Math 441 Topology Autumn 2002
Assignment #7: Due 11/18/02
I. Reading:
•Read Patty, §2.2.
•Skim Patty, §2.6. (We will not cover §§2.3 through 2.5.)
II. Practice problems:
1. Patty, Exercises 2.1 (pp. 65–67) #1, 2, 6, 7, 16, 17, 21.
III. Required problems:
1. Patty, Exercises 2.1 (pp. 65–67) #4.
2. Patty, Exercises 2.1 (pp. 65–67) #5.
3. Patty, Exercises 2.1 (pp. 65–67) #8.
4. Patty, Exercises 2.1 (pp. 65–67) #9.
5. Patty, Exercises 2.1 (pp. 65–67) #15.
6. Patty, Exercises 2.1 (pp. 65–67) #22.
7. (a) Prove the following generalization of the pasting lemma (Theorem 2.15): Let X
and Ybe topological spaces, let A1,...,A
kbe closed subsets of Xsuch that X=
A1∪···∪Ak, and for each i=1,...,k, let fi:Ai→Ybe a continuous function
such that fi|Ai∩Aj=fj|Ai∩Ajfor each iand j. Then there is a unique continuous
function h:X→Ysuch that h|Ai=fifor each i.
(b) By considering the space X=[0,1] ⊂Rwith the usual topology, and the subspaces
A0={0},ai=[1/(i+1),1/i] for i=1,2,..., show that the previous result is false
if the sets {A1,...,A
k}are replaced by an infinite sequence of closed sets.