Topology Definition and 818 Threads

i know that geometric topology is a field that is connected to knot theory, i wonder what are the similarities between the two subjects, and in what subject in particular they overlap?

What is the motivation for a physicist to learn topology? Are there fields of physics that make explicit use of the concept of topology? (which ones) Do the ideas of topology give any insights into any topic of physics? etc.

Hello, Here's a problem that I'm having trouble with: Give an example of metric spaces X and Y and continuous maps f: X->Y and g: Y->X such that f and g are both bijective but X and Y are not homeomorphic. I can find plenty of examples where I can find one such function, but finding...

let A \hookrightarrow B be a continuous inclusion map from A to B. A, B are two topological spaces. A \subset B what can we say about the induced map between topological dual spaces B^* \hookrightarrow A^* ? is it continuous and injective?

Hi, In a euclidean space X with two subsets E and F, the subset E+F is defined as the collection of all x+y, where x E and y F. “+” denotes the addition in the euclidean space. Prove that if E and F are open, then E + F is open. I'd really appreciate your help. Thanks so much!

Let X and X' denote a single set in the two topologies T and T', respectively. Let i: X' -> X be the identity function. a) Show that i is continuous <=> T' is finer than T. Ok I am able to show that for any set in T|X this set is in T'. This is done as follows: Assume i is continuous. For...

in the munkres book, they define A to be a subbasis of X if it is a collection of subsets of X whose union equals X. They define T, the topology generated by the subbasis to be the collection of all unions of finite intersections of elements of A. This definition seems to be flawed because...

Looking for good "Topology" text I'm seeking a good text on Topology to suplement my study of geometrical methods of mathematical physics. For those of you who are learned in this field please post your favorite Topology text. Thanks. :smile: Pete

I would like a little clarification in how to prove that the k topology on R is strictly finer than the standard topology on R. They have a proof of this in Munkres' book. I know how to prove that its finer, but the part that shows it to be strictly finer I am not sure. It says given the basis...

Question. Suppose a particle {o} that is a topological entity [see:http://en.wikipedia.org/wiki/Topology] so that it can take an extended form {...o o o o o o o o o ...} to infinity. Now, suppose the transformation to exist as a wavefunction--is this then a correct view of particle-wave...

I walk into Real Analysis and in the first 3 seconds my thoughts go... ... drop class NOW! When I first took the course, I was under the impression that the chair of the department would be teaching the course. Unfortunately, the professor from last year Complex Analysis is teaching the...

Hi everybody, I'm trying to design and build a matrix converter, but I think I have read contradictory information about which power stage topology could be more recommended. Should I use common collector or emitter configuration for low power systems? with discrete devices or trying to use...

This site has compiled the topology of fermi surfaces for a lot of material. http://www.phys.ufl.edu/fermisurface/ The Science website and the contact e-mail at the end indicate that the people who are responsible for this are at Ohio State (Gokul, do you know any of them?), but it is...

I can't understand it. No matter how much I try, Can anyone explain it step by step, and give some examples. How can it be applied to contruct different shapes?

Hi, I've been studying topology over the last semester and one thing that I was wondering about is why exactly is topology defined the way it is? For a refresher: given a set X we define a topology, T, to be a collection of subsets of X with the following 3 properties: 1) the null set and X...

Hello, I'm reading this book and I've come to a question that has me stumped: I got the first part: since every set in the discrete topology is open, then the inverse image of every open set must be open. Its the second part that's giving me trouble. It seems to me like many functions could...

Let X,Y be two spaces, A a closed subset of X, f:A--->Y a continuous map. We denote by X\cup_fY the quotient space of the disjoint union X\oplus{Y} by the equivalence relation ~ generated by a ~ f(a) for all a in A. This space is called teh attachment of X with Y along A via f. i) If A is a...

Hi all, so I'm finishing my third year as a pure math major and what interests me most is topology. I am thinking I want to go to grad school, but don't know what I would study there. So my question is, what sort of things are there to study/research about topology, how does it relate to...

What does it mean in the statement "Topologically, de Sitter space is R × S^n-1..." What is the topology of a Schwarzschild black hole?

Hi. Can somebody please check my work!? I'm just not sure about 2 things, and if they are wrong, all my work is wrong. 1. Find a counter example for "If S is closed, then cl (int S) = S I chose S = {2}. I am not sure if S = {2} is an closed set? I think it is becasue S ={2} does not have an...

So, if you take a rectangular area XY and wrap the borders up "video game style" (i.e go off the left side, reappear on the right side, go off the top, reappear at the bottom), you get a 2D surface that can be represented by a 3D torus, right? Right. Now, there're two ways you can make a...

I have a few questions to ask. They are simple, but the purpose is to make sure I'm staying on track. First, the definitions that I will use. T2-Space or Hausdorff Space, find http://http://mathworld.wolfram.com/T2-Space.html" here. Separable Space, find...

Just curious if anyone has ever studied what happens when a topology gains new members in the underlying set. How is it incorporated into the existing subsets whose union and intersection are included in the topology? It seems to me that assuming the universe expanded from a singularity, then...

This is simply a self-study. 1 - Can someone help understand what they mean by the following Topological Space: Let T be the family of subsets of R (the reals) defined by: A subset K or R belongs to T if and only if... ...for each r in K there are real numbers a, b such that... a < r...

