The notion of compactness may informally be considered a generalisation of being closed and bounded, and plays an important role in Analysis. Before we state the formal definition, we first have to define what we mean by an open cover of a set.
Definition: open cover
If is an open cover of , any subset of that is also an open cover of is called a subcover of .
Definition: compact set
Let be a metric space. A subset of is said to be compact if and only if every open cover of in contains a finite subcover of . That is, if is an open cover of in , then there are finitely many indices such that .
In , the notion of being compact is ultimately related to the notion of being closed and bounded. This theorem is known as the Heine-Borel theorem, which states that a subset of is compact if and only if it is closed and bounded.
- Rudin, Walter: Principles of Mathematical Analysis, 3rd edition, McGraw Hill, 1976.