Boundedness is not Topological Property

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 homeomorphic.

Then it is not necessarily the case that:

$M_1$ is bounded if and only if $M_2$ is bounded.

That is, boundedness is not a topological property.

Proof

Proof by Counterexample:

Let the metric space $M_1 = \struct {S_1, d}$ such that:

$S_1 = \openint 0 1$ is the open unit interval

$d$ is the usual (Euclidean) metric on $S_1$.

Let the metric space $M_2 = \struct {\R, d}$ such that:

$\R$ is the set of real numbers

$d$ is again the usual (Euclidean) metric on $\R$.

Then $M_1$ is bounded by, for example, $1$.

However, $M_2$ is not bounded.

But from Open Real Interval is Homeomorphic to Real Number Line, we have that $M_1$ and $M_2$ are homeomorphic.

Hence the result.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.6$: Homeomorphisms: Examples $3.6.4$