Download Final exam questions

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, Ā 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 = {U × V | U ∈
TX and V ∈ TY } 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 {Y, ∅}, then the
product topology for X × Y is the trivial topology {X × Y, ∅}.
11. If A ⊂ X, then A0 ⊂ A.
12. If A ⊂ X, then Bd(A) ⊂ A.
13. The point {1} is a limit point of [0, 1) ⊂ R regardless of the topology on R.
14. The point {2} 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 {Xα }α∈J is a collection of
Ytopological spaces
Y and for all α ∈ J, Uα ⊂ Xα is
nonempty and open, then
Uα is open in
Xα with respect to the product
α∈J
α∈J
topology.
19. If {Xα }α∈J !
is a collection of topological spaces and Uα ⊂ Xα for all α ∈ J, then
Y
Y
Y
Int
Uα =
Int(Uα ) in
Xα with the product topology.
α∈J
α∈J
20. A projection function πβ :
α∈J
Y
Xα → Xβ is always open but may not be continuous.
α∈J
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 = {X, ∅}, then X is metrizable.
33. If {Xα }α∈J is a collection of topological spaces
Y and for all α ∈ J, Xα has the discrete
topology, then the product topology on
Xα is the discrete topology.
α∈J
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 intervals 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.