Definition:Homeomorphism/Metric Spaces
This page is about Topological Equivalence in the context of Metric Space. For other uses, see Topological Equivalence.
Definition
Definition 1
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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 $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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 $f \sqbrk U$ is an open set of $M_2$.
Then:
- $f$ is a homeomorphism
- $M_1$ and $M_2$ are homeomorphic.
Definition 3
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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 $f \sqbrk V$ is a closed set of $M_2$.
Then:
- $f$ is a homeomorphism
- $M_1$ and $M_2$ are homeomorphic.
Definition 4
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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 $f \sqbrk N$ is a neighborhood of $\map f a$.
Then:
- $f$ is a homeomorphism
- $M_1$ and $M_2$ are homeomorphic.
Also known as
A homeomorphism between two metric spaces is also known as a topological equivalence.
Two homeomorphic metric spaces can be described as topologically equivalent.
Also see
- Results about homeomorphisms in the context of metric spaces can be found here.