Compactification and Metrisation

Some great notes from Dr. Terence Tao’s lecture on compacity and metric spaces.

Metric spaces are sets with a measure of distance between each of its elements. Compact spaces are spaces in which each sequence (xn) admits a converging sub-sequence. This is called the Bolzano-Weierstrass property and such compact spaces are called sequencially compact. There is a more general way of defining a compact space, by saying that each of its open covers has a finite subcover.