TITLE: Reverse of the statement of uniform continuity on a compact
QUESTION [5 upvotes]: If a set $C$ is compact, then all continuous functions on a set $C$ are uniformly continuous.
Does the reverse hold?
Question If all continuous functions on a set $C$ are uniformly continuous, then $C$ is compact?

REPLY [4 votes]: No. Consider the metric space $(\mathbb{N}, d)$, where $d$ is the Euclidean metric. Then every function is uniformly continuous, but the space is not compact.