# hw2sol - 147 HW2 Solutions 1 In the case of a function f R...

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Unformatted text preview: 147 HW2 Solutions 1.) In the case of a function f : R → R , where R is given the standard topology, prove that the topological definition of continuity is equivalent to the “ εδ definition” you know and love (or not) from calculus. (If you get stuck, one direction of the proof is in Munkres, but write up the proof to both directions in your own words.) Solution: Let f be continuous in the topological sense. Then for each x ∈ R and each > 0, f- 1 (( f ( x )- ,f ( x ) + )) is open. Thus as open intervals give a basis for R , there exists some ( a,b ) containing x such that ( a,b ) ⊂ f- 1 ( V ) and thus f (( a,b )) ⊂ ( f ( x )- ,f ( x )+ ). Let δ < min { x- a,b- x } and we’re done. Let f be continuous in the “ εδ ” sense and V an open set in R . Then for all x ∈ f- 1 ( V ), and any > 0, there exists some δ > 0 such that f (( x- δ,x + δ )) ⊂ ( f ( x )- ,f ( x ) + ). So for each x ∈ f- 1 ( V ) pick an small enough that ( f ( x )- ,f ( x ) + ) ⊂ V . Then the union of the corresponding sets ( x- δ,x + δ ) will be equal to f- 1 ( V ). 2.) Munkres p. 92 #4. A map f : X → Y is said to be an open map if for every open set U of X , the set f ( U ) is open in Y . Show that projections π 1 : X × Y → X and π 2 : X × Y → Y are open maps....
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hw2sol - 147 HW2 Solutions 1 In the case of a function f R...

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