Compactness is one of the most important concepts in topology. A topological space X is called compact if every open cover of X has a finite subcover. This means that whenever we have a collection of open sets that covers the entire space, we can always find a finite number of these sets that still cover the space.
To understand compactness, we need to define two key terms. An open cover is a collection of open sets whose union contains the entire space X. A finite subcover is a finite subcollection of an open cover that still covers the entire space. The visual shows a space X with multiple open sets forming a cover.
Let's look at a classic example of a compact space. Consider the closed interval from 0 to 1 in the real line. This interval is compact because any open cover of this interval has a finite subcover. This is guaranteed by the Heine-Borel theorem, which states that in Euclidean space, a set is compact if and only if it is closed and bounded.