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exer1Math420

# exer1Math420 - R be generated by the basis B = B B = a,b...

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Math 420, Point Set Topology Suggested exercises 1 Problem 1. Consider the following collection of subsets of R B = { B : R \ B is ﬁnite } (a) Show that B is a basis for topology on R . (b) Determined if the generated topological space is Hausdorﬀ. Problem 2. If X = { a, b, c } , let I 1 = {∅ , X, { a } , { a,b }} and I 2 = {∅ , X, { a } , { b,c }} Find the smallest topology containing I 1 and I 2 and the largest topology contained in I 1 and I 2 . Problem 3. Consider the set Y = [ - 1 , 1] as a subset of R with standard topology. Which of the following sets is open in Y . A = { x : 1 2 ≤ | x | < 1 } B = { x : 1 2 < | x | ≤ 1 } C = { x : 1 2 < | x | < 1 } D = { x : 0 < | x | < 1 , and 1 x / Z + } Problem 4. Show that the dictionary topology on R × R is the same as the product topology on R d × R where R d denotes the set R with discrete topology. Problem 5. Show that the T 1 axiom is equivalent to the requirement that the ﬁnite set is closed. Problem 6. Show that the product of two Hausdorf spaces is Hausdorf. Problem 7. Let a topology on

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Unformatted text preview: R be generated by the basis B = { B : B = [ a,b ) , a.b ∈ Q } . 1 Determine the clouser of two sets A = (1 , √ 2) and B = ( √ 2 , 4). Problem 8. Let X and Y be two odered setsin the order topology. Show that if a map f : X → Y is bijective and order preserving then it is a homeomorphism. Problem 9. Let a function f : R → R be continuous from the right. Show that f is continuous as afunctio from R , with stabndard topology, to R with lower limit topology. Problem 10. Let X and Y be topological spaces and A ⊂ X be subspaces of X . Let Y be Hausdorﬀ. Show that if a continuous function f : A → Y may be extended to a continuous function g : ¯ A → Y , then g is uniquely determined by f . 2...
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exer1Math420 - R be generated by the basis B = B B = a,b...

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