Compact Domains

A set $D \subseteq \mathbb{R}^n$ is compact if it is closed (contains all its limit points) and bounded (fits inside a ball of finite radius). Compact sets are geometrically the "finite" or "closed and bounded" subsets of $\mathbb{R}^n$.

Examples of compact sets: closed intervals $[a,b]$, closed balls $\{\mathbf{x} : \|\mathbf{x}\| \leq r\}$, and closed rectangles $[a_1,b_1] \times \cdots \times [a_n,b_n]$. Non-compact examples: open intervals $(a,b)$, all of $\mathbb{R}^n$, half-open intervals $[a,b)$.

Compactness is important for optimization because it ensures the existence of global minima and maxima (Extreme Value Theorem). Without compactness, a continuous function can fail to achieve its minimum or maximum on the domain.