definition – A Set X is compact if every Open cover of X contains a Finite Subcover. definition (vid2)

In Standard topology in $\mathbb{R}^n$, a set is compact if and only if it is bounded and closed (see Heine-Borel theorem)

Real Analysis, Lecture 11: Compact Sets

Relationship of Compact Sets to Closed Sets

Functions on compact sets have a maximum (Supremum) and a minimum (Infimum)

– See more in Compact space –

https://blogs.scientificamerican.com/roots-of-unity/what-does-compactness-really-mean/

"next best thing to being finite"

Theorems

In Standard topology in R, a set is compact if and only if it is bounded and closed (see Heine-Borel theorem)

compact sets in the Subspace topology are also compact in the original topology and viceversa

Compact sets are closed in standard topology of Metric space

A closed set of a compact set is also compact

corollary: A closed set and a compact set, then their intersection is compact

Intersection of nested closed intervals in R^n are not empty – can use to show that R is uncountable

Example

A discrete space is compact if and only if it's Finite

A sequence and its limit point form a compact set

Lemma. A subspace Y of a set X is compact if and only if if ever open covering of Y in X (that is, by open sets in X which contains Y) contains a finite subcovering. (This lemma is needed because he defined open covers using equality , instead of subset inclusion).