Heine-Cantor theorem asserts that every continuous function on a compact metric space is uniformly continuous.
The converse is , if every continous function is uniformly continuous on a metric space X then X is compact. I have got the answer that the converse is false but I am unable to find a counter example.
The other question is if the converse is false then its negation must be true.
Negation- if every continous function is uniformly continuous on a metric space X and X is not compact.
How do we prove this?
Theorem 1 of this old paper gives conditions on a metric space so that all continuous functions on it are uniformly continuous.
It seems to me that $\Bbb Z$ (as a subspace of $\Bbb R$) is an easy example of such a non-compact space. And also $\Bbb Z \cup \{\frac1n: n =1,2,3,\ldots\}$ is one that is non-discrete too. As the theorem 1 quoted above shows, we do need lots of "far away" isolated points for this to happen, and in "nice" metric spaces this will imply compactness.
BTW the negation of the converse is: there exists a non-compact $X$ such that every continuous function on $X$ is uniformly continuous.