525 Assignments (Fall 2017)

  1. p.38 #19 -- "get-acquainted special"

  2. p.40 #2.(b)

  3. p.136 #4

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

  5. 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.

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

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

  8. 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 2.  Determine when a closed graph implies continuity.

  9. p.142 #19

  10. p.170 #20

  11. 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 3.  Prove the long line is locally Euclidean.

  12. 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!)

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

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

  15. p.170 #23

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

  17. p.174 #6

  18. p.192 #5

  19. p.193 #17

  20. p.194 #10

  21. p.195 #5

  22. p.196 #7

  23. Determine whether the line with two origins is path-connected by arguing directly from the definition. If not path-connected, determine the path-components.

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

  25. p.228 #9

  26. p.228 #5 as modified

  27. p.230 #15

  28. p.260 #7

  29. p.261 #4

  30. p.262 #6 -- due Wed 13 Dec by 4:00 pm

All rewrites of ungraded assignments
and any bonus problems
are due by 4:00 pm Mon 11 Dec.

Assignments from Fall '12 and prior.

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