Rewrite the proof of 2.3.8 using ^c notation.
Do a proof of 2.4.4 based on 2.4.6 and 2.4.7.
Call a topological space a Dutch door space if every subset is the union of an open set and a closed set. Show that this is equivalent to every subset being the intersection of an open set and a closed set.
Exercises 3.1: numbers 1 (d) through (i), 3 (f) through (i), 4 (a), (b), (c).
Show that the sets of the form [a, +infinity) and (-infinity, b) form a subbase (see notes of March 2) for the H topology on R.
Corrections on starred problems in this homework set are due Wednesday, April 25
Corrections on items numbered 3, and on 2 (b) in this homework set can be handed in any time up to the final exam which begins at 9 am Monday, May 7.
Problems no longer eligible for extra credit have lines through them below. This includes all problems to which a fully correct solution has been handed back to the person(s) who got it,
If you can't quite get the solution to an extra credit problem but have some ideas, hand them in for partial credit. I will keep adding to your score as you improve your work on it.
1. [worth 10 points, formerly worth 6 and then 8 points] Let lim inf and lim sup be as in the January 31 notes. Prove that lim inf A is the collection of all elements that are in A_i for all but finitely many indices i , while lim sup A is the collection of all elements that are in A_i for infinitely many i. [Note: This shows the former is a subset of the latter.]
2. [worth 3 points] Give an example where the lim inf is a proper subset of the lim sup .
3. Do 8 on the top half of page 26 [4 points].
4. Do 11 on the top half of page 26 [6 points].
5. Show that the definition of ``countable'' given in the notes for February 7 is equivalent to the definition given in the book. [6 points].
6. Prove that every continuous function from [0, 1] to itself has at least one fixpoint: a point p such that f(p) = p. [10 points]
7. Do ``Digging through diagonals,'' number 7 on the handout on Cantor's proof. [10 points]
8. Prove that a topological space is a Dutch door space if, and only if, every dense subset is open. [12 points]
9. Do problem 9 (Grabbing the Brass Ring) in the Mindscapes. (9 points)
10. Do Personal Perspectives 1. in the photocopied handout of Wednesday, March 21. (worth up to 12 points)
11. Show that if (X, T) is arcwise connected, then for any two points x, y of X there is a U_[0,1]-T-continuous function f : [0, 1] -> X such that f(0) = x , f(1) = y. [5 points; 10 points if you can get f to be one-to-one and show it is one-to-one.]
12. Let (X, T) be a topological space, let D be a dense subset of X and let U be an open subset of X. Prove that the closure of (D intersection U) in X is the same as the closure of U in X. [8 points]
13. Let X be the subset {1-1/n : n in Z^+} union {1} of R and let T be the relative topology U_X. Show that (X, T) is a door space, i.e., every subset of X is either open or closed. [6 points]
14. Give an example of a topological space that is a Dutch door space
but not a door space, and show why it is like this. [6 points]
[Hint: there are some finite topological spaces like this.]