Definition. Let A be a subset of X and let C = {Calae be a collection of subsets of X. Then C is a cover of A if and only if A C U, Ca. The collection C is an open cover of A if and only if C is a cover of A and each Ca is open

Definition. Let A be a subset of X and let C = {Calae be a collection of subsets of X.
Then C is a cover of A if and only if A C U, Ca. The collection C is an open cover
of A if and only if C is a cover of A and each Ca is open. A subcover C’ of a cover C of
A is a subcollection of C whose elements form a cover of A.
For instance, the open sets {(-n, n))nen form an open cover of R. A subcover of
this cover is {(-n, n)}n>5, because these sets still cover all of R.
Definition. A space X is compact if and only if every open cover of X has a finite
subcover.
Theorem 6.2. Let C be a compact subset of Rstd. Then C has a maximum point, that is,
there is a point m E C such that for every x E C, x 5 m.

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