One day, I decided to find out in which places topics in Mathematics and Physics were interlinked or used to prove results in each other's topics. Most of Mathematics is applied everywhere in Physics - from Calculus to Group Theory etc. I considered that possibly the only field which is not...

what is it?:confused:

Hi all, I have the following question. Are the following spaces homeomorphic in the real number space with absolute value topology? 1) [a,b) and (a,b] 2) (a,b) and (r,s)U(u,v) where r < s < u < v. For 1), I got that they are not homeomorphic because it fails the topological property that...

This is actually a topology question, but I wasn't sure where to ask it. It's about surfaces, ie, 2D manifolds. I know that the defining property of a surface is that each point has a neighborhood homeomorphic to R2, but I was wondering if this neighborhood is always open in the surface. It...

Can anyone recommend good introductory texts for self-study? I want to teach myself about tensors and about topology. FYI, I have a B.S. in Physics and am a Fellow of the Casualty Actuarial Society. I don't remember my vector calculus and am in the process of relearning - I'm using the book...

wat do u need to learn topology?

In the topic of the topology, how to determine whether or not these collections is the basis for the Eclidean topology, on R squared. 1. the collection of all open squares with sides parallel to the axes. 2. the collection of all open discs. 3. the collection of all open rectangle. 4...

One question I've had lately in my independent study of topology is the problem of how to show two sets are homeomorphic to each other. I am not sure how I would go about doing this in a general, or even specific case. One problem that wants me to demonstrate this is in Mendelson: Prove that...

I am not well 'accustomed' to these kind of proofs so please bear with my stupidity Suppose U and V are both open subsets in Rn. Prove that U intersection V and U union V are open as well. Dfeinition of open is that you cna center a ball about a point a in a set such that that ball is...

Let \mathbb{R}_{l} denote the real numbers with the lower limit topology, that is the topology generated by the basis: \{[a, b)\ |\ a < b,\ a, b \in \mathbb{R}\} Which functions f : \mathbb{R} \to \mathbb{R} are continuous when regarded as functions from \mathbb{R}_l to \mathbb{R}_l? I...

Hi. I was trying to figure out if the following is a metric space in R x R (Cartesian product). D[(x1,y1),(x2,y2)] = min( abs(x1-x2), abs(y1-y2) ) I know there are four properties to confirm that the following is a metric space but I'm having trouble with the "triangle inequality" for...

Hey everyone, First of all, I hope this is an OK place for a topology question. I was debating here or of course set theory, but I guess this is right. Anyway. I'm studying out of Munkres' book, and I'm looking at a certain problem. The problem stated is: THEOREM: If A is a basis for a...

If A and B are two non-closed subsets of X, how would one prove that the closure of A union B= the closure of A union closure of B? Also, what site would you recommend I download TeX from when I get my new computer (Dell, runs on windows)?

How do we unify modal logic with topology or perhaps complex analysis/probability&random variables? Provided the principles of modal logic, is it possible to translate philosophy into computer language and create the real AI? Is it possible to apply modal logic to Shor's computational...

This is a problem 1 from Munkres pg 83. I'm trying to solve for self study. Let X be a topological space; let A be a subset of X. Suppose that for each x belonging in A there is an open set U containing x such that U is a subset of A. Show that A is open in X. I'm not sure exactly how...

I've been trying to explain what topology is, and why it is important, to non-mathematicians. Specifically to other (non-theoretical) physicists. To best explanation I can come up with is along the lines of "generalized geometry one step up from set theory", and that it is important because it...

Been studying some basic algebraic topology lately. Altough interesting in itself, it would also be interesting to hear if it has any important applications in other branches of mathematics or in other fields (physics?).

Given a homeomorphism from a subspace of R^n onto a subspace of R^n if one of the subspaces is open in R^n, is the other one open in R^n too? I know it is trivial, but I can't see any solution.

Let \rho be the uniform metric on \mathbb{R}^\omega For reference, for two points: x = (x_i) and y = (y_i) in \mathbb{R}^\omega \rho(x,y) = \sup_i\{ \min\{|x_i - y_i|, 1\}\} Now, define: U(x,\epsilon) = \prod_i{(x_i - \epsilon, x_i + \epsilon)} \subset \mathbb{R}^\omega I need to...

:mad: Prove that the determinant function of an n x n matrix is an open mapping (from R^{n^2} space to R) proving it to be a continuous mapping is easy, determinant function is a sum of products of projections, which are continuous maps.

Hello all, I am wondering whether it is possible to construct any arbitrary connected 4-manifold out of a sequence of surgeries on a simply connected 4-manifold. That is, suppose we are given a simply connected 4-manifold, and a multiply connected 4-manifold. Is it in general possible to...

Hi, I was trying to help a student with an assignment in topology when I was stumped by a symbol that I had not seen before. Here's the problem. a.) Let (X,\square) be a topological space with A\subseteq X and U\subseteq A. Prove that Bd_A(U)\subseteq A\cap Bd_X(U). The first thing...

I read how in topology you can bend a ractangle into a cylinder and then the cylinder into a torus. I'm a beginner to topology, so how is the torus described topologically? Is it just a set of all points? or an equation?

Can someone please explain to me what topology is? thanks

Can we proove that for any separated topological space, there exists a metric? Seratend.

Can someone explain the concept of Quotient topology. I tried to read it from a book on topology by author "James Munkres" . It was okay but I did not get a feel of what he was trying to do.. He talks about cutting and pasting elements. I kind of got lost in that.. If someone could give me a...