Download Topology Homework 2

Topology
Homework 2
Instructor: W. D. Gillam
Due: October 22, start of class
Instructions. Write your name and “Topology Homework 2” clearly in the upper right
corner. Clearly label the problem numbers. Use a stapler. The bold number next to each
problem indicates its difficulty on a scale of 1 to 3, with 1 being the easiest.
(1) (3) We observed in class that if A is any set then the powerset P(A) (the set of all
subsets of A) is partially ordered by the relation ⊆. So: if S ⊆ P(A) is any subset
(i.e. any family of subsets of A), then (S, ⊆) is a partially ordered set. Show that
every partially ordered set (P, ≤) is isomorphic to one of this form. Hint: You can
take A = P . What is the most obvious subset of P that you might reasonably
associate to an element of p by making using of the ordering relation ≤?
(2) (3) Recall from class that a set V is called a von Neumann ordinal (for the
Hungarian-American mathematician-physicist John (Hungarian: János) von Neumann whom you probably know best for the important role he played in the
development of both the atomic bomb and the computer) iff V is well-ordered by
∈ (used as <) and whenever A ∈ V , A ⊆ V . In the second lecture I hurriedly (and
not very clearly) gave an argument for the following simple fact: Suppose V is a
von Neumann ordinal with maximum α. Then α = V \ {α}. Give a careful proof
of this and conclude that if V and V 0 are von Neumann ordinals with maximum
elements α, α0 (respectively), and V \ {α} = V 0 \ {α0 } then α = α0 (i.e. V = V 0 ).
If you want another exercise with transfinite induction, prove directly that any
two isomorphic (as posets) von Neumann ordinals are equal. (This follows from
the theorem on von Neumann ordinals we proved in class, though you can give a
direct proof where the only remotely interesting argument is the first part of this
exercise.)
Clarification: The topology on R where the basic open sets are the half-open intervals
[a, b) (for real numbers a < b) is called the lower limit topology, not the “K-topology,”
as I mistakenly suggested in class. (This is Munkres’s terminology—it is by no means
standard.) The topological space R, with the lower limit topology, is denoted R` in your
textbook, and is usually called the Sorgenfrey line. (That terminology is fairly standard.)
By definition, the K-topology is the smallest topology (c.f. Exercise 4) on R containing
the open intervals (a, b), as well as each set (a, b) \ K (where K := {1, 1/2, 1/3, . . . }).
(3) (2,3) Prove that the K-topology on R is incomparable to the lower-limit topology
on R (i.e. neither of these two topologies refines the other). More challenging (at
this point): prove that the Sorgenfrey line R` is not homeomorphic to the usual
real line.
(4) (1) Show that the intersection of any family of topologies (on a fixed set X) is
again a topology. Conclude that for any subset A ⊆ P(X) there is a smallest
topology containing A.
(5) (3) Let S := {0, 1} with the topology σ := {∅, {1}, S}. (Some people call this
the “Sierpinski space.”) Let (X, τ ) be any topological space. Show that there is
a bijection between τ and the set of continuous functions (X, τ ) → (S, σ). If you
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are a graduate student show that this bijection is (really: “can be chosen to be”)
natural in (X, τ ) (in some sense that you will explain...).
(6) (2) Section 13 Exercise 7 in the textbook
(7) (1) Let X be a topological space, A ⊆ X any subset. Show that A◦ = X \ (X \ A),
where A◦ denotes the interior of A and A denotes the closure of A in X.