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Math 560 Spring 2014 Final Exam Exercises

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Math 560 Spring 2014 Final Exam Exercises Math 560 Spring 2014 Final Exam Exercises Decide if the following statements are true or false. For \if and only if" statements and equalities which are false, determine if one direction of the implication or inclusion is true. Prove each true statement and find counterexamples for each false statement. Note: Open, closed, and half-open intervals in R refer to intervals taken with respect to the usual order relation. In all problems, X and Y denote topological spaces and ⊂ denotes a subspace. For A ⊂ X, A¯ denotes the closure of A in X, Bd(A) denotes the boundary of A in X, Int(A) denote the interior of A, and A0 denotes the set of limit points of A in X. Important note: Problems 30-49 must be done before you start working through problems 1-29. Exercises adapted from Introduction to Topology by Baker. 1. If f : X ! Y is a function and U and V are subsets of X, then f(U \ V ) = f(U) \ f(V ). 2. Any open interval is an open subset of R regardless of the topology on R. 3. In a topological space (X; T ), any collection of open sets whose union equals X and that is closed under finite intersection is a basis for T . 4. There exists a topological space (X; T ) such that there is no basis for T . 5. There is a topological space (X; T ) such that there is more than one basis for T . 6. There is a topological space (X; T ) such that there is only one basis for T . 7. Let (X; T ) be a topological space with A ⊂ X. If TA is the subspace topology on A, then TA ⊂ T . 8. If (X; TX ) and (Y; TY ) are topological spaces, then the collection S = fU × V j U 2 TX and V 2 TY g is a topology on X × Y . 9. If X has the discrete topology and Y has the discrete topology, then the product topology for X × Y is the discrete topology. 10. If X has the discrete topology and Y has the trivial topology fY; ;g, then the product topology for X × Y is the trivial topology fX × Y; ;g. 11. If A ⊂ X, then A0 ⊂ A. 12. If A ⊂ X, then Bd(A) ⊂ A. 13. The point f1g is a limit point of [0; 1) ⊂ R regardless of the topology on R. 14. The point f2g is not a limit point of [0; 1) ⊂ R regardless of the topology on R. 15. Any constant function f : X ! Y is continuous regardless of the topologies on X and Y . 16. Any two topological spaces with the discrete topology are homeomorphic. 17. If X and Y are homeomorphic, then any bijection f : X ! Y is a homeomorphism. 18. If fXαgα2J is a collection of topological spaces and for all α 2 J, Uα ⊂ Xα is Y Y nonempty and open, then Uα is open in Xα with respect to the product α2J α2J topology. 19. If fXαgα2J is a collection of topological spaces and Uα ⊂ Xα for all α 2 J, then ! Y Y Y Int Uα = Int(Uα) in Xα with the product topology. α2J α2J α2J Y 20. A projection function πβ : Xα ! Xβ is always open but may not be continuous. α2J 21. If R is given the usual topology, [0; 1] ⊂ R and f : R ! Y is injetive, then f([0; 1]) is a connected subset of Y . 22. Subspaces of regular spaces are regular. 23. Subspaces of Hausdorff spaces are Hausdorff. 24. Every normal space is Hausdorff. 25. If a X does not have any nonempty, disjoint, open sets, then X is not normal. 26. If (X; d) is a metric space and T is the topology induced by the metric d, then every open subset of X is an open ball. 27. If d1 and d2 are two different metrics for X, then they induce two different topologies on X. 28. The intersection of two open balls is an open ball. 29. If X is given the discrete topology, then X is metrizable. 30. Closed subsets of normal spaces are normal. 31. Every normal space is regular. 32. If X is given the topology T = fX; ;g, then X is metrizable. 33. If fXαgα2J is a collection of topological spaces and for all α 2 J, Xα has the discrete Y topology, then the product topology on Xα is the discrete topology. α2J 34. If A ⊂ X and f : X ! Y is continuous, then A is connected if and only if f(A) is connected. 35. If X and Y are nonempty, then X × Y is connected if and only if both X and Y are connected. 36. If R has the usual topology and f : R ! R is a bijection, the f is a homeomorphism. 37. Every finite topological space is compact. 38. An interval is a compact subset of R under the usual topology. 39. The compact subspaces of R under the usual topology are precisely the closed in- tervals and singleton sets. 40. The compact subspaces of R under the usual topology are precisely the closed sets which are contained in a closed interval. 41. Every subset of a compact space is compact. 42. Every Hausdorff subset of a compact space is compact. 43. Every closed subset of a compact space is compact. 44. Every compact subset of a Hausdorff space is closed. 45. The union of two compact subsets of a topological space is compact. 46. The union of any collection of compact subsets of a topological space is compact. 47. If R has the usual topology and [0; 1] ⊂ R has the subspace topology, then any continuous function f : [0; 1] ! R assumes a maximum and a minimum. 48. If R has the usual topology and [0; 1) ⊂ R has the subspace topology, then any continuous function f : [0; 1) ! R assumes a maximum and a minimum. 49. If R has the usual topology, then a subset of R is connected if and only if it is compact..
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