Definition:Topological Equivalence
Jump to navigation
Jump to search
Definition
Homeomorphism: 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.
Homeomorphism: Metric Spaces
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.
Topologically equivalent metrics
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
$d_1$ and $d_2$ are topologically equivalent if and only if:
- For all metric spaces $\struct {B, d}$ and $\struct {C, d'}$:
- For all mappings $f: B \to A$ and $g: A \to C$:
- $(1): \quad f$ is $\tuple {d, d_1}$-continuous if and only if $f$ is $\tuple {d, d_2}$-continuous
- $(2): \quad g$ is $\tuple {d_1, d'}$-continuous if and only if $g$ is $\tuple {d_2, d'}$-continuous.
Such mappings $f$ and $g$ can be referred to as homeomorphisms.