Download Topology 640, Midterm exam

March 6, 2006
Topology 640, Midterm exam
(Due on Wednesday, March 22)
1. Let (X, T ) be a topological space. Recall that a subset A ⊆ X is called dense if
A ∩ U 6= ∅ for every non-empty open set U ⊆ X.
(a) Show that A is dense if and only if A = X.
(b) Show that if A1 and A2 are open dense subsets of X then A1 ∩ A2 is also a
dense subset of X.
(c) Show by example that the statement from part (b) above fails if A1 and A2
are not required to be open sets.
2. Show that the product of two metrizable topologies is again metrizable.1
3. Let (X, TX ) be some topological space and let f : X → X be a continuous
function. The fixed-point set of f , denoted by F ix(f ), is defined as
F ix(f ) = {x ∈ X | f (x) = x}
Show that F ix(f ) is a closed set provided X is Hausdorff.
4. Consider R and R2 equipped with the Euclidean topology. Let f : R → R be a
continuous function and let Γf be its graph
Γf = {(x, f (x)) ∈ R2 | x ∈ R} ⊆ R2
Since Γf ⊆ R2 , it becomes a topological space via the subspace topology it inherits from R2 . Show that Γf ∼
= R (recall that ∼
= is the symbol for ‘homeomorphic’).
∞
5. Consider the set R of all sequences of real numbers
R∞ = {(x1 , x2 , x3 , ...) | xi ∈ R}
endowed with the product topology. Let A ⊆ R∞ be the subset of convergent
sequences. Prove or disprove: A is an open set. Hint: The product topology is
metrizable.
6. Let (X, T ) be a topological space and A ⊆ X a subspace of X. We say that A
is a retract of X (or that X retracts onto A) if there is a continuous function
r : X → A such that r(a) = a for all a ∈ A. Such a map r is called a retraction.
(a) Show that every retraction is a quotient map.
(b) Let X = {(x1 , ..., xn ) ∈ Rn |x1 |2 + ... + |xn |2 ≤ 1} ⊆ Rn (with the relative
Euclidean topology) retracts onto A = {(0, 0, ..., 0)}.2
7. Find an example of a function f : R → R which is continuous at precisely one
point. Here both copies of R are given the Euclidean topology.
1Recall
that a topology T on a set X is called metrizable if there is a metric d : X × X → [0, ∞i
whose induced metric topology Td agrees with T .
2In contrast, the boundary ∂X = {(x , ..., x ) ∈ Rn |x |2 + ... + |x |2 = 1} of X in Rn is not
1
n
1
n
a retract of X. This is not so easy to see and is usually the first application one encounters when
studying homotopy and homology groups – two instances of topological invariants.
8. Let (X, TX ) and (Y, TY ) be two topological spaces. Let A be a subspace of X
and let f : A → Y be a continuous function. Moreover, let g, h : A → Y be
continuous function such that g(a) = h(a) for every a ∈ A.
(a) Show that g = h provided Y is Hausdorff.
(b) Show by example that the Hausdorff condition from part (a) cannot be
dropped. In other words, find an example where g = h fails due to a nonHausdorff choice of Y .
9. Let (X, d) be a bounded metric space and let C(X) be the set of all closed subsets
of X. For a given ε > 0 and a closed set A ∈ C(X), let BA (ε) be the open set
[
BA (ε) =
Bx (ε) ⊆ X
a∈A
where as usual Bx (r) = {y ∈ X | d(x, y) < r} (the set BA (ε) should be thought
of as the set of all points in X which are less than ε away from A) .
Let D : C(X) × C(X) → R be the function
D(A1 , A2 ) = inf{ε > 0 | A1 ⊆ BA2 (ε) and A2 ⊆ BA1 (ε)}
Show that D is a metric on C(X) (it is called that Hausdorff metric on C(X)).
10. Let (X, T ) be a first-countable topological space. Prove or disprove the statement: For every point x ∈ X there exists a sequence xn ∈ X − {x} with
limn→∞ xn = x. How about if X is not first-countable?
11. Consider the subspace S 1 = {(x, y) ∈ R2 |x|2 + |y|2 = 1} where R2 is given the
Euclidean topology. Consider the partition P of S 1 given by
P = {(x, y), (−x, −y)} (a, b) ∈ (S 1 )
Show that S 1 /P is homeomorphic to S 1 . The space S 1 /P can geometrically be
thought of as the unit circle where diametrically opposite points get identified.