Chapter 7 Flashcards
an open cover of a subset S of a metric space X
Let X be a metric space and let S be contained in X. If {U_alpha}_alpha in an arbitrary set is a collection of open subsets of X and S is contained in the union, then {U_alpha}_alpha in an arbitrary set is an open cover for S
a compact subset of a metric space X
A set is compact if for all open covers of S there is a finite subcover
the Heine-Borel Theorem
A subset of R is compact iff it is closed and bounded
Subcover for S
Any subcollection of {U_alpha}_alpha in an arbitrary whose union still contains S
Every finite set is _____
compact
Finite Sets have three properties
- Closed
- Bounded
- minimum/maximum of things
Theorem about closed intervals and compactness
A closed interval [a,b] in R with the usual metric is compact when a<b></b>
Every compact set is ________
bounded
Theorem 7.1.8
Every compact set is closed in a metric
Theorem 7.1.9
Every closed subset of a compact set is compact
Characterizations of Compactness Theorem 7.1.11 TFAE
Let X be a metric space
- S contained in X is compact
- Every sqn. in S has a subsequence has a subsequence converging to a point in S
- Every infinite subset of S has a limit point in S
The ________ image of a compact set is _________. Theorem 7.2.1
continuous, compact
Extreme Value Theorem (7.2.2)
Let X be a nonempty, compact metric space. Let f:X to R be continuous. Then, there exists an x,y in X s.t. for all z in X f(x)
Every continuous function into R on a compact set
has a min and max value.
Lemma 7.3.2
If S is a bounded subset of Rn, then for all epsilon >0, S is contained in the Union from i=1 to n of the open balls radius epsilon about x_i
