real analysis – Equicontinuous and compactness

enter image description here

Hello, everyone. I want to discuss about this theorem, which I read in “Principles of Mathematical Analysis” by Rudin, to enrich my knowledge about equicontinuity.
I think “equicontinuity” in the theorem means uniform equicontinuity, is it right?

Then, the theorem said “$K$ is a compact metric space”, why it must be a compact metric space? What will happen if $K$ is not a compact metric space? Will it be equicontinuous at $x$ (not uniformly equicontinuous) if $K$ is not compact?

Thanks for any help.