Topology Definition and 798 Threads

I am a Mathematics and Physics double major, currently in my second year. I really enjoy both subjects, but my interests are progressing towards Theoretical physics/mathematical physics. My academic goal is to improve my understanding of how the universe works and thus I would like to pursue a...

I'm making this thread because in a few weeks I'll be starting with teaching a topology course. I think I did pretty well last time I teached it, but I want to do some new things. The problem with the system in our country is that we hardly assign problems that students should solve. Students...

Can I simply combine the unions into a single interval like I did in (a)? The closed interval [1,4] fills in the (3) hole from (0,3), etc. I did something similar in (b). http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110906_221620.jpg...

I know there are some threads out there already, but none really help me (see my description below). I am a high school student. My highest level of math education is Calculus I. I am currently taking Calculus II (although I already know the integration portion of this course). I have no...

So I need to decide by tomorrow, whether I'll be taking topology or diff geo, (along with real analysis and advanced linear algebra). I've sat in on both classes for the first lecture, and I'm still not certain which class would be more difficult. My diff geo class has no exams, and instead...

In at least one book and one Wikipedia article, I've seen someone specify which sequences are to be considered convergent, and what their limits are, and then claim that this specification defines a topology. I'm assuming that this is a standard way to define a topology. I want to make sure that...

Background: I'm a computer science major, but who has done a lot of math (real analysis, linear/abstract algebra, combinatorics, probab&stats, numerical analysis, linear programming) and currently doing undergraduate research in computational algebra/geometry. I'm taking a graduate level...

Definitions of "topology" and "analysis" How do you define "topology" and "analysis"? I'm tempted to say that topology is the mathematics of...anything that involves limits. (Open and closed sets, continuous functions, etc...they can all be defined in terms of limits). But if that's an...

Can anyone please help me with this because I'm really getting confused. Thanks! In R, we know that fine topology is equivalent to the Euclidean topology as convex functions are continuous. Now if instead of R we consider a subset of it say [0,1], the fine topology induced on [0,1] would...

Background: I'm going to be a junior, having very strong Analysis and Algebra yearlong sequences, in addition to a very intense Topology class, and a graduate Dynamical Systems class. For this coming Fall, I'm sort-of registered for this class, titled "Manifolds and Topology I" (part of a...

And what's considered modern mathematics? I always thought it was 1960s+. Around 50 years ago till now is what i considered modern math. Anyway, how important is topology? I've heard people say "the idea of evolution to biology is the same as the ideas of topology to mathematics." So is it...

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).

Homework Statement If A and B are finite, show that the set of all functions f: A --> B is finite. Homework Equations finite unions and finite caretesian products of finite sets are finite The Attempt at a Solution If f: A -> B is finite, then there exists m functions fm mapping to...

So Tychonoff theorem states products of compact sets are compact in the product topology. is this true for the box topology? counterexample?

Hi everyone, First my background, I'm a junior in physics and math. I've taken griffiths QM, EM (first semester only so far), mechanics and such. In terms of math, I've taken an applied algebra and linear algebra course. I've learned some GR from Sean Carroll's text and a short course in GR...

James came to a place where there was a bridge, supported by parabolic arcs. In the middle waving a transparent gelatinous substance in the form of spherical shell of exotic matter. He had come to " delighted well", a horizontal formation, which is much talk and little experienced. Slowly James...

hi all, i am studying from croom's introduction to topology book. i came across such a question. and i don't have a clue as to how to start . Let X be a metric space with metric d and A a non-empty subset of X. define f:X->IR by : f(x): d(x,A), x E X (x is an element of X) show that f is...

Hi! Can someone recommend some books on point set topology for undergraduates? I am going to use it this summer for preview and also during the fall because the instructor is not going to use a textbook. Thank you!

Let X be any infinite set. The countable closed topology is defined to be the topology having as its closed sets X and all countable subsets of X. Prove that this is indeed a topology on X. Any help would be greatly appreciated. Thanks!

Homework Statement When doesn't the reflexive relation hold? In order for aRb to be true, aRa must hold and the other two conditions. Homework Equations The Attempt at a Solution I am new to topology and am not really taking a course in topology. To me it looks like a is always equivalent to...

Homework Statement I am trying to solve part d of problem 9 in chapter 2 of Rudin's Principles of Mathematical Analysis. The problem is: Let E* denote the set of all interior points of a set E (in a metric space X). Prove the complement of E* is the closure of the complement of E. I will...

Hey guys, So my prof assigned John Milnor's book about topology from a differentiable viewpoint for out topology and geometry class. I was wondering if anyone had a book they could recommend as an introduction into topology because I don't think professor Milnor's book really is an...

I'm about to take my higher upper division classes to complete my mathematics major. I've decided that I'd like to take either Intro to Topology or Intro to Analysis as my topics of choice, along with the required Linear Algebra course that most graduate schools are looking for in applicants...

I recently Googled "spacetime topology" and found that the topology of Minkowski spacetime is generally described as that of an R4 manifold. This is not my field, but I'm surprised. Perhaps mathematically the (---+) "Lorentz signature" can be taken as a secondary characteristic of the...

Topology Excluded set << Original text of post restored by Mentors >>

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...

I am studying topology. There I learn that the open sets given by the metric can be used to define a topology, e.g. the usual metric topology on R^n given by the Euclidean metric. Now I try to understand the topology of (flat) spacetime. Is there a metric? The Lorentz 'metric' gives...

