Compactness theorem question

Question

Well in my topology notes I have:

$A \subset$ $\mathbb{R^n}$ is compact iff $A$ is closed and enclosed.

But it doesn't specify the topology, my question is if this is true for any topology or only with the usual topology of $\mathbb{R^n}$.

Answer 1

This is true in the standard topology of $\mathbb R^n$ and in some other topologies, but not true in all topologies

For example, in the discrete metric, every set is closed and bounded, but only finite sets are compact. Therefore, any infinite set is closed, enclosed and non-compact.