# Compactness

The Wikipedia article for compact spaces is large, but extremely informative. I want to distill the important points of what it means to be compact in the context of optimization.

The motivation for something to be compact is a generalization of being closed and bounded. Indeed, any subset $A \subseteq \mathbb R$ that is both closed and bounded is compact. Take this as a definition for now, and we’ll expand on it and re-adjust our thinking. Recall that $\mathbb R$ is both closed and open, but it is definitely not bounded, as it has no upper or lower bound. Without loss of generality, we will only consider closed intervals, i.e. $[a,b]$. Being closed means that $A$ contains its endpoints, but in general it means that a sequence with terms in $A$ has a limit that also stays within $A$. We would like it if compactness would mean just that – that sequences with terms in $A$ have their limit point in $A$. Is the ‘definition’ of compactness we used at the beginning of this paragraph consistent with this logic? Yes and no, and we’ll delve into that now.
Continue reading “Optimization, Step 4: Compactness”