mathAnalysis5-07-draft

# mathAnalysis5-07-draft - P r e l i m i n a r y d r a f t o...

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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #5: THU, SEP 13, 2007 PRINT DATE: AUGUST 21, 2007 (ANALYSIS5) Contents 1. Analysis (cont) 1 1.8. Topology of R n (cont) 1 1.8.5. Closure of a Set 1 1.8.6. Boundary of a Set 1 1.8.7. Compact Sets 2 1. Analysis (cont) 1.8. Topology of R n (cont) 1.8.5. Closure of a Set. The analog of the interior of a set is the closure of a set. Definition: Let A ⊂ X . The closure of A in X , denoted cl( A ) or ¯ A in X is the intersection of all closed sets containing A . Theorem: For A ⊂ X , A is closed in X iff A = cl( A ) in X . 1.8.6. Boundary of a Set. A point x is a boundary point of a set A ⊂ X if there are points arbitrarily close to x that are in A and if there are points arbitrarily close to x that are in X but not in A . 1 2 LECTURE #5: THU, SEP 13, 2007 PRINT DATE: AUGUST 21, 2007 (ANALYSIS5) Example: The set A = braceleftbig ( x,y ) ∈ R 2 : x 2 + y 2 < 1 bracerightbig . Its boundary is the circle braceleftbig ( x,y ) ∈ R 2 : x 2 + y 2 = 1 bracerightbig That is, the boundary is the border between A and X \ A . Definition: The set of boundary points of A in X , denoted bd( A ), is the set cl( A ) ∩ cl( X \ A ). That is, Theorem: Let A ⊂ X . A point x ∈ bd( A ) iff ∀ epsilon1 > 0, ∃ y,z ∈ B ( x,epsilon1 ) such that y ∈ A and z ∈ X \ A . Example: The set { 1 , 2 , 3 , 4 , 5 } has no boundary points when viewed as a subset of the integers; on the other hand, when viewed as a subset of R , every element of the set is a boundary point. Theorem: A set A ⊂ X is closed in X iff A contains all of its boundary points. Note the difference between a boundary point and an accumulation point. Take the set A = { } ⊂ R . 0 is a boundary point of A but not an accumulation point. On the other hand, every element of the interval A = (0 , 1) ⊂ R is an accumulation point of A , but A contains none of the boundary points of A . Though boundary points and accumulation points are resoundingly different in general , there is a close connection between the two concepts: Theorem: Given a set A ⊂ X , a point x ∈ X that does not belong to A is a boundary point of A in X iff it is an accumulation point of A in X ....
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mathAnalysis5-07-draft - P r e l i m i n a r y d r a f t o...

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