exer3Math420

# exer3Math420 - A be a subset of X Deﬁne d x,A = glb d x,a...

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Math 420, Point Set Topology Suggested exercises Problem 1. Let I and I 0 be two topologies on X . Show that if X is a compact Hausdorﬀ space under both I and I 0 , then either I = I 0 or they are not comparable. Problem 2. Is the interval [0 , 1] compact in the lower limit topology R l ? Problem 3. Show that if f : X Y is continuous, where X is compact and Y is Hausdorﬀ, then f is closed map (that is, f carries closed sets to closed sets). Problem 4. Prove that if X is ordered set in which every closed interval is compact, then X has the least upper bound property. Problem 5. Let ( X,d ) be a metric space and
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Unformatted text preview: A be a subset of X . Deﬁne d ( x,A ) = glb { d ( x,a ) : x ∈ A } . a. Show that f : X → R , deﬁned by f ( x ) = d ( x,A ), is continuous function. b. Is it true that for some a ∈ A , we have d ( x,A ) = d ( x,a )? Is this true if A is closed? If A is compact? Problem 6. Show that the set [0 , 1] is not limit point compact as a subspace of R l . Problem 7. Let X be limit point compact. If A is a closed subset of X , is A necessarily limit point compact? 1...
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