Compact Sets and Open Covers, Real Analysis II

I introduce the concept of compact sets in a metric space by exploring the idea of open covers. A set is compact if every possible open cover has a finite subcover, meaning that you can reduce the potentially infinite collection of open sets to a finite number that still fully covers the set.

The lecture begins by explaining open covers and finite subcovers, using examples such as an infinite cover of the closed interval from 0 to 10 which can be reduced to a finite subcollection of open intervals.

Next, we explore cases where a set does not have a finite subcover. One example is an open cover of the open interval from 0 to 1 for which no finite subcover can be found. This is followed by the integers Z, the Archimedean set, and the rationals between 0 and 5. These sets are not compact.

The lecture then formally defines compact sets and demonstrates that finite sets in any metric space are compact, while infinite sets in a discrete metric space are never compact.

Finally, we conclude by proving that every compact set is closed.

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[yopenzo]

Thank you, Bevin. Math is fun, brilliant, and smart. Teachers like you prove this statement without... contradiction.