Lipschitz Equivalent Metric Spaces are Homeomorphic
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Theorem
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $M_1$ and $M_2$ be Lipschitz equivalent.
Then $M_1$ and $M_2$ are homeomorphic.
Proof
Let $M_1$ and $M_2$ be Lipschitz equivalent.
Then, by definition, $\exists h, k \in \R_{>0}$ such that:
- $\forall x, y \in A_1: h \map {d_1} {x, y} \le \map {d_2} {\map f x, \map f y} \le k \map {d_1} {x, y}$
From the definition of open $\epsilon$-ball:
\(\ds y\) | \(\in\) | \(\ds \map {B_{h \epsilon} } {\map f x; d_2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_2} {\map f x, \map f y}\) | \(<\) | \(\ds h \epsilon\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_1} {x, y}\) | \(\le\) | \(\ds \frac {\map {d_2} {\map f x, \map f y} } h < \epsilon\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(\in\) | \(\ds \map {B_\epsilon} {x; d_1}\) |
and:
\(\ds y\) | \(\in\) | \(\ds \map {B_{\epsilon / k} } {x; d_1}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_1} {x, y}\) | \(<\) | \(\ds \frac \epsilon k\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_2} {\map f x, \map f y}\) | \(\le\) | \(\ds k \map {d_1} {x, y} < \epsilon\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(\in\) | \(\ds \map {B_\epsilon} {\map f x; d_2}\) |
Thus:
- $\map {B_{h \epsilon} } {\map f x; d_2} \subseteq \map {B_\epsilon} {x; d_1}$
- $\map {B_{\epsilon / k} } {x; d_1} \subseteq \map {B_\epsilon} {\map f x; d_2}$
$\Box$
Now, suppose $U$ is $d_2$-open.
Let $x \in U$.
Then:
- $\exists \epsilon \in \R_{>0}: \map {B_\epsilon} {\map f x; d_2} \subseteq U$.
Hence:
- $\map {B_{\epsilon / k} } {x; d_1} \subseteq U$
Thus $U$ is $d_1$-open.
Similarly, suppose $U$ is $d_1$-open.
Let $x \in U$.
Then:
- $\exists \epsilon \in \R_{>0}: \map {B_\epsilon} {x; d_1} \subseteq U$
Hence:
- $\map {B_{h \epsilon} } {\map f x; d_2} \subseteq U$
Thus $U$ is $d_2$-open.
The result follows by definition of homeomorphic metric spaces.
$\blacksquare$
Also see
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.4$: Equivalent metrics: Definition $2.4.8$