Definition:Homeomorphism

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This page is about homeomorphisms, the isomorphisms in topology. For other uses, see Definition:Isomorphism.

Not to be confused with Definition:Homomorphism.

Definition

Topological Spaces

Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces.

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


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


Metric Spaces

The same definition applies to metric spaces:


Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $f: A_1 \to A_2$ be a bijection such that:

$f$ is continuous from $M_1$ to $M_2$
$f^{-1}$ is continuous from $M_2$ to $M_1$.


Then:

$f$ is a homeomorphism
$M_1$ and $M_2$ are homeomorphic.


Manifolds

The same definition applies to manifolds:


Let $X$ and $Y$ be manifolds.

A homeomorphism of $X$ to $Y$ is a continuous bijection such that the inverse is also continuous.


Also known as

Also known as:

  • a topological equivalence, usually used when the spaces in question are metric spaces
  • an isomorphism, usually used when the spaces in question are manifolds.


Caution

Not to be confused with homomorphism.


Also see

  • Results about Homeomorphisms can be found here.