ABSTRACT
This chapter explores an important and useful connection that some infinite sets have with finite sets. This new property may seem strange at first, but it captures a key property that is preserved under continuous mappings. A collection (or family) of open sets is a formal way to describe a set whose elements are open sets. The most interesting collections of open sets often have infinitely many open sets in the collection. When dealing with a set that we know is compact, we know that every open cover (no matter how bizarre) will admit a finite subcover. We can leverage this fact to great effect by cleverly choosing open covers of our compact sets. The chapter also provides theorems and proofs for compact sets.
