Topology Definition and 800 Threads

Mod note: This thread contains an off-topic discussion from the thread https://www.physicsforums.com/showthread.php?p=4216768 So a notion of distance is used... I wonder how.

The question comes from the Munkres text, p. 133 #3. Let Xn be a metric space with metric dn, for n ε Z+. Part (a) defines a metric by the equation ρ(x,y)=max{d1(x,y),...,dn(x,y)}. Then, the problem askes to show that ρ is a metric for the product space X1 x ... x Xn. When I originally...

which books do you think are good to beginners in topology ? for someone don't know any thing in topology and little set theory ?

To all who have taken an introduction course to topology and abstract algebra, which did you prefer and why? Does the preference of one course over the other reflect a certain from of intuition that we rely on for reasoning or heuristics for problem solving? For these classes, I used...

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...

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...

I am trying to visualize the subsppace topology that is generated when you take the Rationals as a subset of the Reals. So if we have ℝ with the standard topology, open sets in a subspace topology induced by Q would be the intersection of every open set O in ℝ with Q. Since each open set...

I want to say yes. I am having trouble convincing myself though. Can anyone give me a very small nudge in the right direction? Thanks!

Homework Statement Let ##A \subset X##; let ##f:A \mapsto Y## be continuous; let ##Y## be Hausdorff. Show that if ##f## may be extended to a continuous function ##g: \overline{A} \mapsto Y##, then ##g## is uniquely determined by ##f##. Homework Equations The Attempt at a Solution...

I was working on some algebraic topology matters, thinkgs like the connected sum of some surfaces is some other surface. And for this study, I was using the Munkres's famous textbook 'Topology' the algebraic topology part. My qeustions are as follows: Q1) Munkres introduces 'labelling scheme'...

Homework Statement Find the fundamental group of T^{n}, the torus with n holes, by finding the planar representation of T^{n}. Homework Equations I'm just having a hard time finding the planar representation of T^{n}. I can't picture it.The Attempt at a Solution I can see how the picture...

Does anyone have any good reference to exercises concerning these topics? I would like to understand them better. Thank you.

This isn't really hw, just me being confused over some examples. I have 'learned' the basic definitions of neighborhood, limit point, closed, and closure but have some trouble accepting the following examples. 1. For Q in R, Q is not closed. The set of all limit points of Q is R, so its...

Hi all! I'd like to ask for some opinions on a book. I'm currently taking an undergraduate course in topology. We're using the book A Combinatorial Introduction to Topology, by Michael Henle, and so far I have mixed feelings about it, feelings that my class and professor seem to share. 1...

I am trying to understand the theorem: Let f:S->T be a transformation of the space S into the space T. A necessary and sufficient condition that f be continuous is that if O is any open subset of T, then its inverse image f^{-1}(O) is open in S. First off, I don't really understand what...

Hi everyone, I was wondering if I could some advice from anyone who has some experience with higher level general relativity. Any help would be greatly appreciated! Some background: I'm currently working through Robert Wald's General Relativity and am struggling a lot with the "advanced...

Homework Statement Prove that every Hausdorff topology on a finite set is discrete. I'm trying to understand a proof of this, but it's throwing me off--here's why: Homework Equations To be Hausdorff means for any two distinct points, there exists disjoint neighborhoods for those points...

Homework Statement Let X:=ℝn with the Euclidean Topology. Is X first countable? Find a nested neighborhood basis for X at 5. Homework Equations If X is a topological space and p\inX, a collection Bp of neighborhoods of p is called a neighborhood basis for X at p if every neighborhood...

Homework Statement Show that the set of odd integers is dense in the digital line topology on \mathbb{Z} The Attempt at a Solution if m in Z is odd then it gets mapped to the set {m}=> open . So is the digital line topology just the integers. If I was given any 2 integers I could...

Homework Statement Prove that T1={U subset of X: X\U is finite or is all of X} is a topology. Homework Equations DeMorgan's Laws will be useful. Empty set is defined as finite, and X is an arbitrary infinite set. The Attempt at a Solution 1) X/X = empty set, finite. Thus X is in T1...

Homework Statement Let X be a set and p is in X, show the collection T, consisting of the empty set and all the subsets of X containing p is a topology on X. Homework Equations? A topology T on X is a collection of subsets of X. i) X is open ii) the intersection of finitely...

What is the relationship between topology, functional analysis, and group theory? All three seem to overlap, and I can't quite see how to distinguish them / what they're each for.

Homework Statement Show [0,1] is not open in ℝ Homework Equations [0,1] is open if and only if ℝ\[0,1] is closed. The Attempt at a Solution ℝ\[0,1] = (-∞,0) U (1,∞), this set is open. Despite the if and only if statement this is enough to say that [0,1] is not open in ℝ. Is this correct?

Homework Statement Let E be a nonempty set of real numbers which is bounded above. Let y=sup E. Then y \in E closure. Hence y \in E if E is closed.Homework Equations E closure = E' \cup E where E' is the set of all limit points of E. The Attempt at a Solution By the definition of closure, y...

Surely sets with the same cardinality are homeomorphic if we assign both of them the discrete topology. What's preventing us from doing that? For example, (0,1) and (2,3) \cup (4,5) have the same cardinality. With the natural subspace topology they are not homeomorphic - as one is connected...

1. Hello, I'm reading through Munkres and I was doing this problem. 16.8) If L is a straight line in the plane, describe the topology L inherits as a subspace of ℝl × ℝ and as a subspace of ℝl × ℝl (where ℝl is the lower limit topology). Homework Equations The Attempt at a Solution I've...

Hello, Some people on PF are currently self-studying calculus and topology. So we thought we might make a post here so that interested people could join us. We are doing the following books: Book of Proof by Hammack (freely available on http://www.people.vcu.edu/~rhammack/BookOfProof/)...

