For a space X and metric space Y, if X is compact then uniform and compact convergence topology coincides

Question

In the book Topology by Munkres, there is a theorem: Let $X$ be a space and $(Y,d)$ be a metric space. For the function space $Y^X$ ,one has the following inclusions of topologies: $uniform\supset compact convergence \supset pointwise convergence $. I have done this. The next part says: If $X$ is compact, the first two coincide and if $X$ is discrete the second two coincide. I am stuck in the next part. Please help.

Answer 1

Compact convergence means uniform convergence on compact sets. If $X$ is already compact, then there is uniform convergence on the entire space $X$, so both topologies coincide.