn't create hard corners?
61446
Fixed point of interior of closed disk
93776
Bijective mapping between $\mathbb{R}^2$ and $\mathbb{R}$?
93471
Space of homeomorphisms Homeo$(S^1)$ of $S^1$ deformation retracts onto $O(2)$
10350
Automorphisms of the complex plane
12764
$\Bbb RP^2$ as the union of a Möbius band and a disc
160
homeomorphism non-example
140713
Finding the Mobius Transformation that maps open unit disk onto itself
147350
Homeomorphism between $\prod X_i$ and the Cantor set