Hello all, In the Fall I am planning on taking Real Analysis, Abstract Algebra and doing an independent study in something(my professor has yet to get back to me on what he is willing to do it in). My question is would it be too much of a workload to try and do another independent study in...

Hey everybody, I just wanted to ask a general question about Topology. I am planning on taking a General Topology course in Spring 2013 and first of all I don't know what it is. I am finishing up Differential Equations 1 right now with an A. By the spring I will have taken linear algebra 2...

Why is it that the set A={1/n:n is counting number} is not a closed set? We see that no matter how small our ε is, ε-neighborhood will always contain a point not in A (one reason is that Q* is dense in ℝ), thus, all the elements in A is boundary point, and we know that by definition, if...

I'm sure this has already been a thread but I'm currently taking my first analysis course and I was wondering (because the tiny bit I've been introduced to so far is so interesting) what the best intro books to topology would be. Thanks!

Hi, I am not very strong in maths, so sorry if these sounds simple. If I have a 3D geometry of a pipe which has its surface defined by triangles (such as that in Computational Fluid Dynamics or Finite Element Analysis) and I have the coordinate points for all the triangles, how can I...

I am reading Munkres book on Topology, Part II - Algegraic Topology Chapter 9 on the Fundamental Group. On page 348 Munkres gives the following Lemma concerned with the homomorphism of fundamental groups induced by inclusions": " Lemma 55.1. If A is a retract of X, then the homomorphism...

On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachement giving Munkres pages 333-334) "Suppose that h: X \rightarrow Y is a continuous map that carries the point x_0 of X to the point y_0 of Y. We denote this fact by writing: h: ( X...

Proving a Set is Closed (Topology) Homework Statement Let Y be an ordered set in the order topology with f,g:X\rightarrow Y continuous. Show that the set A = \{x:f(x)\leq g(x)\} is closed in X. Homework Equations The Attempt at a Solution I cannot for the life of me figure...

consider the set P={1/n:n is counting number}, my classmate said that P is equal to (0,1] but actually i don't agree with him since (0,1] contains irrational numbers. is he correct? also, is it possible for a set not to contain both interior and boundary points?

"Is the electron a photon with toroidal topology?" - what is that? Hello, there s a paper from 1996 http://members.chello.nl/~n.benschop/electron.pdf I have no knowledge to understand the paper, but I am very interested in how two photons can produce an electron. I would like to try to read...

Hello, I'm an undergraduate who's going to be a senior this coming fall. I'm currently triple majoring in Mathematics/Engineering Physics/Biological Engineering. I'm also looking to enter graduate school in applied mathematics. My schedule for this last year all fits together quite well, except...

I have Munkres' book on Topology. For higher Physics (beyond standard model, string theory, etc.) I know we need to have an understanding of differential geometry, etc. that assume knowledge in topology. My question is how much should I study from Munkres' book? I know that it is useful to...

Homework Statement If the set \Z of integers is equipped with the relative topology inherited from ℝ, and κ:\Z→\Z_n (where κ is a canonical map and \Z_n is the residue class modulo n) what topology/topologies on \Z_n will render κ globally continuous? Homework Equations The Attempt...

Hi everybody. Next year I will start an undergraduate course on algebraic topology. Which book would you suggest as a good introduction to this matter ? My first options are the following: 1.- "A First Course in Algebraic Topology" by Czes Kosniowski 2.- "Algebraic Topology: An...

Homework Statement Suppose that (X,\tau) is the co-finite topological space on X. I : Suppose A is a finite subset of X, show that (A,\tau) is discrete topological space on A. II : Suppose A is an infinite subset of X, show that (A,\tau) inherits co-finite topology from (X,\tau). The...

Homework Statement Let X be an infinite set with the cofinite topology. Show that the product topology on XxX (X cross X) is strictly finer than the cofinite topology on XxX. Homework Equations None The Attempt at a Solution So we know that a set U in X is open if X-U is finite...

I know that my question is not very clear, so I'll try my best to clarify it. Firstly, by general topology I mean point-set topology, because that's the only form of topology that I've encountered so far. In point-set topology, they teach us a lot of new definitions like open sets (that are...

Geometry - Topology; "What is the difference?" It is certainly important for a good understanding of a lot of modern problems. So I think it could be important to explain clearly the difference(s) between these two notions. Can you help me?

It's an elective, I've been told that point-set topology isn't what I think it is. That is, there isn't much geometry in the introductory class and it's mostly a review of real analysis. How is the difficulty of this course? What is the typical workload? Or are these contingent upon the...

Given an interior operator on the power set of a set X, i.e. a map \phi such that, for all subsets A,B of X, (IO 1)\enspace \phi X = X; (IO 2)\enspace \phi A \subseteq A; (IO 3)\enspace \phi^2A = \phi A; (IO 4)\enspace \phi(A \cap B) = \phi A \cap \phi B, I'm trying to show that the set...

Given the following sequence in the product space R^ω, such that the coordinates of x_n are 1/n, x1 = {1, 1, 1, ...} x2 = {1/2, 1/2, 1/2, ...} x3 = {1/3, 1/3, 1/3, ...} ... the basis in the box topology can be written as ∏(-1/n, 1/n). However, as n becomes infinitely large, the basis...

Where should I start studying topology for analysis? I'm completely new to the subject of topology, and I found there are different areas of topology, but my concern is the one that mostly maps to analysis concepts. Besides I know Munkre's Topology is the standard, but I'm not specializing in...

Homework Statement The problem is as follows: Homework Equations We are using the definition that D is dense if its closure is the whole space. Proofs using this definitions would be best as we were not taught any equivalent ones. Not sure if relevant but just in case: ~ is an...