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Math 636 Topology Paulo Lima-Filho Assignment # 3 (Due date: Monday, Oct. 11 , 1999) 1 Practice Problems (You do not need to hand these in.) Additional Problem 1. a If T and T 0 are topologies on X, with T finer than T 0 , is it true that a compact set in T is also compact in T 0 ? What about the converse? Provide counterexamples whenever necessary. b Describe the compact sets in the lower limit topology on R. (The reals with the lower limit topology R` is often called the Sorgenfrey line.) c Is the Sorgenfrey line connected? Locally connected? Describe its components. Additional Problem 2. 1. Let X be a (nonempty) compact Hausdorff space. If every point of X is an acumulation point of X, then X in uncountable. (HINT: Step 1 Show that if U is a non-empty subset of X and x ∈ X, there is a (non-empty) open set V contained in U such that V does not contain x. Note that x may not be in U. Step 2 Given f : N → X show that f is not surjective. Start with U = X a non-empty open set and use previous step to find a non-empty open set V1 such that V 1 does not contain f (1). Successively define a nested sequence V 1 ⊃ V 2 ⊃ · · · . Use the compactness of X to find a point not in the image of f .) 2. Use previous problem to conclude that any closed interval [a, b] in the real line is uncountable. Additional Problem 3. A space X satisfies the countable chain condition iff every disjoint family of open sets is countable. Show that a separable space satisfies the separable chain condition, but not conversely. Fall 1999 Assignment # 3 Page 1 of ?? Math 636 Topology Paulo Lima-Filho Additional Problem 4. Let {[an , bn ] | n ∈ N} be a collection of intervals in R such that [an+1 , bn+1 ] ⊂ [an , bn ] for all n. Is it true that ∩n [an , bn ] 6= ∅? Explain. Additional Problem 5. Here we construct the Cantor set and study its basic properties. Let A0 be the closed interval [0, 1] ⊂ R. Let A1 be the set obtained from A0 by deleting its “middle third” ( 13 , 23 ). Let A2 be the set obtained from A1 by deleting its “middle thirds” ( 19 , 29 ) and ( 79 , 89 ). In general, define An by [ 1 + 3k 2 + 3k . An = An−1 , 3n 3n k≥0 The intersection C = ∩n∈N An is the subspace of [0, 1] called the Cantor set. a A space X is totally disconnected if its only connected subspaces are the one-point subsets. Show that C is totally disconnected. b Show that C is compact. c Show that each An is a union of finitely many disjoint closed intervals of length 1/3n , and show that the end pointsof these intervals lie in C. d Show that every point of C is a limit point of C. e Conclude that C is uncountable. Fall 1999 Assignment # 3 Page 2 of ?? Math 636 2 Topology Paulo Lima-Filho The Problems You must hand these in. Problem 1. One says that a Hausdorff space is countably compact if every countable cover has a finite subcover. a Give an example of a countably compact space which is not compact. b Show that X is countably compact if and only if every family of closed subsets having the finite intersectin property also has the countable intersection property. c If Y is first countable and X is countably compact, then a continuous bijection f : X → Y is a homeomorphism. Problem 2. Given a subset A ⊂ X of metric space X, define diam(A) = sup{dist(x, y) | x, y ∈ A}. One says that A is bounded if diam(A) < ∞. Let A be a compact subspace of X. Show that A is closed and bounded. Give two non-homeomorphic examples of metric spaces where closed and bounded does not imply compactness. Problem 3. Show that a closed subspace of a normal space is normal. Problem 4. A topological space X is a door space if every subset is either open of closed. Show that a Hausdorff door space has at most one accumulation point, and if x is a point which is not an accumulation point, then {x} is open. Give an example of a non-discrete door space. Problem 5. A point x in a space X is an accumulation point for a subset A ⊂ X if every neighborhood of x intersects A at a point other than x. In other words U ∩ (A − {x}) 6= ∅ for every neighborhood U of x. 1. Show that a subset A of X is closed if and only if it contains all its accumulation points. 2. Prove that x is an accumulation point of A if and only if there is a net Φ : D → X converging to x and satisfying Φ(α) ∈ A − {x} for all α ∈ D. Fall 1999 Assignment # 3 Page 3 of ??