We know that a Compact set is closed. However a finite discreet set is compact but not closed (contradicting the theorem?) We know that a Compact set is closed. We also know that a finite discreet set is compact (as every cover has a finite sub cover). However a finite discreet set is not closed (contradicting the theorem?). I am sure I am missing something here, or the theorem has certain conditions embedded.

In a metric space, the following two things are true: any compact set is closed, and any finite set is both closed and compact.

Your statement that a finite set might not be closed is not true for metric spaces.