# What is Munkres: Definition and 43 Discussions

James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including Topology (an undergraduate-level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology. He is also the author of Elementary Linear Algebra.
Munkres completed his undergraduate education at Nebraska Wesleyan University and received his Ph.D. from the University of Michigan in 1956; his advisor was Edwin E. Moise. Earlier in his career he taught at the University of Michigan and at Princeton University.Among Munkres' contributions to mathematics is the development of what is sometimes called the Munkres assignment algorithm. A significant contribution in topology is his obstruction theory for the smoothing of homeomorphisms. These developments establish a connection between the John Milnor groups of differentiable structures on spheres and the smoothing methods of classical analysis.
He was elected to the 2018 class of fellows of the American Mathematical Society.

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1. ### I Munkres Chapter 5: Problem involving the Tychonoff Theorem

Hi, In Chapter 5 Munkres proves the Tychonoff Theorem and after proving the theorem the first exercise is: Let ##X## be a space. Let ##\mathcal{D}## be a collection of subsets of ##X## that is maximal with respect to finite intersection property (a) Show that ##x\in\overline{D}## for every...
2. ### Munkres Topology Ch. 1 section 2 Ex #1

Mostly I need to clear up a few basic things about functions and their inverses, the problem seems easy enough. Ok, so for (a) I have $$f^{-1}(f(A_0))= \left\{ f^{-1}(f(a)) | a\in A_0\right\}$$ but here I’m not certain if ##f^{-1}## is allowed to be multi-valued or not, the text says that if...
3. ### Munkres Topology ch1 ex #9 - Generalized DeMorgan's Laws

So formulating them was easy, just set ##C:=D\cup E## in (1) and set ##C:=D\cap E## in (2) to see the pattern, if ##\mathfrak{B}## is a non-empty collection of sets, the generalized laws are $$A-\bigcup_{B\in\mathfrak{B}} B = \bigcap_{B\in\mathfrak{B}}(A-B)\quad (3)$$...
4. ### Munkres Topology Ch 1 ex#7) part (c) — basic set theory Q

Obviously the parenthetical part of the definition of ##F## means ##B\subset C## but we are not allowed to use ##\subset##. I do not know how to express implication with only union, intersection, and set minus without the side relation ##B\cap C = B\Leftrightarrow B\subset C##. This is using the...
5. ### MHB Topology Munkres Chapter 1 exercise 2 e- Set theory

Dear Everyone I am having some difficulties on exercise 2e from Topology 2nd ed by J. Munkres . Here are the directions: determine which of the following states are true for all sets A, B, C, and D. If a double implication fails, determine whether one or the other one of the possible...
6. ### MHB Topology Munkres Chapter 1 exercise 2 b and c- Set theory equivalent statements

Dear Every one, I am having some difficulties on exercise 2b and 2c from Topology 2nd ed by J. Munkres . Here are the directions: determine which of the following states are true for all sets $A$, $B$, $C$, and $D$. If a double implication fails, determine whether one or the other one of the...
7. ### I Topology vs Differential Geometry

Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with...
8. ### I Munkres-Analysis on Manifolds: Theorem 20.1

Hello. I am studying Analysis on Manifolds by Munkres. I have a problem with a proof in section 20. It states that: Let A be an n by n matrix. Let h:R^n->R^n be the linear transformation h(x)=A x. Let S be a rectifiable set (the boundary of S BdS has measure 0) in R^n. Then v(h(S))=|detA|v(S)...
9. ### I What chapters of Munkres Topology are essential?

I would like to know which chapters in Munkres Topology textbook are essential for a physicist. My background in topology is limited to the topology in baby Rudin, Kreyzig's functional, and handwavy topology in intro GR books. I feel like the entire book isn't necessary, but I could be mistaken.
10. ### Topology How are Kelley, Dugundji, and Willard compared to Munkres?

Dear all, I recently found the topology textbooks written by Kelley, Dugundji, and Willard, which I heard that they are more concise and motivational than Munkres, which is a required text for my current topology course. I actually do not like Munkres as he is very verbose, and his problems...
11. ### Knot Theory & Topology w/ Munkres & "The Knot Book" by Adams

I will be doing a presentation on some knot theory stuff next semester (graduate seminar), and also studying for our Topology qualifier and taking Algebraic topology. My textbook for topology is Munkres (of course!) and the book I am studying knot theory from is Colin Adams wonderful work "The...
12. ### Analysis Where to Purchase Munkres' Analysis on Manifolds" (Hardcover

Dear Physics Forum friends, I am currently trying to purchase Munkres' Analysis on Manifolds to replace the vector-calculus chapters of Rudin-PMA, which is quite unreadable compared to his excellent chapters 1-8. I know that there is a paperback-edition for Munkres, but I heard that the...
13. ### Covering Maps and Liftings - Munkres Theorem 54.4

