Definition:Topologically Equivalent Metrics/Definition 1

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Definition

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

  • Results about topologically equivalent metrics can be found here.


Sources