Lipschitz Equivalent Metric Spaces are Homeomorphic

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

• Lipschitz Equivalent Metrics are Topologically Equivalent

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.4$: Equivalent metrics: Definition $2.4.8$