MATH 561-TOPOLOGY-FALL 2020-A. FREIRE

Math 467 (Honors Toplogy), fall 2019

Syllabus

General Topology References

COURSE LOG:

W 8/19 (F2F) From text: sectoons 12, 13, 15, 16, 17, 19, 30 (read)
Introduction of problems in problem set 1

PROBLEM SET 1

Lecture 8-19 (scan)

F 8/21(online)  Problems in set 1 (start)

Problem 1.1 (Sam)
Problem 1.2 (Tariq)
Problem 1.1.5 (Kelechi)
Problem 1.3 (Jacob)
Problem 1.4 (Patrick)
3 topologies on the upper half plane (scan)

M 8/24 (online)

Problem 1.5 (Jun)
Problem 1.7.8 (Bryan)
Problem 1.9 (Bernardo)
Problem 1.15 (Matthew)
Also discussed: 14, 17, 18, 22, 23, 24.
Lecture 8-24 (scan)
(includes the example: "Hoelder spaces are not separable")

W 8/26 (F2F)
Discussion of: 16, 19, 20, 21, Hoelder spaces not separable.
Lecture 8-26 scan (5 pages)

F 8/28 (online)
Lecture 8-28 scan (3 pages)
Problem 1.10 (Eli)
Problem 1.11 (Tariq)
Problem 1.12 (Sam)
Problem 1.13 (Kelechi)

Extension to the closure, regular spaces (problem set 2)
PROBLEM SET 2

M 8/31 (online)--read Munkres sections 20, 21, 31, 32, 33
Problem 2.1 (Matthew)
Problem 2.7 (Jun)
Problem 2.8 (Bryan)
Problem 2.9 (Patrick)
Problem 2.10 (Jacob)
Problem 2.2 & remark on 2.8 (scan, 2 pages)

W 9/2 Problems 2.4 (Eli), 2.5 (F)
2.4 (Eli)
Normal spaces: problems 2.14, 2.11, 2.13 (F)
Urysohn's Lemma
Lecture 9-2 (scan, 4 pages)

F 9/4 (online)
2.6 (Larissa)
2.12 (Sam, F)
2.15 (Tariq)
2.16 (F)
Tietze extension (F)[Munkres, no. 35]
Lecture 9-4 (scan) 6 pages

M 9/7 (online) Baire property (#48 in [Munkres])
2.17 (Matthew)
2.18 (Kelechi)
2.19 (Jacob)
metric completion via embeddings
Notes on Baire spaces
Lecture 9-7 scan (3 pages)
PROBLEM SET 3
1-F 2-Bernardo 3-Eli 4-F  5, 5.5, 6, 7: review of compact spaces (open)
8-Patrick 9-Jun 10-Bryan 11-Jacob 12-Larissa 13-Tariq
14-Kelechi 15-Matthew 16-Sam 17-Eli 18, 19, 20-open.

W 9/9 (F2F) Hilbert cube, Urysohn metrization theorem(Munkres, no. 34)
Part I: embeddings into product spaces
Lecture 9-9 scan (4 pages)

F 9/11 (online) Urysohn metrization (Part II)/complete metrizability and G-delta sets
Lecture 9-11 scan (5 pages)
(includes 3 exercises)

M 9/14 (online) Problems from set 3 (students)
Lecture 9-14 (2 pages)
3.3 (Eli)
3.5, 3,6, 3.7 (Jacob)

W 9/16 (online) Compactness vs. sequential compactness
Lecture 9-16 (3 pages)
(Read in [Munkres]: no. 28, no. 45)
3.8 (Patrick)
3.9 (Jun)
3.10 (Bryan)

F 9/18 Problem set 4: Compactness and local compactness-(version: 10/5)
Lecture 9-18 (2 pages)
3.13 (Tariq)
3-14 (Kelechi)

M 9/21 Compactness (cont.)--Problems from set 3
3-15 (Matthew)
3.16 (Sam)
3-17 (Eli)
Lecture 9-21 (2 pages)

W 9/23 (in person)-Tychonoff's theorem
Lecture 9-23 (3 pages)

F 9/25  Spaces of maps of separable metric spaces with compact domain are separable/
Locally compact, connected metric spaces are separable. (Problems 12, 13 on compactness list.)
Lecture 9-25 (3 pages)

M 9/28  Locally compact spaces
Lecture 9-28 (3 pages)
from compactness-loc. compactness list (problem set 4, see 9/18):
6-Bernardo 9-Larissa 14-Sam 15--Kelechi  16-Eli 17-Jun 18-F 19-F 20-Tariq 21-F, 22-F
4-6 (Bernardo)
4-15 (Kelechi)
4-17 (Jun)

W 9/30  Compactifications in general, Stone-Cech (read [Munkres, no. 38])
Lecture 9-30 (4 pages)
Set 5: Notes and problems on compactifications (version: 10/5)

F 10/2 Complete metrizability (4 problems)
(repeats part of 9-11 lecture notes.) Goal: a subset of a complete metric space is
completely metrizable iff it is a G-delta subset.
Ex. 1-Jacob Ex. 2-Patrick Ex.3-Matthew Ex.4-Bryan
Ex.1-(Jacob)
Ex.2 (Patrick)
Ex. 3 (Matthew)
Ex. 4 (Bryan)

M 10/5:proper maps (set 4, problems: 14, 19, 20, 21, 22)
Set 5 assignments:
1-F; 2+3: Eli 4: Bernardo 5: Sam 6: Kelechi 8: Jun 11: Matthew 12:Bryan
Lecture 10-5 (problems 4-19, 4-21, 4-22, 4-23)
4-20 (Tariq)

