This video is about the Alexander Subbase Theorem, sometimes called the Alexander Subbase Lemma. It's useful for proving Tychonoff's Theorem, which asserts that an arbitrary product of compact sets is compact, and is equivalent to the Axiom of Choice. This video is mostly self-contained, with refreshers on important definitions like base and subbase of a topology, an open cover of a space, and what a compact topological space is. The majority of this video is dedicated to carefully proving the Alexander Subbase Theorem, filling in details that most textbooks assume the reader will fill-in. I hope you find it helpful, and thanks for watching!