Download Problem Set #1 - University of Chicago Math

Problem Set # 1 / MATH 26200
Due: Friday, January 13, 2017, at 15:00:00 CST
I.
A. Let N be a non-negative integer
prove
n and
o that the number of topologies on a set of cardinality
2N −1 −1
N is bounded above by max 1, 4
.
B. Prove that the bound is sharp (i.e., it gives the exact number of topologies) if and only if N ≤ 2.
II. Consider a nested collection T of sets which contains the empty set as a member and whose union is
X.
A. Prove that T can be viewed as either a sub-basis or a basis for a topology.
B. Prove that, if T is a finite set, then, in either case, the topology generated by T is T itself.
C. Prove by example that, if T is a infinite set, then, in either case, the topology generated by T
may not be equal to T .
III. Which, if any, among the relations “is finer than”, “is coarser than”, and “is comparable to” forms
an equivalence relations among the class of topologies? For those which do not form an equivalence
relation, which of the three axioms of an equivalence relation do they satisfy?
IV. Complete the proof of Lemma 13.4 of your text by showing that the topologies of R` and RK are not
comparable.
V. Let X be a topological space and let U ⊆ X. Prove that U is open if and only if for each x ∈ U there
is an open subset Ux of X such that x ∈ Ux ⊆ U .
VI. Suppose Tα is a topology on X for each α ∈ J. Define
\
[
T :=
Tα and S :=
Tα .
α∈J
α∈J
A. Prove that T forms a topology on X and that it is the largest topology on X that is contained in
all of the Tα .
B. Give an example to show that S may not be a topology on X.
C. Prove that S forms a sub-basis for a topology on X and that this topology is the smallest topology
on X that contains all of the Tα .
VII. Let X be a set.
A. Prove that if B is a basis for a topology on X, then the topology generated by B is the intersection
of all topologies on X which contain B.
B. Prove that if S is a sub-basis for a topology on X, then the topology generated by S is the
intersection of all topologies on X which contain S.
VIII. Prove that if X is a topological space, Y ⊆ X, and Z ⊆ Y , then the topology that Z inherits as a
subspace of Y is the same as the topology it inherits as a subspace of X.
IX. A map f : X → Y is said to be an open map if f (U ) is open in Y whenever U is open in X. Prove
that πj : X1 × X2 × · · · × XN → Xj , the jth projection map, is an open map for 1 ≤ j ≤ N .
X. Prove that the dictionary-order topology on R × R is the same as the product topology on Rd × R,
where Rd denotes R equipped with the discrete topology.
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XI. Define I := [0, 1] ⊆ R. Compare the topologies of I × I when it is equipped with
A. the product topology;
B. the dictionary-order topology; and
C. the subspace topology it inherits from R × R equipped with the dictionary-order topology.
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