# Category:Definitions/Homeomorphisms

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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:

- $\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \left[{U}\right] \in \tau_\beta$

### 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 9 pages are in this category, out of 9 total.

### H

- Definition:Homeomorphic Topological Spaces
- Definition:Homeomorphism (Topological Spaces)
- Definition:Homeomorphism/Manifolds
- Definition:Homeomorphism/Topological Spaces
- Definition:Homeomorphism/Topological Spaces/Definition 1
- Definition:Homeomorphism/Topological Spaces/Definition 2
- Definition:Homeomorphism/Topological Spaces/Definition 3
- Definition:Homeomorphism/Topological Spaces/Definition 4