**4500.001, Spring 2002**

**Introduction to Topology**

**Conley**

the University of North Texas. |

**INSTRUCTOR:** Charles Conley, GAB 475,
(940)
565-3326

**OFFICE HOURS:** Tuesday and Thursday 1:00-3:00, and
by
appointment

**CLASS MEETS:** TR 11:00-12:20, GAB 204

**EXAMS, HOMEWORK, AND GRADING:** There will be two 100 point
midterms, on Tuesday, Feb. 19 and Tuesday, April 2, and a comprehensive
180 point final
on Tuesday, May 7, 10:30-12:30. There will also be twelve 10
point
homeworks, due Thursdays at the beginning of class. There will be
no
make-up exams except for emergencies, and late homework will be worth
half-credit.

**TEXT AND PREREQUISITES: **The text is *Topology,* second
edition, by James Munkres. The prerequisite is Real Analysis
(Math 2510-20).

**TOPICS:** We will cover most of the first four
chapters of Munkres' book, and as time permits, parts of the fifth,
seventh, and eighth.
We will begin with some of the set theory and logic underlying
topology,
followed by the definition of abstract topological spaces. The
concepts
of continuous functions, compactness, and connectedness will be
central,
and rigorous proofs will be emphasized. Most of our examples will
be
metric spaces, but we will also look at some of the standard
counter-examples associated to the separability axioms. We will
conclude with some discussion
of function spaces, one of the motivations for the original
theory.
The contents of Part~I of the book are given below.
Unfortunately
we will not have time to go into Part~II, which concerns algebraic
topology
and homotopy theory.

*Chapter 1:* Set Theory and Logic

*Chapter 2:* Topological Spaces and Continuous Functions

*Chapter 3:* Connectedness and Compactness

*Chapter 4:* Countability and Separation Axioms

*Chapter 5:* The Tychonoff Theorem

*Chapter 6:* Metrization Theorems and Paracompactness

*Chapter 7:* Complete Metric Spaces and Function Spaces

*Chapter 8:* Baire Spaces and Dimension Theory

FINAL EXAM: Tuesday, May 7,
10:30-12:30

HOMEWORK 12, Due Thursday, May 2:

Section 26: 7

Section 27: 2de

Section 28: 2, 3b, 7a

HOMEWORK 11, Due Thursday, April 25:

Section 26: 4, 5, 6

Section 27: 2abc, 4

HOMEWORK 10, Due Thursday, April 18:

Section 23: 1

Section 24: 1, 2, 3, and the first sentence of 11
for
R^2

Section 25: The second sentence of 1

Section 26: 1a, 2a, 3

HOMEWORK 9, Due Thursday, April 11:

Section 21: 6, 8 for X = Y = R

Section 23: 2, 4 for R_Z, 5, 7, 9, 12

EXAM 2: Tuesday, April 2

HOMEWORK 8, Due Tuesday, April 2:

Section 18: 11 for F: R^2 --> R and 13 for X =
R^2, Y = R

Section 21: 1, 2, and 3a for n = 2

There are also 3 extra problems I handed out on a
worksheet.

HOMEWORK 7, Due Thursday, Mar. 14:

Section 18: 4 for X=Y=R, 5, 7a, 8 for R, 10 for
A=B=C=D=R, 11 for X=Y=Z=R

Section 20: 1a for R^2, 2, 3a

HOMEWORK 6, Due Thursday, Mar. 7:

Section 17: 10 for R, 11 for R x R, 12 for R, 16b
for
R, R_l, and R_Z

Section 18: 1, 2 for f from R^2 to R^2, 6

Note: 5 and 7a in Section 18 are
postponed
to Homework 7.

HOMEWORK 5, Due Thursday, Feb. 28:

Section 17: 2, 3, 4, 6, 9: do these with X and Y
equal
to **R**, the real line. Also do 16a for **R**, **R**_L,
and
**R**_Z, 17 for **R**_L, and 18CD.

Note: problems 10, 11, 12, and 16b
are
postponed until Homework 6.

EXAM 1: Tuesday, Feb. 19.
The review
sheet will be given out on Feb. 12 and Feb. 14 will be a review day.

HOMEWORK 4, Due Thursday, Feb. 14 (due date changed to Feb. 19, the day of the test):

Section 13: 4c, 7 (but skip the second topology),
8

Section 14: 1, 3, 4, 6, 7, 9

HOMEWORK 3, Due Thursday, Feb. 7:

Section 10: 1, 2, 3, 5, 6

Section 13: 1, 3

HOMEWORK 2, Due Thursday, Jan. 31:

Section 6: 1b, 2, 3, 7

Section 7: 1, 3, 5aefij

Section 9: 1, 2b, 3, 8

HOMEWORK 1, Due Thursday, Jan. 24:

Section 1: 1, 2e, 3a, 4d, 7F, 8, 10c

Section 2: 1, 2, 4abcd, 5ab, 6

Section 3: 1, 2, 4, 9, 11, 12, 13

Section 4: 2k, 8b

Section 5: 1, 4ad, 5cd