Topology: Example of a compact set but its closure not compact Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.

I think the simplest example would probably be the "particular point topology" on an infinite set. 

Let $X = \{p\} \cup \mathbb{N}$, topologized so that the nonempty open sets are the sets containing $p$.

The singleton $\{p\}$ is compact (finite sets are always are). However, $\overline{ \{p\}} =X$, and $X$ is not compact because the open cover $X = \bigcup_{n\in \mathbb{N} } \{p , n\}$ has no finite subcover. 
