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Midterm Examination Maths 7510 Tues., 14 Oct., 2008
Read each question carefully. Organize your answers clearly. Write your answers using
complete sentences. Write solutions on the right hand pages of the blue book.
Problem 1. [16 pts] Let {xn } be a sequence of points in a topological space X.
(1) Define: limit of the sequence {xn }. Ans:
An element x is a limit point provided
for every open set U containing x all but a finite number of {xn } are in U .
(2) State a condition and prove a result that guarantees that the limit of the sequence
{xn } is unique. Ans:
If X is Hausdorff then the limit of a sequence is unique. (If
x and y are distinct limits, then invoke the Hausdorff property to choose respective
open sets separating the two point. Choose the maximum of the respective integers
N and M such that all xi are in the respective open sets. All these points are now
in the intersection which is empty, contradicting the inequality of the limits.
Problem 2. [16 pts]
(1) Given a collection {Tα }, of topologies on X, show that the intersection is a topology
on X. Ans:
Yes. The four properties for a topology follow in a straight forward
manner, e.g. the empty set is in any topology hence in the common intersection.
(2) Is the union of a collection of topologies also a topology on X? (Justify your
answer.) Ans:
No. Consider X = {a, b, c}, T1 = {φ, {a}, , X}, T2 = {φ, {c}, X}.
The union of the topologies is not closed under the (arbitrary) union of a collection
of open sets.
Problem 3. [16 pts]
(1) Show that the diagonal, ∆ = {(x, x) : x ∈ X}, is closed in X × S if and only if
X is Hausdorff. Ans:
If the diagonal is closed and (x, y) : x 6= y is not on the
diagonal, there is a basic open set U × V about (x, y) in the complement of the
diagonal. Then U, V form a disjoint (U × V misses the diagonal) open set pair
separating x and y, and X is Haudorff. Reverse the procedure if X is Hausdorf.
(2) Let f : Z → X1 × X2 . Show that f is continuous if and only if all the coordinate
functions fi := πi ◦ f are continuous. Ans: The projections are continuous hence,
if f is continuous, so are the composites with the projections. Conversely, if the
coordinate functions are continuous, then the inverse image of a basic open set,
U × V in the product, is the intersection of the inverse images under the coordinate
functions (which is therefore open.)
Problem 4. [18 pts]
(1) State the definition of quotient map. Ans:
A map p : X → Y is an quotient
map provided U is open in Y iff p−1 (U ) is open in X. (or the equivalent definition
using saturated open sets in X.)
(2) Prove that the composition of two quotient maps is a quotient map. Ans:
Easy
check using (q ◦ p)−1 (U ) = p)−1 (q −1 (U )).
(3) Give an example of a quotient map which is not open. Ans:
See example one,
p. 137.
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Problem 5. [16 pts] Consider the topological space X := [0, 1] × [0, 1] with the lexicographic order and provided with the order topology.
(1) Describe a simple basis for the subspace topology on [0, 1] × {0}. Ans:
The
open intervals (a × c, b × d) in the order topology intersect to give intervals of the
form (a, b] on the subspace [0, 1] × {0}. In addition, there is a minimal element
0 × 0 and a maximal element 1 × 1 for which the corresponding basis elements have
intersections of the form {0} × 0, [0, b), (a, 1], [0, 1]. Hence a basis for the topology
on the unit square induces the upper limit topology on the interval.
(2) Is X connected? (Justify your answer.) Ans:
Yes, it is a linear continuum (state
the conditions necessary) and apply the theorem 24.1 (state the theorem.)
Problem 6. [16 pts]
(1) If A is connected and f : A → B is a continuous function, prove that f (A) is
connected. Ans:
If U and V form an open partition of f (A) then the inverse
images form a partition of the connected set A which is connected. Hence the
partition of A is trivial and the original partition is not a separation.
(2) If the cartesian product, X × Y , is connected, prove that X and Y are connected.
Ans:
The projections are continuous with targets the spaces we wish to show
connected. Apply the first part of the question.
1. Bonus Question
Problem 7. [10 pts] Let X be the real numbers with the finite complement topology.
Find the closure of the integers. Ans:
The closure of the integers is X, the only infinite
closed set in the finite complement topology.