W 10/7 :Heine-Borel property/ locally compact normed linear spaces are finite-dimensional/
rigidity of compact+Hausdorff
Proper metric spaces (includes 3 problems)
Lecture 10-7 (4 pages)

F 10/9: problems from set 5
Lecture 10-9
5-4 (Bernardo)
5-5 (Sam)
5-8 (Jun)
(problems 5-1, 5-2, 5-3, 5-4, 5-5, 5-8)

M 10/12 problems: 5-6, 5-11, 5-12
from "proper metric spaces" handout: ex.1 (Patrick), ex.2 (Tariq), ex.3 (Jacob)
Lecture 10-12
5-11 (Matthew)
5-12 (Bryan)
Ex.1 (Patrick)
Ex.2 (Tariq)
Ex. 3 (Jacob)

W 10/14 MIDTERM (material up to 10/9)--online.
General Topology Review Map
(Make sure you know the definitions and equivalent conditions for each concept,
and review the proofs of the various implications.)
midterm
solutions

F 10/16  Spaces of maps, Ascoli's theorem [Munkres no. 46+47]
Ascoli-Arzela Notes (includes 14 problems=problem set 6)
Lecture 10-16

Assignments from Ascoli-Arzela notes:
Sam: Ex. 2, Ex. 4 Bernardo: Prob. 3, Prob.4 Eli: Prob. 7, Prob. 8
Kelechi: Prob 9  Patrick: Prob. 10  Jun: Prob. 11

M 10/19 Topologies in spaces of maps [Munkres 46+47]
Lecture 10-19

W 10/21 Stone-Weierstrass theorem (in person)
Problem Set 7
Patrick: 8/18. Matthew 7/7.5/7.7 Tariq: 2 Sam: 5/6 Bryan: 1/17
Eli: 9/10 Jacob: 12, 14 Kelechi: 19 Jun: 21/22 Bernardo: 20/23
Notes on Stone-Weierstrass  (includes proof of Weierstrass approximation, and 3 problems)

F  10/23 Compact-open topology on spaces of maps
problems from Ascoli-Arzela notes
Ex. 2 (Sam)
Prob. 3, 4 (Bernardo)
Prob 8 (Kelechi)
Prob 10 (Patrick)
6-11 (Jun)

M 10/26 connectedness, path connectedness
[Munkres: 23-24-25]
Lecture 10/26
Problem set 8: connectedness
7-1, 7-17 (Bryan)
7-2 (Tariq)
7-5 (Sam)
7-6 (Sam)
7-7, 7-7.5, 7-7.7 (Matthew)
7-8 (Patrick)
7-12, 7-14 (Jacob)
7-18 (Patrick)

W 10/28 connected spaces, path connectedness
Ref: [Munkres, Willard]
Lecture 10/28

F 10/30 Cantor surjectivity theorem
Ref:[ Pugh, Willard]
Lecture 10/30

M 11/2 Problem session (from problem set 7)
Problems: 1, 2, 6, 7, 8, 12

W 11/4 Cantor spaces: Moore-Kline characterization theorem
Ref: [Pugh]
Lecture 11-4
Handuot 9: Cantor set, Cantor spaces (incomplete)

F 11/6 Peano spaces/more problems from set 7
Lecture 11-6
7-20, 7-23 (Bernardo)
7-9 (Eli)
7-10 (Eli)
7-19 (Kelechi)
7-21, 7-22 (Jun)

M 11/9 Quotient topology, quotient maps [Munkres no.22]/characterization results
Lecture 11-9
Topological Uniqueness of Spaces of Reals (paper)

Assignments from problem set 8:
2=Bernardo 6=Sam 9=Tariq 11=Kelechi 12=Matthew 13=Eli 14=Bryan 15=Jun 16=Patrick 17=Jacob
8-2 (Bernardo)
8-6 (Sam)
8-9 (Tariq)
8-11 (Kelechi)
8-12 (Matthew)
8-13 (Eli)
8-14 (Bryan)
8-16 (Patrick)
8-17 (Jacob)

W 11/11 Covering dimension/subsets of euclidean space/read: partition of unity [no.36]
Lecture 11-11

F 11/13 Covering dimension: embedding theorem
Lecture 11-13

M 11/16 Problem session (from problem set 8)
Connectivity of matrix groups
(alternative solution to problem 8-14)

W 11/18 Fundamental group  [Munkres 52,53]
Lecture 11-18

F 11/20 Basic properties of fund. group and free homotopy
Lecture 11-20

M 11/23 Covering spaces and the fundamental group of the circle
Lecture 11-23
Practice problems from [Munkres]:
p. 335: 4, 5/p. 341: 1, 5, 6/p. 347: 7, 8

FINAL EXAM: Friday, December 4, 10:30AM--12:45PM (online)

Included in final: material since 10/16:

Topologies in spaces of maps: Arzela-Ascoli, Stone-Weierstrass
Connected and path-connected spaces
Cantor spaces: Cantor surjectivity thm, Moore-Kline Theorem, Peano spaces
Quotient topology, quotient maps, perfect maps
Covering dimension: subsets of euclidean space, embedding theoorem
Homotopic maps, fundamental group, covering spaces: first definitions and results

final exam     solutions

Included in course (from [Munkres]):
Ch. 2: except 14
Ch. 3: all
Ch. 4: except 36
Ch. 5: all
Ch. 7: all
Ch. 8: except 49
Ch. 9: 51 to 54