Definition:Topologically Equivalent Metrics

This page is about Topological Equivalence in the context of Metric Space. For other uses, see Topological Equivalence.

Definition

Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.

Definition 1

$d_1$ 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.

Definition 2

$d_1$ and $d_2$ are topologically equivalent if and only if:

$U \subseteq A$ is $d_1$-open if and only if $U \subseteq A$ is $d_2$-open.

Also see

• Equivalence of Definitions of Topologically Equivalent Metrics

• Topological Equivalence is Equivalence Relation

• Definition:Homeomorphism (Metric Spaces)

• Results about topologically equivalent metrics can be found here.