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Section Summary
Important ideas that we discussed in this section include the following.
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 \(\R^n\) are exactly the subsets that are closed and bounded.
