Math 730 homework assignments
2nd homework, due Sept. 1:
- Show that a function is bijective if, and only if, it has a 2-sided inverse.
- 2B1
- 2B2*
- 2D
- 2E1
- 2E2*
3rd homework, due Sept. 8:
- Show that every isometry is continuous. [4 points]
- Show that if || . || is an F-pseudonorm on a vector space V, then
d(x, y) = || x - y || defines a metric on V. [8 points]
- 2C1 [6 points]
- 2C2 [8 points]
- 2J2* [12 points]
4th homework, due Sept. 15:
-
Show that a set in a pseudometric space is open iff it is a union of open disks. You may assume 2D. [4 points]
- 2F2* [12 points]
- Finish the proof of Theorem 3.4, beginning with "Conversely,..." [8 points]
5th homework, due Sept. 22:
-
* Show that the complements of "closed" sets (A = A^) in a Cech closure space constitute a topology. [12 points]
- 3A1 [8 points]
- 3E2 [8 points]
- 3E3 [6 points]
6th homework, due Sept. 29:
-
- 3A2: "Describe" means "give explicit formulas" [8 points]
- 3C: You may use R^2 instead of R [8 points]
- 4A3: proofs optional [4 points]
- 4B1 [8 points]
7th homework, due Oct. 6:
-
- Let B be a base for a topology on a space X. Show that a function from
a space Y to X is continuous if, and only if, the preimage of every member
B of B is open in Y. [6 points]
- 6A1 [8 points]
- 6A2 [8 points]
- 6C* [8 points]
9th homework, due Oct. 20:
-
- 6B3 [4 points]
- 7B [4 points]
- 8D1, [6 points]
- 8D2, [6 points]
10th homework, due Oct. 27:
-
- 8D3* [8 points]
- 8H1 [6 points]
11th homework, due Nov. 3:
-
- 9B1 [4 points]
- 9B2 [6 points]
- 9C2 [2 points]
- 9H1 [10 points]
12th homework, due Nov. 10:
-
- 13B1 [4 points]
- 13B2 [8 points]
13th and 14th homework, due Nov 24:
-
- 13D1 [4 points]
- 13D3* [6 points]
- 13E2 [4 points]
- 13E4* [8 points, omit part in parentheses]
- 13E5 [4 points, result 3 only]