Topology

Test File

Fall 1998

 

Test #1

 

1.) Define the following terms.

a.) Topological space b.) L.U.B. of A R

c.) Closed Set d.) Open Set

e.) Closure of a Set f.) Dense Set

g.) Limit Point h.) Interior of a Set

2.) Determine if the following are topologies for the given set. Prove your answer.

a.) X = R, T = {U R {2, 4} U}

b.) X is an infinite set, T = {U X U = X or U is finite}

3.) Prove the following.

a.) Let (X, T) be a topological space. If A, B X then A B A B. Give an example to show that the reverse inclusion is not necessarily true.

b.) A set, A, is closed if and only if for every point, x, not in A, there exists an open set, Ux, such that x Ux X - A.

4.) True-False - Write "True" only if the statement is always true. Otherwise, write "False" and give an example to show that it is false.

a.) In a topological space, (X, T), all subsets of X are either open or closed.

b.) In a topological space, (X, T), open sets can also be closed.

c.) In a topological space, the closure of a set is closed.

d.) In a topological space, (X, T), a union of open sets is open.

e.) In a topological space, a union of closed sets is open.

1.) Determine if the following are topologies for the given set. Prove your answer.

a.) X, T = {U X U = or A U, where A is a fixed subset of X}

b.) X, T where T is the intersection of a collection of topologies for X.

c.) X, Let A be a subset of X and let S be a topology for X. T = {U (V A) U, V S}

2.) Prove the following.

a.) Between any two real numbers is an irrational number. [Note: This can be done two ways. You can construct the irrational number or you can use a counting argument.]

b.) The set of rationals is dense in the usual topology on the real line.

c.) Let X be a finite set. The discrete topology and the cofinite topology are the same.

3.) Do the following

a.) Consider the real line with the usual topology. Find the set of limit points and the closure for each of the following sets.

(i) (3, 4)

(ii) (3, 4) {7}

(iii) {rationals}

(iv)

b.) Consider the set of integers with the cofinite topology. Find the set of limit points and closure for:

(i) {2, 3, . . . , 100}

(ii) {positive integers}

4.) Let A and B be two fixed non-empty subsets of a set X. Let T be the collection of all subsets of X that are empty, contain A or contain B. Is T a topology for X? If not, would requiring A and B to be disjoint make a difference? If not, would requiring A and B to have nonempty intersection make a difference?

 

Test #2

 

1.) Define the following terms.

a.) Homeomorphism b.) One-to-one function

c.) Topological property d.) Compact

e.) Hausdorff f.) Regular space

g.) T3 h.) T1 i.) T0 j.) converges

2.) Prove the following.

a.) The property of being Hausdorff is a topological property.

b.) Limits are unique in Hausdorff spaces.

c.) The continuous image of a compact space is compact.

d.) The real line with the usual topology is Hausdorff.

e.) The real line with the usual topology is not compact.

3.) Find a base for the discrete topology on R.

1.) Prove the following.

a.) Any topological space is homeomorphic to itself.

b.) If f is 1-1 and onto the f(Ac) = f(A)c.

c.) Let B = {(x - 1/n, x + 1/n) x R, n a positive integer} is a base for the usual topology on R.

d.) Let A be a fixed subset of X, where X has a topology, T. Prove that the collection TA = {U A U T} is a topology for A.

e.) Prove that the topology on the integers induced by the usual interval topology is the discrete topology.

f.) If A is a closed subset of a normal space, X, then the induced topology on A is normal.

g.) The cofinite topology is compact.

2.) Let X = {a, b, c}, T = {, X, {a}, {b}, {a, b}}, Y = {x, y}, S = {Y, , {x}}. Define f from X to Y by f(a) = f(b) = x and f(c) = y. Is f continuous? Is f open? Is f a homeomorphism?

 

Final Exam

 

A.) Determine if the following are topologies for the given sets. Prove your answer.

1.) X = R, T = {(a, b) a, b R}

2.) X, T = {U X U = or A U, where A is a fixed subset of X}

B.) Prove the following.

1.) Let f be a function from the metric space (X, d) to the metric space (Y, p). If f is continuous then {f(xn )} converges to f(x) whenever {xn} converges to x.

