compact set

Definition (in reals)

A set of real numbers $S$ is compact if for every covering $O$ of $S$ by open sets, $S$ is covered by some finite set of members of $O$.

Definition (compact topological space)

A topological space $(X, \mathcal{T})$ is said to be compact if every open cover of $X$ has a finite subcover.

Notes

• Heine-Borel theorem says that for real sets, compact iff closed and bounded.

See also

• topology

References

1. https://math.mit.edu/~djk/calculus_beginners/chapter16/section02.html
2. https://math.stackexchange.com/questions/1614133/the-two-definitions-of-a-compact-set
3. https://en.wikipedia.org/wiki/Compact_space
4. https://ncatlab.org/nlab/show/compact+space
5. https://www.math.toronto.edu/ivan/mat327/docs/notes/16-compact.pdf
6. https://en.wikipedia.org/wiki/Heine–Borel_theorem