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Unformatted text preview: 147 HW4 Solutions 17.15 Show the T 1 axiom is equivalent to the condition that for each pair of points of X , each has a neighborhood not containing the other. T 1 axiom: All finite point sets are closed. Solution: Assume the T 1 axiom. Then for any pair of poinits x,y ∈ X , X { y } is a neighborhood of x not containing y . Assume that for each pair of points of X , each has a neighborhood not containing the other. Let x ∈ X . The for all y 6 = x in X . There exists a neighborhood U y of y not containing x . Thus { x } = X S y 6 = x U y is a closed set. As finite intersections of closed sets are closed, we have the T 1 axiom. 17.20 Find the boundary and the interior of each of the following subsets of R 2 : (a) A = { x × y  y = 0 } Solution: This is the xaxis. As it contains no basis elements ( a,b ) × ( c,d ), its interior is empty and (from 17.19a), as it’s closed, it is its own boundary....
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This note was uploaded on 05/19/2010 for the course MAT mat147 taught by Professor Derekwise during the Spring '10 term at UC Davis.
 Spring '10
 derekwise

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