Could someone clarify, and/or point me to some reference on: Lorentz metric is not really a metric in the sense of metric spaces of a topology course since it admits negative values. If I use it to define the usual open sphere about a point, that sphere includes the entire light-cone through...

Just want to ask for recommendations for good math books on 1) groups, modules, rings - all the basic algebra stuff but for a physicist 2) topological spaces, compactness, ... I need books for a theoretical physicist to read up on these topics so that I could study, say, algebraic...

I want to learn topology but I can't find any good resources (except for thoughtspacezero on youtube, which got a bit difficult to understand after a few videos). Can you suggest some materials? Thanks in advance!

Homework Statement Let (X,T) be a metrizable space such that every metric that generates T is bounded. Prove that X is compact. The Attempt at a Solution I was thinking about this problem a bit before I headed off to work and wanted to get you guys' thoughts and/or ideas. At first I...

Am i right in saying that consenses amongst the cosmological community is that the curvature of space is 0? So it is flat euclidean geometric space (I hope I am using correct terminology) which is neither +/- in curvature but exactly 0? Am I also right in saying that flat cosmological models...

I need to write a paper on something to do with (general) topology, and we are encouraged to try and relate it to something that we enjoy. I really like probability (at least basic probability + stochastic processes), and I'm wondering if someone might suggest topic(s) that might be of interest...

Hello, all, the most important results that I know in this topic is the Gauss-Bonnet Theorem (and hence the classification of compact orientable surfaces) and also the Poincare-Hopf index theorem. But there are still some fundamental problems I don't understand. For example, is the...

I 've been reading about Homotopy , homology and abstract lie groups and diff.forms and I would like to see those beautiful ideas applied on a Nonabelian Gauge Theory . Any recommendations for a textbook that apply these ideas to gauge theory ? Text books on particle Physics and QFT do not...

Homework Statement Given N = {1, 2, 3, ...} and En = {n, n+1, ...} (n in N) define a topology on N with T = {empty set, En} Determine the accumulation points of {4, 13, 28, 37}, find the closure of {7, 24, 47, 85} and {3, 6, 9, 12, ...}. Also determine the dense subsets of N. Homework...

Could you please check the statement of the theorem and the proof? If the proof is more or less correct, can it be improved? Theorem Let be a topological space and be the discrete space. The space is connected if and only if for any continuous functions , the function is not onto...

Homework Statement a) Let I be a subset of the real line. Prove I is an interval if and only if it contains each point between any two of its points. b) Let Ia be a collection of intervals on the real line such that the intersection of the collection is nonempty. Show the union of the...

Homework Statement Prove that the topologist's comb is pathwise connected but not locally connected. Homework Equations For A = {1/n : n = 1,2,3...}, the topologist's comb (C) is defined as: C = (I X {0}) U ({0} X I) U (A X I) The Attempt at a Solution Consider the point p =...

Let f: X \rightarrow Y be a function. The graph of f is a subset of X x Y given by G = {(x,f(x) | x \in X } . Show that if f is continuous and Y is Hausdorff, then G is closed in X x Y. Any tips on how to start? Is it saying that f: X \rightarrow Y = G ?

If AxB is a compact subset of XxY contained in an open set W in XxY, then there exist open sets U in X and V in Y with AxB contained in UxV contained in W. Is this true for all spaces XxY? Or does it hold for only regular spaces?

Homework Statement Let T be the lower-limit topology on R. Is (R,T) connected? Prove your answer. The Attempt at a Solution Since there exists a proper subset V of R such that V is both open and closed (since all intervals of the form [a,b) are open and closed), then (R,T) is...

My professor and I were discussing the emergence of the Swartzschild solution from topological considerations, corresponding to the manipulations of a point singularity. He pointed out to me that mass nowhere enters into the considerations, and so classifying black holes according to mass is...

Homework Statement Give an example of a topological space (X,T) that is separable and Hausdorff, with a subspace (A,T_A) that is not separable. The Attempt at a Solution Let X = R and T be a topology on X whose basis elements are open intervals intersected with the rationals and...

When one is considering a the real numbers with the extension of an infinitesimal, implying that it is possible for a number to be the highest number lower than a number, would .999... then be the highest number less than 1? (Similar to *R, but I'm generalizing my hypothesis to include any...

Homework Statement Let f be a real-valued function defined and continuous on the set of real numbers. Which of the following must be true of the set S={f(c):0<c<1}? I. S is a connected subset of the real numbers. II. S is an open subset of the real numbers. III. S is a bounded subset of the...

Homework Statement Show that any countable subset of N with the discrete topology cannot be dense in N. Homework Equations Informally a set is dense if, for every point in X, the point is either in A or arbitrarily "close" to a member of A The Attempt at a Solution I was thinking...

Greetings all, doing more problems for my test tomorrow. I'm not sure how to start these two.. 1.) I'm trying to show that if X and Y are Hausdorff spaces, then the product space X x Y is also Hausdorff. So, I know that I must have distinct x1 and x2 \in X, with disjoint neighborhoods U1...

Here's two more question I'm working on in test prep. 2.) Let Y = [-1,1] have the standard topology. Which of the following sets are open in Y, and which are open in R. A= (1,1/2) \cup (-1/2,-1) B= (1,1/2] \cup [-1/2,-1) C= [1,1/2) \cup (-1/2,-1] D= [1,1/2] \cup [-1/2,-1] E= \cup...

Greetings all, I have a test upcoming next week and a lot of problems to solve. I asked in a previous thread whether or not I should just continue to adding to the thread and it seems the answer was yes. I appreciate any help! This is a tough subject. Anyways, the first one I'm working on. I...