Topology Definition and 800 Threads

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...

My question comes from homework from a section on the Tychonoff Theorem. This is the question: Now I have an idea about how to go about this. I know that Q is compact since I = [0,1] is and the Tychonoff Theorem states that the product of compact spaces is compact. I then know that f(Q)...

I'm trying to prove some stuff that involves the projection map, say p:X x Y ->X. But I need to know if it's continuous. If a map is continuous, then the preimage of a open/closed set is open/closed. The problem is, what do open sets in X x Y look like? I know what the basis elements are...

:bugeye: Does anybody have any suggestions on how to prove that a topological space X is COUNTABLY compact (i.e. every COUNTABLE open cover has a finite subcover), IF AND ONLY IF, EVERY NESTED SEQUENCE of closed nonempty subsets of X has a nonempty intersection? I also need hints on how to...

Hi, I am typing up my topology homework and I want to make the cool looking t I see in the book. How do I accomplish this? Thanks!

I'm really just having trouble figuring out what a question is asking. Here's the question: My problem is really just that in proving that say regularity is a local property, I'm not sure what to use as a subspace. I could take a given base set and then consider the rest of the base sets...

I can not figure out what is wrong with this : suppose one deals with a multiple connected universe, such as a torus. In order to make it simple, let us imagine we consider two very massive objects in this topology, say two well separated clusters of galaxies whose distances are large compared...

How useful is topology for physics? And what are soome good books for learning topology. I find a lot of the definitions in textbooks way too abstract and not giving examples of the topological spaces they are defining. Drop some titles if you have a moment.

What exactly is topology? I know it's used a lot in modern physics, but what other applications does it have? Now, a little bit on the theoretical side, what's difference between point-set, algebraic, geometric and differential topology? Can anyone provide an example problem on each? What...

If S is a set with the discrete topology and f:S->T is any transformation of S into a topologized set T, then f is continuous. Can someone help me prove this? I have no idea where to even begin.

Hi, I am not sure if this would be the right place to post this but i know that it is a mathematical concept. I have read a bit about topology in the latest scientific american, and it really intrigued me. I am fascinated by this idea. Therefore i ask if you would kindly point me in the...

can somebody tell me how to unlink 2 linked tori using continuous deformations only? Also are there any free software tools for visualizing topological operations?

I have an idea of what topology is but I am clueless as to what applications it has? Anybody have any idea what topology is used for?

In a quantum gravity discussion ("Chunkymorphism" thread) some issues of basic topology and measure theory came up. Might be fun to have a thread for such discussions. for instance the statement was made, apparently concerning the real line (or perhaps more generally) that a countable set...

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

a line segment (including its endpoints) is toplogically equivalent to a point. consider S a nonempty totally ordered (ie a set with a relation <= such that <= is reflexive, transitive, x<=y and y<=x imply x=y, and every pair is comparable--i think that's what total order means anyway) set...

Greetings, Having decided on a physics/math double major next year, I decided to get a head start this summer. After tackling some classical mechanics, my next target is topology. My problem is the following: I have been informed that there are two approaches to the subject, one...

What are the main differences in approach between standard? topology and algebraic topology?