Definition:Homeomorphism/Metric Spaces

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This page is about topological equivalence of metric spaces. For other uses, see Definition:Topological Equivalence.

Definition

Definition 1

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.


Definition 2

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:

for all $U \subseteq A_1$, $U$ is an open set of $M_1$ if and only if $f \left[{U}\right]$ is an open set of $M_2$.


Then:

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


Definition 3

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:

for all $V \subseteq A_1$, $V$ is a closed set of $M_1$ if and only if $f \left[{V}\right]$ is a closed set of $M_2$.


Then:

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


Definition 4

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:

for all $a \in A_1$ and $N \subseteq A_1$, $N$ is a neighborhood of $a$ if and only if $f \left[{N}\right]$ is a neighborhood of $f \left({a}\right)$.


Then:

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


Also known as

A homeomorphism is also known as a topological equivalence.

Two homeomorphic metric spaces can be described as topologically equivalent.


Also see

  • Results about homeomorphic metric spaces can be found here.