Browse Course Material
Course info.
- Prof. James Munkres
Departments
- Mathematics
As Taught In
- Topology and Geometry
Learning Resource Types
Introduction to topology, assignments.
There are two kinds of assignments: Weekly exercises that are not to be handed in for grading but are intended to prepare students for the exams, and problem sets that are to be handed in and graded.
Problem Sets
The problem sets are assigned from the textbook: Munkres, James R. Topology . 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 28 December 1999. ISBN: 0131816292.
Problem set 0 is a “diagnostic” problem set. It is designed to determine whether you are comfortable enough with the language of set theory to begin the study of topology. We will grade it in class at the second session. lf you miss more than 8 answers, you should probably take another proof-based course before trying this one. The grade on this problem set is intended for advising purposes only.
For problem sets 1-4, the solutions are to be written out carefully and legibly, in good mathematical style. “Careful” has an obvious meaning. “Legible” means to do it in LATEX or (if handwritten) in ink and double-spaced. “Good mathematical style” means the style demanded by editors of math journals. Please read these comments about what the mathematics profession means by “good mathematical style ( PDF ).” Follow them!
Note well: your first written solutions constitute your first draft; this is not acceptable. It will need to be rewritten, to clean up grammar and sentence structure and exposition. Treat it like a paper in a humanities class. (Unless you hand in sloppy papers there too!)
Weekly Exercises
The exercises are assigned from the textbook: Munkres, James R. Topology . 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 28 December 1999. ISBN: 0131816292. Collaboration on the weekly exercises is encouraged; you can learn a good deal from your fellow students. But the work on the problem sets is to be strictly your own. If you can’t do all of a problem, do what you can and write “here’s where I got stuck” “Faking it” is much worse than saying “I couldn’t do it.”
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Math 461 Homework 6 Solutions 10/16/2020 (b)The \=)" direction in the previous part is still true in the box topology: this topology is ner than the product topology which implies the coordinate projections are still continuous in this case and hence the same proof works. The other direction is false as the following counterexample demonstrates ...
Dec 28, 1999 · The problem sets are assigned from the textbook: Munkres, James R. Topology. 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 28 December 1999. ISBN: 0131816292. Problem set 0 is a “diagnostic” problem set. It is designed to determine whether you are comfortable enough with the language of set theory to begin the study of topology.
MATH231BR: ADVANCED ALGEBRAIC TOPOLOGY { HOMEWORK 1 SPRING 2018, HARVARD UNIVERSITY The following problem sheet contains 25 problems. We will drop the ve problems with lowest score, so e ectively you only need to hand in solutions to 20 of these. That is, you should feel free to skip those ve problems you nd too easy/hard/tedious.
MATH GU4053: Introduction to Algebraic Topology Homework 4 Solution Exercise 1.3.24. GivenacoveringspaceactionofagroupGonapath-connected,locally
Homework 2 MTH 869 Algebraic Topology Joshua Ruiter May 3, 2020 Proposition 0.1 (Exercise 1.1.3). Let Xbe a path-connected space. Then ˇ 1(X) is abelian if and only if all basepoint-change homomorphisms h depend only on the endpoints of the path h. Proof. First, suppose that ˇ 1(X) is abelian. Let h 1;h 2: I !X be paths with h 1(0) = h 2(0 ...
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Homework 1 Algebraic Topology Joshua Ruiter February 12, 2018 Note: When unspeci ed, a map is assumed to be continuous. Lemma 0.1 (not assigned, just stated for clarity). Composition of continuous maps is continuous. De nition 0.1. Let X;Y be spaces and f : X!Y be a continuous map. Consider the space (X I) tY, and de ne an equivalence relation ...
MATH GU4053: Introduction to Algebraic Topology Homework 3 Solution Exercise 1.3.5. Let X be the subspace of R2 consisting of the four sides of the square [0;1] [0;1], together with the segments of the vertical lines x = 1 2;1 3;1 4;::: inside the square. Show that for every covering space Xe !X, there is some neighborhood of the
The space [0;1]! with the product topology is metrizable. True We derived a similar result in class for R! with the product topology (see page 125 Theorem 20.5). Any nite T 1-space is discrete. True If you have a nite topological space with n elements, and all (n 1)-element subsets are open, then any subset is open (exercise).
ALGEBRAIC TOPOLOGY HOMEWORK PROBLEMS WINTER QUARTER 2011 3 (12) The Klein bottle KB is the quotient space obtained from the square I2 via the boundary identifications (0,y) ∼ (1,1 − y) and (x,0) ∼ (x,1). Prove that KB is a surface. (13) Let Abe a non-degenerate closed annulus in the plane and define an equivalence