Homeomorphism

Definition

A function $f:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ between two topological spaces is a homeomorphism if it satisfies the following conditions:

• $f$ is Bijective
• $f$ and its inverse function $f^{-1}$ are continuous ($f,f^{-1}\in C^0$)
  • Equivalent: $U\in {\mathcal{T}}_X\iff f(U)\in {\mathcal{T}}_Y$

Homeomorphic

Consider the two topological spaces $(X,{\mathcal{T}}_X),(Y,{\mathcal{T}}_Y)$. If there exists a homeomorphism between $X$ and $Y$, then $X$ is homeomorphic to $Y$