Bert Mendelson "Introduction to topology" 3rd edition, Dover 1990.

Martin D. Crossley "Essential topology", Springer 2005.

Week

Material covered (with page numbers)

Homework

Aug 21-25

Guest lecturer : Prof Pries

• Topic of her choice

As much as she wants to set.
Due Wednesday 30th August.

Aug 28-Sep 1

Mendelson Chapter 2 : Metric spaces

• Metric spaces (30-33)

Please review Chapter 1 of Mendelson.
Page 6, exercise 2.
Page 9, exercises 2 and 5.
Page 11, exercise 1.
Try to find "natural" functions d:XxX->R which fail to satisfy at least one of the four metric space conditions.
Due Wednesday 6th September.

Sep 4-8
No class Monday
Happy Labor Day!

Chapter 2 : Metric spaces

• Continuity (35-39)
• Open balls and neighbourhoods (40-45)

If (X,d) is a metric space, prove that d'(x,y):=d(x,y)/(1+d(x,y)) also defines a metric on X, but d''(x,y):=(d(x,y))^2 may not be a metric.
Page 34, exercises 4, 8.
Page 39, exercises 1, 3.
Page 45, exercises 6, 8.
Due Wednesday 13th September.

Sep 11-15

Chapter 2 : Metric spaces

• Open sets and closed sets (52-57)
• Equivalence of metric spaces (58-64)
• Topological spaces (71-74)

Page 57, exercises 2, 3, 6.
(A sequence x_1, x_2, x_3, ... of points converges to x if lim d(x,x_n)=0.)
Page 65, exercises 2, 6.
Page 74, exercises 1, 5.
Due Wednesday 20th September.

Sep 18-22

Chapter 3 : Topological spaces

• Neighbourhoods (75-80)
• Closure, interior, boundary (81-86)
• Functions, continuity, homeomorphism (87-91)

Page 86, exercises 1, 2, 5, 7, 8, 12.
Due Wednesday 27th September.

Sep 25-29

Chapter 3 : Topological spaces

• Subspaces (92-96)
• Products (97-100)
• Identification topologies (101-105)

Page 96, exercises 4, 7.
Page 100, exercises 2, 5.
Page 106, exercise 2.
Due Wednesday 4th October.
Practise Midterm Exam: postscript, pdf.

Oct 2-6
Midterm
Friday 6th

Chapter 4 : Connectedness

• Connectedness (113-118)

Page 118, exercises 3, 5.
Page 122, exercises 2, 3.
Due Wednesday 11th October.

Oct 9-13

Chapter 4 : Connectedness

• Connectedness on the real line (119-121)
• Applications (122-128)
• Components (130-133)
• Path-connectedness (133-138)

Page 129, exercises 2, 3.
Page 133, exercises 2, 3, 5.
Page 138, exercise 6.
Due Wednesday 18th October.

Oct 16-20

Chapter 4 : Connectedness

• Homotopic paths (139-149)

• Compact topological spaces (158-164)

Page 150, exercises 4, 5 (see exercise 1 for the definition of f_*).
Page 164, exercises 3, 5.
Page 168, exercise 2.
Page 171, exercise 1.
Due Friday 27th October.

Oct 23-27

Chapter 5 : Compactness

• Compact subsets of the real line (164-167)
• Products of compact spaces (168-171)
• Compact metric spaces (172-178)
• The Bolzano-Weierstrass property (179-184)

Page 178, exercises 1, 2, 4, 5.
Page 185, exercises 1, 6, 9 (we say X satisfies the second axiom of countability if it has a countable basis for open sets).
Due Friday 3rd November.

Oct 30-Nov 3

Crossley Chapter 6 : Homotopy

• Homotopy (91-96)
• Homotopy equivalence (96-101)
• The circle (102-110)

Page 116 (Crossley), exercises 6.1, 6.3, 6.4, 6.5, 6.6.
Due Friday 10th November.

Nov 6-10

Chapter 6 : Homotopy

• Brouwer's fixed-point theorem revisited (110-111)
• Vector fields (112-115)

We will discuss topics for final projects in class.

Nov 13-17

Mendelson Chapter 5

• Section 7 : Surfaces by identification (186-198)

Continue with final project.

Nov 20-24

Thanksgiving

Continue with final project.

Nov 27-Dec 1

Crossley Chapter 7 : The Euler number

• Simplicial complexes (117-120)
• The Euler number (121-123)

Continue with final project.

Dec 4-8

Continue with final project.