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Unformatted text preview: Lecture 2 1.6 Compactness As usual, throughout this section we let ( X, d ) be a metric space. We also remind you from last lecture we defined the open set U ( x o , λ ) = { x ∈ X : d ( x, x o ) < λ } . (1.10) Remark. If U ( x o , λ ) ⊆ U ( x 1 , λ 1 ), then λ 1 > d ( x o , x 1 ). Remark. If A i ⊆ U ( x o , λ i ) for i = 1 , 2, then A 1 ∪ A 2 ⊆ U ( x o , λ 1 + λ 2 ). Before we define compactness, we first define the notions of boundedness and covering. Definition 1.19. A subset A of X is bounded if A ⊆ U ( x o , λ ) for some λ . Definition 1.20. Let A ⊆ X . A collection of subsets { U α ⊆ X, α ∈ I } is a cover of A if A ⊂ U α . α ∈ I Now we turn to the notion of compactness. First, we only consider compact sets as subsets of R n . For any subset A ⊆ R n , A is compact ⇐⇒ A is closed and bounded . The above statement holds true for R n but not for general metric spaces. To motivate the definition of compactness for the general case, we give the HeineBorel Theorem. HeineBorel (HB) Theorem. Let A ⊆ R n be compact and let { U α , α ∈ I } be a cover of A by open sets. Then a finite number of U α ’s already cover A . The property that a finite number of the U α ’s cover A is called the HeineBorel (HB) property . So, the HB Theorem can be restated as follows: If A is compact in R n , then A has the HB property. Sketch of Proof. First, we check the HB Theorem for some simple compact subsets of R n . Consider rectangles Q = I 1 × ··· × I n ⊂ R n , where I k = [ a k , b k ] for each k . Starting with one dimension, it can by shown by induction that these rectangles have the HB property....
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
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