I am reading Munkres book Topology. Currently, I am studying Section 54: The Fundamental Group of the Circle and need help with a minor point in the proof of Theorem 54.4 Theorem 54.4 and its proof reads as follows: In the proof we read:"If E is path connected, then, given e_1 \in...
14. ### MHB Finding inverse of F in Munkres' Topology Ch.2 EX 5 pg 106

In Munkres book "Topology" (Second Edition), Munkres proves that a function F is a homeomorphism ... I need help in determining how to find the inverse of F ... so that I feel I have a full understanding of all aspects of the example ... Example 5 reads as follows:Wishing to understand all...
15. ### Interpretation of 'homologous to' - Munkres analysis of homology grou

I am reading James Munkres' book, Elements of Algebraic Topology. Theorem 6.2 on page 35 concerns the homology groups of the 2-dimensional torus. I would appreciate some help with interpreting the term 'homologous to' as it relates to a part of the proof of Munkres' Theorem 6-2 concerning...
16. ### Munkres HW Problem Help Regarding Quotient Spaces and Collapsing

This is how the problem appears in my book(Munkres 2nd edition Topology, sect. 22 pg 145) 6. Recall that R_K denotes the real line in the K-topology. Let Y be the quotient space obtained from R_K by collapsing the set K to a point; let p : R_K → Y be the quotient map. (a) Show that Y...
17. ### MHB Munkres' 'Analysis on Manifolds' Question.

In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads:QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$. Let $M$ be the set of all the points $\mathbf x$ such that $f(\mathbf x)=\mathbf 0$ and $N$ be the set of all the points $\mathbf x$ such...
18. ### Analysis Analysis On Manifolds by James R. Munkres

Author: James R. Munkres Title: Analysis On Manifolds Amazon Link: https://www.amazon.com/dp/0201315963/?tag=pfamazon01-20 Prerequisities: Contents:
19. ### Munkres states that equicontinuity depends on metrics

So munkres states that equicontinuity depends on the metric and not only on the topology. I'm a little confused by this. Is he saying that if we take C(X,Y) where the topology on Y can be generated by metrics d and p, then a set of functions F might be equicontinuous in one and not the other...
20. ### Topology Topology by James Munkres | Prerequisites, Level & TOC

Author: James Munkres Title: Topology Amazon link https://www.amazon.com/dp/0131816292/?tag=pfamazon01-20 Prerequisities: Being acquainted with proofs and rigorous mathematics. An encounter with rigorous calculus or analysis is a plus. Level: Undergrad Table of Contents: Preface A Note...
21. ### Munkres Topology - Chapter 7 - Complete Metric Spaces and Function Spaces

Hello, I was wondering if it was possible (or advisable) to read Chapter 7 of Munkres (Complete Metric Spaces and Function Spaces) without having done Tietze Extension Theorem, the Imbeddings of Manifolds section, the entirety of Chapter 5 (Tychonoff Theorem) and the entirety of Chapter 6...
22. ### Criticize my proof (metric topology, Munkres)

Homework Statement Let ##X## be a metric space with metric ##d##. Show that ##d: X \times X \mapsto \mathbb{R}## is continuous.Homework Equations The Attempt at a Solution Please try to poke holes in my proof, and if it is correct, please let me know if there's any more efficient way to do it...
23. ### Explanation of uniform topology theorem in Munkres

Hi all, I'm looking for some help in understanding one of the theorems stated in section 20 of Munkres. The theorem is as follows: The uniform topology on ##\mathbb{R}^J## (where ##J## is some arbitrary index set) is finer than the product topology and coarser than the box topology; these...
24. ### Topology: Munkres - Urysohn lemma

Hi, the problem I am referencing is section 33 problem 4. Let X be normal. There exists a continuous function f: X -> [0,1] such that fx=0 for x in A and fx >0 for x not in A, if and only if A is a closed G(delta) set in X. My question is about the <= direction. So let B be the...
25. ### Finite Supremum and Infinium Solution for Munkres Text Question 20.7

I can't seem to find a answer for 20.7 anywhere. Unfourtantly, I do not have the skills to latex the problem out, so I only hope someone looks in the book. My solution is that the supremum of the set of a_i 's must be finite above and the infinium is finite and greater then zero , and the b_i...
26. ### How does the second paragraph prove that \tau' is the same as \tau?

Lemma 13.2: Let X be a topological space. Suppose that C is a collection of open sets of X such that for each open set U of X and each x in U, there is an element c of C such that x\in c\subset U. Then C is a basis for the topology of X. Proof: The first paragraph is trivial, it just shows...
27. ### Answer for Munkres Topology Problem 17.18 - Share Your Solution!