2.) Let X be a set. Find a base for the indiscrete topology on X. Prove that it is a base.

C.) True-False

1.) In a topological space, (X, T), all subsets of X are either open or closed.

2.) In a topological space, (X, T), open sets can also be closed.

3.) In a topological space, the closure of a set is closed.

4.) Every non-empty topological space has at least 2 distinct open sets.

5.) In a topological space, (X, T), any intersection of closed sets is closed.

6.) In a topological space, (X, T), a union of open sets is open.

7.) In a topological space, a union of closed sets can be open.

8.) For any set X, there is a metric.

9.) Two different metrics on same set give different metric tops.

10.) Every metric space is normal.

11.) Every homeomorphic image of a metric space is a metric space.

12.) Every Cauchy sequence converges.

13.) Every convergent sequence in a metric space is Cauchy.

14.) Complete implies compact.

15.) In a metric space, the intersection of open sets is an open set.

16.) Completeness is a topological property.

17.) The indiscrete topology on a space with more than one point is not Hausdorff.

18.) The discrete topology is Hausdorff.

19.) T4 implies regular.

20.) Singletons are closed in a T0 space.

21.) Subspaces of regular spaces are regular.

22.) Normal spaces are regular.

23.) Normal implies T0.

24.) Any finite topological space is compact.

25.) Any compact space is finite.

26.) Compact implies connected.

27.) Connected implies compact.

28.) The union of two compact subspaces of a space is compact.

29.) The union of compact subspaces of a space is compact.

30.) If every proper subspace of a space is connected then the space is connected.

31.) If a space is connected then every proper subspace is connected.

32.) The property of being second countable is a topological property.

33.) The real line with the right order topology is Hausdorff.

34.) Singletons are closed in a T1 space.

35.) The discrete topology is metrizable.

36.) Every metric space is Hausdorff.

37.) Closed subsets of a compact space are compact.

38.) The indiscrete topology is not metrizable.

39.) Compactness is a topological property.

40.) Any space with the cofinite topology is compact if and only if the space is finite.

41.) The real line with the right order topology is T4.

42.) The topology on the integers induced by the usual topology is the discrete topology.

43.) If f is a homeomorphism of (X, T) onto (Y, R) and g is a homeomorphism of (Y, R) onto (Z, S), then gf is a homeomorphism.

44.) Let f be an open function from (X, T) onto (Y, S). If X is connected then Y is connected.

45.) Let f be an open function from (X, T) onto (Y, S). If Y is connected then X is connected.

46.) Let T and S be two topologies on a set X. Then the union of T and S is a topology on X.

47.) The real line with the usual topology is Hausdorff.

48.) The real line with the usual topology is not compact.

49.) Subspaces of compact spaces are compact.

50.) Let d((x1, y1), (x2, y2)) = max{x1 - x2, y1 - y2}. Then d is a metric for R2.

51.) Limits are unique in Hausdorff spaces.

52.) Let X be an infinite set with the cofinite topology. There are proper dense subsets.

53.) Let f be a function from a topological space (X, T) to a set Y with the indiscrete topology. Then f is continuous.

54.) The set of rationals is dense in the usual topology on the real line.

55.) Let X be a finite set. The discrete topology and the cofinite topology are the same.

56.) Every metric space is connected.

57.) Second countable implies separable.

58.) Separable implies second countable.

59.) Any topological space is homeomorphic to itself.

60.) Any continuous mapping from a metric space to itself is a homeomorphism.

61.) Every metric space is complete.

62.) It is possible for a space to have two different topologies on it, one of which is compact and the other not compact.

63.) In a T1 space, there are always sequences that converge to more than one point.

64.) If X is an infinite set, then T = {U X U = or U is infinite} is a topology for X.

65.) A subspace of a T2 space is T2.

66.) A subspace of a normal space is normal.

67.) Connectedness is a topological property.

68.) Not every topological space is metrizable.

69.) If a space, X, has more than one point and has the discrete topology there are no proper dense subsets.

70.) A space is disconnected if there exists a set that is both open and closed.