Definition:Topological Equivalence

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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.


Also see