### 725 Assignments (Fall 2010)

1. p.38 #19

2. p.40 #2.(b)

3. p.41 #5

4. p.136 #4

5. Fourth Ex on p.3 of the Nets and Filters handout.

6. Third Ex on p.4 of the Nets and Filters handout.

BONUS  Discuss the composition of associations (or derivations): filter to nets to filters, and net to filter to nets.

7. Ex on p.6 of the Nets and Filters handout.

8. p.137 #7 and p.138 #8 as modified

9. Refer to p.141 #9.
1. Give an example to show that must be Hausdorff for the claim to hold.
2. Prove the modified claim.
3. Prove or construct a counterexample: if the graph is closed, the function is continuous.

BONUS  Determine when a closed graph implies continuity.

10. p.142 #19

11. p.170 #20

12. A topological space is said to be locally Euclidean if and only if every point in has a neighborhood homeomorphic to an open set in some Euclidean space.
1. Check each separation axiom (= 0,1,2,3,4,5) for a locally Euclidean space.
2. Prove the line with two origins is locally Euclidean.

BONUS  Prove the long line is locally Euclidean.

13. 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.)

14. By identifying opposite edges of a rectangle (or square), one obtains three spaces. One is the torus; determine the other two.

15. Is a locally Euclidean space first countable? Second countable? Separable?

16. p.170 #23

17. Express the real projective plane as D² attached to S¹.

18. p.174 #6

19. p.192 #5

20. p.193 #17

21. p.194 #10

22. p.195 #5

23. p.196 #7

24. Determine whether (a) the line with two origins and (b) Sierpinski space are path-connected by arguing directly from the definition in each case. If not path-connected, determine the path-components.

25. A compact subset of a Hausdorff space is closed, and a closed subset of a compact space is compact. Show that neither holds conversely.

26. p.228 #9

27. p.228 #5.(b)

These last three are due by 4:30 pm Wed 15 Dec.

28. p.260 #7

29. p.261 #4

30. p.262 #6

All rewrites of ungraded assignments
and any bonus problems
are due by 3:00 pm Mon 13 Dec.

Assignments from Fall '08 and prior.

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