Topology: An Inquiry-Based Approach

A cover of a subset $A$ of a topological space $X$ is any collection of subsets of $X$ whose union contains $A\text{.}$ An open cover is a cover consisting of open sets.

A subcover of a cover of a set $A$ is a subset of the cover such that the union of the sets in the subcover also contains $A\text{.}$

A subset $A$ of a topological space is compact if every open cover of $A$ has a finite subcover.

A continuous function from a compact topological space to the real numbers must attain a maximum and minimum value.

The Heine-Borel Theorem states that the compact subsets of ${\mathbb{R}}^n$ are exactly the subsets that are closed and bounded.