Homework 12. Due Monday 16 November. Problem 7.2.

Homework 11. Due Friday 13 November. Problem 6.9.

Homework 10. Due Wednesday 21 October. Problem 4.14

Homework 9. Due Monday 19 October. Problem 4.12.

Homework 8. Due Wednesday 14 October. Problem 4.11.

Homework 7. Due Friday 9 October.

• Can a finite set have a limit point?
• Give an example of an infinite set with no limit points.
• Give an example of a set S such that S’ = S. Here S’ is the set of limit points of S.
• Problem 4.6. You may wish to use the following outline (filling in carefully all details).
  Suppose \(p\in X \) and for every \( \epsilon >0 \) there is an \(s \in S’\) such that \( 0< d(p,s)<\epsilon.\) For positive integers \(n, \) choose \(s_n \in S'\) such that \(0<d(s_n,p)< \frac 1n. \) Show, for each positive integer \(n,\) there exists \(q_n \in S\) such that \(d(s_n,q_n) < d(s_n,p). \) Conclude \( p \in S' \). Conclude, if \( p\notin S^\prime \) then there is an \( \epsilon>0\) such that \( N_\epsilon(p)\cap S^\prime =\emptyset.\) Conclude \( S^\prime \) is closed.

Homework 6. Due Wednesday 7 October. Exercise 4.2.

Homework 5. Due Friday 18 September.

In the metric space \( \mathbb R^2 \) with the Euclidean distance, show that the set \( E=\{(x_1,x_2): |x_1|\ne |x_2|\} \) is open. Show the set \( F=\{(x_1,x_2): x_1=x_2\} \) is not open. If instead \( \mathbb R^2 \) is give the discrete metric is \( E \) open? Is \( F \) open?

Homework 4. Due Friday 11 September. Problem 2.6.

Homework 3. Due Wednesday 9 September. Problem 2.3.

Homework 2. Due Wednesday 2 September. Problem 1.6. Namely, show that the set \( \mathcal F\) of finite subsets of \( \mathbb N^+ \) is at most countable. If you are looking for a strategy, here is a suggestion: Observe \( \mathcal F = \cup_{n=1}^\infty P(J_n) \) and use Problems 1.4 and 1.5.

Homework 1. Due Monday 31 August. Problem 1.2.