We often utilize the concept of a compact set.

Compact Set: $\mathcal{X} \subseteq \mathbb{R}^d$ is compact iff $\mathcal{X}$ is closed and bounded.

This is the Heine-Borel Theorem. What this means:

1. Closed: The limits are included in the set. So given $a,b \in \mathbb{R}$, $[a,b]$ is a closed set whereas $(a,b)$ is an open set because the limits are not included.
2. Bounded: All elements of the set must be able to be bounded by some real number $M$ where the distance from one point in the set to another must be $\leq M$. I.e., the set cannot extend in one direction infinitely.

Some examples of this, where $a,b \in \mathbb{R}$:

• $[a,b]$ is closed and bounded
• $[a,\infty)$ is closed and not bounded
• $(a,b)$ is not closed but bounded
• $(a,\infty)$ is not closed and not bounded

This also leads to the Weierstrass Extreme Value Theorem:

• If $\mathcal{X}$ is compact and $f$ is continuous, then $\exist x \in \mathcal{X}: f(x)= \sup_{y \in \mathcal{X}} f(y)$
  • I.e., the largest value of the function (defined over our compact set) can be reached (equivalently) by some value in our set. So the extreme value is included in our domain.