Uniform Continuity on Metric Space does not imply Compactness

Theorem

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $f: A_1 \to A_2$ be a uniformly continuous mapping on $A_1$.

Then $M_1$ does not necessarily have to be a compact metric space.

Proof

Let $M_1 = \struct {A_1, d_1}$ be any metric space which is not compact.

Let $I_{M_1}: M_1 \to M_1$ be the identity mapping.

From Identity Mapping is Uniformly Continuous, $I_{M_1}$ is uniformly continuous on $M_1$.

Hence the result.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.8$: Compactness and Uniform Continuity: Remark $5.8.3$