lec10 - Math 117 – 10 Compact sets Definition 10.1 A...

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Unformatted text preview: Math 117 – May 05, 2011 10 Compact sets Definition 10.1. A family of sets F is called an open cover of a set S , if all the sets in F are open, and their union contains S . A subset S ⊆ F is called a subcover of S , if it is also an open cover of S . Using the notion of a cover and a subcover, we can define compact sets. Definition 10.2. A set S is called compact, if every open cover of S contains a finite subcover. Although the definition is abstract and cumbersome to work with, in the space of real numbers R, there is a simpler characterization of compact sets, given by the Heine-Borel theorem below, for the proof of which one needs the following lemma. Lemma 10.3. If S ⊂ R is nonempty, closed and bounded, then S has a maximum and a minimum. Using this lemma one can then establish the following. Theorem 10.4 (Heince-Borel). A subset S ⊂ R is compact iff S is closed and bounded. The above characterization of compact subsets of R allows one to establish useful results for special subsets of R. Theorem 10.5 (Bolzano-Weiestrass). If a bounded set S ⊂ R has infinitely many points, then there exists an accumulation point of S in R. The above theorem of course implies that only finite or unbounded sets may not have accumulation points in R. Some other useful results are given below. Theorem 10.6. Let F = {Kα : α ∈ A } be a family of compact subsets of R, such that the intersection of any finite subfamily of F is nonempty. Then ∩α∈A Kα = ∅. And as a simple corollary of this theorem one has. Corollary 10.7 (Nested intervals). Let F = {An |n ∈ N} be a family of closed bounded intervals in R, such that An+1 ⊆ An for all n ∈ N. Then ∩∞ An = ∅ n=1 . ...
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