Existence of an metric or a topology so that every subset is compact

Question

Let $X$ be a infinite set. Is there a metric on $X$ such that every sub set of $X$ is compact? What about a topology on $X$?

I think that if we can answer first question then we can answer the second question. I can't answer first question, but I can answer the second question: we can use indiscrete topology $\{ \emptyset , X\}$.If exist this meter must every subset of $X$ both open and close

Answer 1

If $X$ is Hausdorff and every subset of $X$ is compact, then every subset of $X$ is closed, and hence every subset of $X$ is open. So we necessarily get the discrete topology. Since $X$ with the discrete topology is itself compact, we must have that it's finite, thus contradicting your assumption.

So no, as soon as $X$ is Hausdorff, and metric spaces are Hausdorff, this cannot happen.

Answer 2

The answer to the first question is No. If this were possible for a given infinite set $X$, then every subset would be closed. So every subset would also be open, including one point subsets. That means that it' s the discrete topology.