Topology Definition and 798 Threads

Homework Statement X is the space of all real numbers topology A={empty set} U {R} U {(-infinity,x];x in R}The Attempt at a Solution Is it because (-infinity, x] is not an open set usuing the usual metric on R but is using a metric allowed as it was not specified in the question. If not then...

I have the following A\subset\mathbb{R}^{n} is dense then A isn't bounded. Is this true? I know that A is dense iff \bar{A}=\mathbb{R}^{n} and that A is bounded iff \exists \epsilon>0\mid B_{\epsilon}(0)\supset A. How to proof it? Or there is an counterexample?

Homework Statement Show that if {K_n} is a decreasing family of compact connected sets in a metric space, then their intersection is connected as well. Illustrate with an example why 'compact' is necessary instead of just 'closed'. The Attempt at a Solution Well, I have a example for...

Homework Statement The problem is to show that if A is closed in R^n, and x is outside of A, then there is a point y in A such that d(y,x) = d(x,A). My method of solution involves letting b = d(x,A) = inf{d(x,z) : z in A} and considering a closed ball of radius b + delta centered on x, and...

Please read the following problem first: Suppose n > 1 and let S^n be the n-sphere in R^{n+1}. Let e be the unit-coordinate vector (1,0,...,0) on S^n. Prove that the fundamental group pi_1(S^n;e) is the trivial group. Okay, now my question is what does the notation "pi_1(S^n;e)" mean...

Homework Statement The lemma to prove is that "If [-R,R]^{n-1} is compact, then [-R,R]^n is too. To help us, we have two other lemmas already proven: L1: "[a,b] is compact." L2: "If A is R^n is compact and x_0 is in R^m, then A x {x_0} is compact." The Attempt at a Solution I found a proof...

Homework Statement The lemma sets out to show that if A in R^n is compact and x_0 is in R^m, then A x {x_0} is compact in R^n x R^m. They say, "Let \mathcal{U} be an open cover of A x {x_0} and \mathcal{V}=\{V\subset \mathbb{R}^n:V=\{y:(y,x_0)\in U\}, \ \mbox{for some} \ U\in \mathcal{U}\}...

Defn: the discrete topology on X is defined by letting the topology consist of all open subsets of X. Why do they use the word discrete in the term discrete topology? Is it because there are subsets such that each subset contain only one point in the space. And these collection of subsets are...

Is a continuous 1-1 and onto mapping from the Euclidean 3-space to itself a homeomorphism? i.e. Is its inverse also continous?

Homework Statement Does a basis exist for all sets with a topology? In other words do all topological spaces with topology T contain a basis for T?

If a question at the end specifies wrt the normal topology, what does it mean? What would be an example of a topology that is not normal?

Homework Statement Prove that every nonempty proper subset of Rn has a nonempty boundry. The Attempt at a Solution First of all, I let S be an nonempty subset of Rn and S does not equal Rn. I tried to go about this in 2 different ways: 1) let x be in S and show that B(r,x) ∩ S ≠ ø and...

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.

I have a course on this in the following year and was just wondering what kind of texts are useful for a course on elementary topology. The course description is this: "Set Theory, metric spaces and general topology. Compactness, connectedness. Urysohn's Lemma and Tietze's Theorem. Baire...

I am interested in learning set theory. It is an independent study. I already have previous knowledge of logic and deduction. Does anyone know of any good resources for learning set theory? Also, the reason I plan on learning set theory is so I can learn topology afterward, so any learning...

Homework Statement Consider the sets A=\{(t,\sin(1/t))\in \mathbb{R}^2:t\in(0,1]\}, B=\{(0,s)\in\mathbb{R}^2:s\in[-1,1]\}. Let X=A\cup B. We consider on X the topology induced by the open ball topology of R². a) Is X connected? b) Is X path connected? The Attempt at a Solution a) I found...

Homework Statement What is the fundamental group of A where A is the 2-sphere with two disjoint disks removed. It has the same homotopy type as a familiar space.Homework Equations The Attempt at a Solution When I first looked at this problem, and saw how it was drawn out (in Munkres book,) it...

Question 1 says "Is the bouquet* of two 2-spheres a surface"? How does this question even makes sense? A surface is a paracompact Hausdorff 2-manifold w/o boundaries, and a manifold is a topological space plus an atlas. Here, no atlas is provided! *...

Are there any good online introductions to topology?

I read on wiki* that a (pointed) topological space is simply connected iff its fundamental group is trivial. But I don't see how this in accordance with the R² caracterisation that U is simply connected iff it is path-connected and has no holes in it. Take the closed unit-disk with a point of...

Homework Statement Defn: A subset A of a metric space (X, d) is NOWHERE DENSE if its closure has empty interior. Now I am told that this implies 1. A is nowhere dense iff closure of A does not contain any non-empty open set and 2. A is nowhere dense iff each non-empty open set has a...

Homework Statement Question: Prove Int(A) is an open set, given Int(A) is the set of all interior pts of A where x is an interior pt of A if it is the centre of an open ball in A. Homework Equations None The Attempt at a Solution Attempted Soln: Suppose x is an element of...

Homework Statement Our professor gave us a few exercises at the end of class the other day and it is possible that he wrote the problem wrong, or I copied it wrong, but in any case, something's fishy about it. It says Show that there exists a topology on the set \mathcal{D} of all lines of...

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...

I had to ask myself two simple problems in differential topology: 1) Why is the rank of a diffeomorphism (on a manifold of dimension m) of rank m? 2) Why is a chart on a manifold an embedding? These are actually quite obvious so textbooks don't even bother proving it. So I've...

Alright, instead of starting a new thread everytime I have a question, I will just post it in here. Note: These are not from assignments. Note: Most of these questions can be found in Topology by Munkres. I will make a mention when it is, and where it is. So, here is the first one...

I am looking at an independent study next semester and I want to do something with point-set topology and analysis, so I have been looking at the book Introduction to Topology and Modern Analysis by Simmons, which I have heard good things about. For a little background I am currently taking...

I have just found that topology is very interesting. I just want to know how one studies topology. do they go in the order of Point-set Topology, Algebraic Topology, then Differential Topology? My ultimate goal is to understand Calculus on Manifolds and Morse Theory. Is it possible to jump to...

i think that i read that the compactness theorem in logic has a similar theorem in topology. i wanted to inquire, are there any other theorems in logic which have similar, dual theorems in topology or other branches in maths?

I'm familiar with the idea that there are very strong reasons to believe that possible spacetimes (M,g) for the universe can have restricted topologies. For example, I believe Hawking proved during the 70s that, given a four-dimensional manifold M and a Lorentzian metric g, then (M,g) can be...

Prove: If p:X->Y is a quotient map and if Z is a locally compact Hausdorff space, then the map m: p x i : X x Z -> Y x Z is a quotient map. Note: i is the identity map on Z i assume. There is a few lines of hints talking about using the tube lemma and saturated neighborhoods which i don't...

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?