18.901 INTRODUCTION TO TOPOLOGY

tue, thu 2:30-4, room 4-257.

Text: James Munkres: Topology (Second Edition)

Lecturer: George Lusztig

Office hour: tuesdays 1:30-2:30; thursdays 4-4:30 (room 2-365).

Grader: Robin Elliott .

There will be no final exam. There will be two quizzes.

50%/25%/25% of the final grade will come from the homework/quiz 1/quiz 2.

Sections to be covered:

1,2,3 (omit order relation),12,13 (omit 13.3 and page 82), 15,16 (omit 16.4), 17,18,19 (omit 19.1, box topology),20,21,23,24,25 (omit 25.3,25.4,25.5), 26, 27, 29. 31,32,33,34,35,43,44,45(omit 45.2-45.5), 48(omit 48.4), 51,52,53,54,56,59.

For the first quiz you should know all definitions and proofs that are not more than 15 lines in the following sections

1,2,3 (omit order relation),12,13 (omit 13.3 and page 82), 15,16 (omit 16.4), 17,18,19 (omit 19.1, box topology),20,21,23,24,25 (omit 25.3,25.4,25.5), 26, 27, 29.

Also revise the homework exercises.

Homework 1 (due 09/14) p.14,15 #2,7,8; p.20 #2; p.100 #2,3; p.101 #11; p.112 #11,12.

Homework 2 (due 09/21) p.111 #5; p.112 #13; p.126 #1(a),3; p.133 #1,3(a).

Homework 3 (due 09/28) p.152 #2,4,9.

Homework 4 (due 10/5) p.157 #1(a), 1(c); p.171 #3,5; p.172 #12.

Homework 5 (due 10/12) p.178 #6(b),(c),(d),(e) ("isolated point" is defined on p.176).

Homework 6 (due 10/24(!)) p.186 #1,8; p.205 #1,2,3.

Homework 7. (due 10/31) p.212,#1,#3. (In #3, use definition and properties of d(x,A) on page 175.)

Homework 8. (due 11/07) p.223,#2,#4,#5.

Homework 9. (due 11/14) p.270, #4,#5,p.271 #10.

Homework 10.(due 11/21) p.298 #1, p.299 #3.

Homework 11. (due 11/30!) p.334 #1,#2, p.335 #4,#5.

The first quiz (lasting one hour) was on Tuesday October 17.

The second quiz (lasting one hour) will be on Tuesday November 28.

Possible questions for second quiz. Additional questions may be related to problems in Homework 7,8,9.

1) Show that a subspace of a regular space is regular (See 31.2(b))

2) Show that a product of regular spaces is regular (See 31.2(b))

3) Show that a metrizable space is normal (See 32.2)

4) Show that a compact regular space is normal (See 32.3)

5) Let A be a closed subset of a normal space X. Assuming Tietze's theorem for maps into a closed interval prove Tietze's theorem for maps into R. (See 35.1(b))

6) Prove the existence of a finite partition of unity for a finite open covering of a normal space (See 36.1)

7) Let X be a metric space in which every Cauchy sequence has a convergent subsequence. Show that X is complete. (See 43.1)

8) Show that the euclidean space R^n is complete for the euclidean metric. (See 43.2)

9) Let Y be a complete matric space with distance d such that d(y,y') is less than or equal to 1 for any y,y'. Let X be a set and let Y^X be the set of all maps from X to Y. Show that Y^X has a natural structure of complete metric space. (See 43.5)

10) Draw the first three maps from [0,1] to [0,1]^2 in the sequence of maps used in class to construct Peano's curve. (See p.272,273).

11) Let X be a totally bounded complete metric space. Show that any sequence in X has a convergent subsequence (See 45.1)

12) Show that any compact Hausdorff space is a Baire space. (See 48.2)
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