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
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.4$: Equivalent metrics: Definition $2.4.1$