# Math 522 homework

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## Weeks 3-4 (Due Tuesday, September 22)

- (5.2) Prove the well-ordering principle: If $A$ is any set, then there exists a binary relation $\leq$ which is a well-order of $A$. [Use Zorn’s Lemma!]
- (5.3) If $A$ is an infinite set, show that $\abs{A}$ is equal to $\aleph_\alpha$ for some ordinal $\alpha$.
- (6.2) Give an example of an open subset of $\omega^\omega$ which is not closed.
- (6.3) Show that $\omega^\omega$ is homeomorphic to its product with itself $\omega^\omega\times\omega^\omega$.

## Week 2 (Due Thursday, September 10)

- (3.2) Prove that the properties (a)–(c) of a measure imply
*continuity from below*: if $A_n$ is an increasing sequence of sets and $A=\bigcup A_n$, then $m(A)=\sup m(A_n)$. Then prove *continuity from above*: if $A_n$ is a decreasing sequence of sets, $m(A_n)$ is finite, and $A=\bigcap A_n$, then $m(A)=\inf m(A_n)$.
- (3.4) Prove directly from the definition of null set that the null sets are closed under countable unions. (The definition of $A$ is null: for all $\epsilon>0$ there exist intervals $I_n$ such that $A\subset\bigcup I_n$ and $\sum l(I_n)\lt\epsilon$.)
- (4.1) Show that the following sets are all in bijection with one another: $\mathbb R$, $(0,1)$, $(0,\infty)$, $\mathcal P(\mathbb N)$, and $\set{A\in P(\mathbb N)\mid A\text{ is infinite}}$.
- (4.2) Which of the following categories satisfy the analog of the Cantor–Schroder–Bernstein theorem? (That is, monomorphisms $A\to B\to A$ implies isomorphism $A\cong B$.) linear orders with order-preserving maps; groups with group homomorphisms; topological spaces with continuous maps; topological spaces with piecewise continuous maps.

## Week 1 (Due Tuesday, September 1)

- (1.1) With the definition of $C+C’$ for Dedekind cuts, show that addition is commutative and associative.
- (1.4) Show that any two complete ordered fields are isomorphic as ordered fields. [Hint: observe that both must contain a copy of $\mathbb Q$ which is dense.]
- (2.1) Compute the sum of the lengths of all of the intervals removed from $[0,1]$ in the construction of the Cantor set. What if some fraction other than $1/3$ is removed at each stage?
- (2.2) Prove proposition 2.4 in the notes: $A$ is nowhere dense iff $\bar A$ contains no intervals of positive-length iff $A$ is non-dense in every open set.