- p.38 #19 -- "get-acquainted special"
- p.40 #2.(b)
- p.136 #4
- Fourth Ex on p.3 of the
*Nets and Filters*handout. - Third Ex on p.4 of the
*Nets and Filters*handout.BONUS 1. Discuss the composition of associations (or derivations): filter to nets to filters, and net to filter to nets.

- Ex on p.6 of the
*Nets and Filters*handout. - p.137 #7 and p.138 #8 as modified.
- Refer to p.141 #9.
- Give an example to show that
*Y*must be Hausdorff for the claim to hold. - Prove the modified claim.
- Prove or construct a counterexample: if the graph is closed, the function is continuous.

BONUS 2. Determine when a closed graph implies continuity.

- Give an example to show that
- p.142 #19
- p.170 #20
- A topological space is said to be
*locally Euclidean*if and only if every point*x*in*X*has a neighborhood homeomorphic to an open set in some Euclidean space.- Check each separation axiom (
*i*= 0,1,2,3,4,5) for a locally Euclidean space. - Prove the line with two origins is locally Euclidean.

BONUS 3. Prove the long line is locally Euclidean.

- Check each separation axiom (
- Express the line with two origins as a quotient space of the disjoint
union of two copies of the Euclidean line. (Be sure to verify that the
topology is correct!)
- By identifying opposite edges of a rectangle (or square), one obtains
three spaces. One is the torus; determine the other two.
- Is a locally Euclidean space first countable? Second countable?
Separable?
- p.170 #23
- Express the real projective plane as D² attached to S¹.
- p.174 #6
- p.192 #5
- p.193 #17
- p.194 #10
- p.195 #5
- p.196 #7
- Determine whether the line with two origins is path-connected by
arguing directly from the definition. If not path-connected, determine the
path-components.
- A compact subset of a Hausdorff space is closed, and a closed
subset of a compact space is compact. Show that neither holds
conversely.
- p.228 #9
- p.228 #5 as modified
- p.230 #15
- p.260 #7
- p.261 #4
- p.262 #6 -- due Wed 13 Dec by 4:00 pm

and any bonus problems

are due by 4:00 pm Mon 11 Dec.