Would anyone care to share their answer for problem 17.18 in munkres intro topology book? no need for indepth explanation, just the answer will work (computational problem).
28. ### The meaning of different in Munkres' Topology

The meaning of "different" in Munkres' Topology Hi, I'm working on problem 20.8(b) (page 127f) in Munkres' "Topology", the problem is to show that four topologies are "different". Does different in this context mean that they are unequal - in which case one can contain the other, or...
29. ### Recursive definition from Munkres Topology

Hi all, just joined PF; I have a question about Munkres's Topology (2ed), Section 9 #7(c): Homework Statement Find a sequence A_1, A_2, \ldots of infinite sets, such that for each n \in \mathbb{Z_+}, the set A_{n+1} has greater cardinality than A_n. Homework Equations...
30. ### What is the Acyclic Carrier Theorem and how can it be applied using chain maps?

This is theorem 13.6 in Munkres' Elements of Algebraic Topology. I'm trying to go through this, but I can't prove it. Can someone do this one please? Btw, its "Choose a partial ordering of the vertices of K that induces a linear ordering on the vertices of each simplex of K. Define...
31. ### Question regarding Munkres' proof of the Tychonoff theorem

So, as the title suggests, there's point about the proof of the Tychonoff theorem I don't quite get. The theorem in Munkres is based on the "closed set and finite intersection" formulation of compactness, which, after doing some google-ing, I found out to be a less formal version of the...
32. ### Understanding Equivalence Classes in the Plane

Homework Statement Define two points (x_{0}, y_{0}) and (x_{1}, y_{1}) of the plane to be equivalent if y_{0} - x_{0} ^2 = y_{1} -x_{1}^2. Check that this is an equivalence relation and describe the equivalence classes. Homework Equations The Attempt at a Solution I can...
33. ### Simple Topology problem (Munkres)

Hey guys, I'm reading Munkres book (2nd edition) and am caught on a problem out of Ch. 2. The problem states: If {Ta} is a family of topologies on X, show that (intersection)Ta is a topology on X. Is UTa a topology on X? Sorry for crappy notation; I don't know my way around the symbols...
34. ### Help with 2 problems about compact/pointwise convergence from Munkres - Topology

Hi everyone, I am stuck with 2 problems from Munkres' book and I would appreciate if someone helped me solve them. Thank you in advance. Here they are: 1. Consider the sequence of continuous functions fn : ℝ -> ℝ defined by fn(x) = x/n . In which of the following three topologies does...
35. ### Reading Analysis on Manifolds by Munkres

Show that U(x0, ε) is an open set. I'm reading Analysis on Manifolds by Munkres. This question is in the review on Topology section. And I've just recently been introduced to basic-basic topology from Principles of Mathematical Analysis by Rudin. I'm not really certain where to begin...
36. ### Munkres' Topology: 2nd Edition Spine Misprint?

I have no idea where else to ask this: I have the 2nd edition of the book and I noticed that on the spine of the book it says "Secon Edition" instead of "Second". I'm just wondering if this is on every copy of the second edition of the book? Or, is your copy like this?
37. ### Why Does Munkres Use 3 Delta Instead of Delta in Exercise 22.12?

[SOLVED] Munkres Exercise 22.12 Homework Statement Please stop reading unless you have Munkres topology textbook. Why does Munkres use 3 \delta instead of \delta in part b? I get that replacing the 3 \delta by \delta works fine. Homework Equations The Attempt at a Solution
38. ### Theorem 20.5 in Munkres Topology Book

[SOLVED] munkres question Homework Statement Please stop reading unless you have Munkres topology book. In Theorem 20.5, what is i? Are they assuming the index set omega is countable? Homework Equations The Attempt at a Solution
39. ### Is Munkres Lemma 81.1 Misinterpreted Regarding Normal Subgroups?

[SOLVED] munkres lemma 81.1 Homework Statement Please stop reading this if you do not have Munkres. The statement of this lemma implies that pi_1(B,b_0)/H_0 is a group. That does not make sense to me because H_0 is not necessarily normal in pi_1(B,b_0). Homework Equations The...
40. ### How do you pronounce the name of the author of the popular topology book?

Does anyone know how to pronounce the name of the author of the popular topology book?
41. ### Set theory in Munkres Topology

In Munkres' Topology he defines a Cartesian product AxB to be all (a,b) such that a is in A and b is in B. He says that this is a primative way of looking at things. And then defines it to be {{a},{a,b}} He says that if a = b then {a,b} will just be {a,a} = {a} and therefore will only be...
42. ### Topology by Munkres: Reviews & Resources

I'm planning on buying this book, but since its so expensive I'm looking for as much information as possible on it. So you're input would be nice. Also if you have some pdf files with a chapter or so, that would really help.
43. ### Studying I'm doing EVERY exercise in munkres' topology textbook

i think I've accelerated my learning enough, and now I'm going to start doing problems, problems, and more problems to strengthen my mathematical thinking. this thread will be devoted to munkres' well-used topology textbook. I've done all the problems in chapter 1 so far, and i haven't gotten...