Definition:Topological Equivalence
in the process of being refactored, or:
has been identified as a candidate for refactoring. In particular:
Might be worth putting the alternative definition into the proof page for Equivalence of Definitions of Topological Equivalence and just using that page as a proof of an equivalent definition. Also I'd call this page "Topologically Equivalent Metrics", or something.
Until this has been finished, please leave {{refactor}} in the code.
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Metric
Let $A$ be a set upon which there are two metrics imposed: $d_1$ and $d_2$.
Then $d_1$ and $d_2$ are topologically equivalent iff:
$U \subseteq A$ is $d_1$-open $\iff$ $U \subseteq A$ is $d_2$-open.
Alternative Definition
Let $A$ be a set upon which there are two metrics imposed: $d_1$ and $d_2$.
Let $\left({B, d}\right)$ and $\left({C, d\,'}\right)$ be any metric spaces.
Let $f: B \to A$ and $g: A \to C$ be any mappings such that:
- $f$ is $\left({d, d_1}\right)$-continuous iff $f$ is $\left({d, d_2}\right)$-continuous;
- $g$ is $\left({d_1, d\,'}\right)$-continuous iff $g$ is $\left({d_2, d\,'}\right)$-continuous.
Then $d_1$ and $d_2$ are topologically equivalent.
Topological equivalence is clearly an equivalence relation.
Equivalence of Definitions
The above two definitions are equivalent.
Metric Spaces
Let $M$ and $M'$ be metric spaces.
Let $f: M \to M'$ be a bijection such that both $f$ and $f^{-1}$ are continuous.
Then $f$ is a topological equivalence.
Otherwise known as a homeomorphism.
Sources
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definition $2.4.1$, Proposition $2.4.2$
