Download Math 3390 Introduction to topology, Assignment 2. Due October 26

Math 3390
Introduction to topology, Assignment 2.
Due October 26.
Question 1: Suppose that f : R → R is continuous. Show that the − δ definition of continuity
of f is equivalent to our definition of continuity in terms of local bases.
Question 2: Fix a set X. Let X0 denote the set X equipped with a topology τ0 , and let X1 denote
the set X equipped with a topology τ1 . Consider the identity map id : X0 → X1 .
1. Show that id is continuous if and only if τ1 ⊂ τ0 .
2. Show that id is a homeomorphism if and only if τ0 = τ1 .
Question 3: By the Baire category theorem, R cannot be written as a countable union of closed
sets having empty interior. Show that “closed” is necessary in this statement, by expressing R as a
countable union of sets (not necessarily closed) having empty interior.
Question 4: Given a family {Xi }i∈I of topological spaces, prove that the projection maps
Y
Xi → Xi
pi :
i∈I
are open maps.
Question 5: Consider the space X = [0, 1]/ ∼, where ∼ is the relation whose equivalence classes
are all singletons, except for the equivalence class {0, 1}. (So 0 is identified with 1, and there are no
other identifications). Show that X is homeomorphic to the subspace {(x, y) | x2 + y 2 = 1} of R2 .
Question 6: Quotient maps need not be open maps or closed maps. Here is an example that
shows this fact: Let p1 : R × R → R denote the projection onto the first coordinate. Let A be the
sucspace of R × R consisting of (x, y) with either x ≥ 0 or y = 0, or both. Let q : A → R denote
the restriction of p1 to the set A. Show that q is a quotient map which is neither open nor closed.
Question 7: Set X = {0, 1}. List all the topologies on the set X, and show that for every one of
them arises as a quotient of Q (here, Q ⊂ R is given the subspace topology).
Question 8: Consider the following theorem and proof by user Andrey Gogolev on Mathoverflow:
Theorem: Suppose that f : R → R is an infinitely differentiable continuous function. Assume
that for each x ∈ R there exists n ≥ 0 such that f (n) (x) = 0. Then f is a polynomial.
Proof: Assume f is not a polynomial. Consider the closed sets
Sn = {x ∈ R | f (n) (x) = 0}
and
X = {x ∈ R | for every interval (a, b) containing x, the restriction f |(a,b) is not a polynomial.}
2
We equip the set X with the subspace topology, it is non-empty and has no isolated points. Then
apply the Baire category theorem to X, which is equal to the union
X=
∞
[
(X ∩ Sn )
n=1
The Baire category theorem says, then, that there exists an interval (a, b) and n ≥ 0 so that (a, b)∩X
is nonempty and (a, b) ∩ X ⊂ Sn . For every x ∈ (a, b) ∩ X, since it is an accumulation point of X
we also have x ∈ Sm for all m ≥ n.
The set ((a, b) \ X) is open so it is a union of intervals, choose a maximal one (c, e). Then f
must be a polynomial of some degree on (c, e), say it is a polynomial of degree d on (c, e). Then
f (d) = constant 6= 0 on [c, e]. So d < n (since either c or e is in X).
Thus f (n) = 0 on (a, b) which contradicts (a, b) ∩ X being nonempty (here we use the fact that
an infinitely differentiable function whose n-th derivative is zero on an interval is a polynomial).
1. Prove that the sets Sn and X are closed, as claimed.
2. Explain in detail why an application of the Baire category theorem gives an interval (a, b) and
an integer n as claimed.
3. Prove that ((a, b) \ X) is open. Explain why this means it is a union of intervals.
Challenge problem (not for credit): Show that if D is a countable dense subset of R, then
there is no function f : R → R which is continuous at each point d ∈ D and discontinuous at each
point x ∈ R \ D. (Sketch: The proof proceeds in two steps. First show that the set C of points
where f is continuous can be expressed as a countable intersection of open sets in R. This can be
done T
by setting Un to be the union of all sets U of R satisfying diameter(f (U )) < 1/n and showing
C = Un . Then show that D cannot
T be expressed as a countable intersection of open sets in R.
This can be done by setting D = Wn where Wn is open, setting Vd = R \ {d} and showing that
both Wn and Vd are dense in R.)
Challenge problem (not for credit): A space is a Baire space if it cannot be written as a
countable union of nowhere dense subsets. Determine whether or not the Sorgenfrey line is a Baire
space.
Challenge problem (not for credit): Let X be a Hausdorff space and ∼ any equivalence relation
on X. When is the quotient X/ ∼ Hausdorff? (There is no easy answer to this! Can you come up
with any conditions on X or ∼ that would make X/ ∼ Hausdorff?)