Intro to Analysis in-class_Part_26

# Intro to Analysis in-class_Part_26 - continuous. 3.3.3...

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3.3 Compact sets 57 Since B c doesn’t cover any part of B , we can throw it out and still have that { U i } n i =1 is an open cover of B . This is a ﬁnite subcover for B , i.e., B is compact. Example 3.3.2. Deﬁne A n = (0 , 1 n ). Then T A n = . 3.3.1 Key properties of compactness K R is compact iﬀ 1. Every sequence { x n } ⊆ K has a limit point x K . 2. K is closed and bounded. 3. Every open cover of K has a ﬁnite subcover. Later, after we’ve seen continuity, we’ll also want Theorem 3.3.13. Let f : X R be continuous, and let K X be compact. Then there exist points m,M K such that f ( m ) f ( x ) f ( M ) , x K . Example 3.3.3. f ( x ) : (0 , 1) R by f ( x ) = x 2 or f ( x ) = 1 x . Theorem 3.3.14. Let f : K Y be continuous, where K is compact. Then f ( K ) is compact in Y . Theorem 3.3.15. On a compact set, any continuous function is automatically uniformly

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Unformatted text preview: continuous. 3.3.3 Exercise: #4,8 Recommended: #3,6,10 #3 is short-answer; a rigorous proof is not required. 1. If F is closed and K is compact, then F K is compact. 58 Math 413 Topology of the Real Line 2. Suppose K = { K } is a collection of compact sets. if K has the property that the intersection of every nite subcollection is nonempty, then prove that T K is nonempty. (Try contradiction or DeMorgans.) 3. Suppose that every point of the nonempty closed set A is a limit point of A . Show that A is uncountable. (Try contradiction, and use the previous problem.)...
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## This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_26 - continuous. 3.3.3...

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