**Topology**

**Test File**

**Fall 1998**

**Test #1 **

1.) Define the following terms.

a.) Topological space b.) L.U.B.
of A Í 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 Í R
½
{2, 4} Í U}

b.) X is an infinite set, T = {U Í X
½ U
= X or U is finite}

3.) Prove the following.

a.) Let (X, T) be a topological space. If A, B Í X then A Ç B Ê A
Ç
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, U_{x}, such that x Î U_{x}
Í 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 Í X ½ U
= Æ or A Í 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 È
(V Ç A) ½ U, V Î
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) È {7}

(iii) {rationals}

(iv) Æ

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.) T_{3 }h.) T_{1 }i.) T_{0 }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(A^{c})
= f(A)^{c}.

c.) Let **B** = {(x - 1/n, x + 1/n) ½ x
Î
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 T_{A} = {U Ç A ½ U
Î
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 = {Æ,
X, {a}, {b}, {a, b}}, Y = {x, y}, S =
{Y, Æ, {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) ½
a, b Î R}

2.) X, T = {U Í X ½ U
= Æ or A Í 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(x_{n })} converges to f(x) whenever
{x_{n}} 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.) T_{4} implies regular.

20.) Singletons are closed in a T_{0}
space.

21.) Subspaces of regular spaces are regular.

22.) Normal spaces are regular.

23.) Normal implies T_{0}.

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 T_{1}
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 T_{4}.

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 g‘f 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((x_{1}, y_{1}), (x_{2},
y_{2})) = max{½x_{1} - x_{2}½, ½y_{1} - y_{2}½}. Then d is a metric for R^{2}.

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 T_{1} space, there are always
sequences that converge to more than one point.

64.) If X is an infinite set, then T = {U Í X ½ U
= Æ or U is infinite} is a topology for X.

65.) A subspace of a T_{2} space is T_{2}.

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.