 # Topology Problems

## Transcribed Text

1. Recall that a subset A of a topological space (X, T) is closed provide X − A ∈ T. In (X, T), prove that a) ∅, X are closed b) Finite unions of closed sets are closed c) Arbitrary intersections of closed sets are closed. 2. Suppose that Y is a subspace of the topological space X. We say A ⊂ Y is closed in the subspace topology Y provided Y − A is open in Y . a) Show by example that a closed subset of the subspace Y is not necessarily closed a subset of X. b) Prove that A is closed in the subspace topology on Y if and only if A = Y ∩ C for some closed subset C ⊂ X. 3. Let X be a topological space and let A ⊂ X. Then - The interior of A, denoted Å, is the union of all open subsets contained in A. - The closure of A, denoted A¯, is the intersection of all closed sets containing A. - The boundary of A, denoted ∂A, is the intersection A¯ ∩ X − A. Observe that Å ⊂ A ⊂ A. ¯ 4. Consider A = [0, 1] ⊂ R. What are the interior, closure and boundary of A ? Repeat the question for the subset B = {1, 1/2, 1/4, 1/8, . . .} 5. Prove the following statements: a) Å is open. b) A¯ is closed. c) Å ∩ ∂A = ∅. d) A¯ = Å ∪ ∂A. 6. Let A be the subset of the topological space X. Prove that x ∈ A¯ if and only if, for every open set U containing x, U ∩ A 6= ∅. (Hint: the contrapositive in both directions.) 7. Let A be a subset of the topological space X, and suppose β is a basis for the topology on X. Prove that x ∈ A¯ if and only if, for every B ∈ β containing x, B ∪ A 6= ∅ (Hint: use the previous result.) 8. Let A be a subset of a topological space X. Then x ∈ X is a limit point for A if, for every open set U containing x, U ∩ A contains at least one element y not equal to x itself. a) What are the limit points of (0, 1] ⊂ R ? b) What are the limit points of { n n+1 | n ∈ N} ⊂ R ? c) What are the limit points of S 1 ⊂ R 2 ? Here S 1 is the unit circle in R 2 ; assume R 2 has the usual topology (generated by the δ-ball basis). d) What are the limit points of Z ⊂ R ? e) What are the limit points of {x ∈ R|x > 0} ? 19. Given the subset A of the topological space X, let A0 denote the set of limit points of A. Prove that A¯ = A 0 ∪ A. As a corollary, prove that A ⊂ X is closed if and only if A0 ⊂ A (i.e. A contains all of its limit points). 10. Show that the topological space X is not connected if it contains a proper subset A which is both open and closed. Then use this result to characterize connected topological spaces. 11. We dened a path-connected topological space X to be any space in which, for any pair of points x, y, there exists a continuous function f : [0, 1] → X with f(0) = x and f(1) = y. Prove that if X is path-connected and g : X → Y is a continuous function between topological spaces, then the image f(X) of X is path connected. 12. Consider the topological space R 2 in the usual topology (generated by the δ-ball basis). Recall that a path connected space is connected (but the converse is false). a) Let L ⊂ R 2 be any straight line. Is R 2 − L connected ? b) Is R 2 − S 1 connected ? c) Let A be any nite subset of R 2 . Is R 2 − A connected ? d) Z × Z is the subset of R 2 consisting of all points with integer coordinates (aka the integer lattice). Is R 2 − Z × Z connected ?

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