# Category:Definitions/Homeomorphisms

This category contains definitions related to Homeomorphisms.
Related results can be found in Category:Homeomorphisms.

Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces.

Let $f: T_\alpha \to T_\beta$ be a bijection.

### Definition 1

$f$ is a homeomorphism if and only if both $f$ and $f^{-1}$ are continuous.

### Definition 2

$f$ is a homeomorphism if and only if:

$\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \sqbrk U \in \tau_\beta$

### Definition 3

$f$ is a homeomorphism if and only if $f$ is both an open mapping and a continuous mapping.

### Definition 4

$f$ is a homeomorphism if and only if $f$ is both a closed mapping and a continuous mapping.

## Pages in category "Definitions/Homeomorphisms"

The following 10 pages are in this category, out of 10 total.