18.901 INTRODUCTION TO TOPOLOGY

tue, thu 9:30-11

Text: James Munkres: Topology (Second Edition)

Lecturer: George Lusztig

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

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 (omit 27.5,27.6,27.7,27.8).

Also revise the homework exercises.

Here are some sample questions for quiz 1 (there will be six questions).

(1) Let X be a subspace of R (real numbers with standard topology). Show that X is compact if and only if it is closed and bounded. (20 points)

(2) Show that a closed subspace of a compact space is compact.

(3) Let X be a compact topological space and let f:X\to R\$ be a continuous function. Show that there exists c\in X such that f(x) is less than or equal to f(c) for all x\in X.

(4) What is a basis for a topology?

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

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

Homework 3 (due 02/26) p.152 #2,4,9.

Homework 4 (due 03/5) p.157 #1(a), 1(c); p.171 #3,5; p.172 #12.

Homework 5 (due 3/12) p.178 #6(b),(c),(d),(e) ("isolated point" is defined on p.176).

Homework 6 (due 4/02) p.186 #1,8; p.205 #1,2,3. (In Ex.8 on p.186, Z_+ is the subset {1,2,3,...} of the real numbers with the topology induced from the real numbers.) The first quiz will be on Thursday March 14. It will be closed book. It will last one hour.

The second quiz will be on Tuesday April 30. It will be closed book. It will last one hour.