The Heine-Borel Theorem is a fundamental result in real analysis that characterizes compact subsets of Euclidean space. It states that a subset of R^n (n-dimensional real space) is compact if and only if it is both closed and bounded. This means that the set contains all its limit points (closed) and fits within some finite region of space (bounded). Compactness is an important property in mathematics because it allows for the generalization of various theorems, such as the Extreme Value Theorem, which guarantees that a continuous function on a compact set attains its maximum and minimum values. The Heine-Borel Theorem thus provides a crucial link between topology and analysis.