Topology Definition and 818 Threads

Homework Statement I am asked to show that T=[particular point topology on R^2 ((0,0) being the particular point)] is equal to T'=[topology on R^2 from taking the product of R in the particular point topology (0 being particular point) with itself]. The Attempt at a Solution I'm...

Hi, I've read this article (http://www.space.com/scienceastronomy/mystery_monday_040524.html), which says: "In the new study, researchers examined primordial radiation imprinted on the cosmos. Among their conclusions is that it is less likely that there is some crazy cosmic "hall of mirrors"...

I'm totally stuck on these two. The first is... Let A be a subset of X; suppose r:X->A is a continuous map from X to A such that r(a)=a for each a e A. If a_0 e A, show that... r* : Pi_1(X,a_0) -> Pi_1(A,a_0) ...is surjective. Note: Pi_1 is the first homotopy group and r* is the...

Homework Statement What is the torus excluding a disc homeomorphic to? What is the boundary of a torus (excluding a disc)?The Attempt at a Solution RP^2 X RP^2? As a guess.

Good Morning, I am trying to prove that any 2 open intervals (a,b) and (c,d) are equivalent. Show that f(x) = ((d-c)/(b-a))*(x-a)+c is one-to-one and onto (c,d). a,b,c,d belong to the set of Real numbers with a<b and c<d. Let f: (a,b)->(c,d) be a linear function which i graphed to help me...

Homework Statement In topology, a f: X -> Y is continuous when U is open in Y implies that f^{-1}(U) is open in X Doesn't that mean that a continuous function must be surjective i.e. it must span all of Y since every point y in Y is in an open set and that open set must have a pre-image...

I need to show if the finite complement topology,T_3, and the topology having all sets (-inf,a) = {x|x<a} as basis ,T_5, are comparable. I've shown that T_3 is not strictly finer than T_5. But I'm not sure about other case. I need help.

Homework Statement I always get confused between countably many vs. uncountable. I suppose if one can index the points of a set , then it is countable. 1)So, anything that is finitie is countable. Anything that is infinite is also countable? Then what is uncountable, something that...

I am not a math student but I have read some basic stuff about topology just because it sounded interesting, and I was wondering if people could name some uses of it, because it does not seem to have very many. Also on a related question, how often are new branches of mathematics "invented"...

Many abstract mathematical concepts have their intuitive correspondences or geometrical meanings. such as differentiable is corresponding to "smooth", determinant is corresponding to "volumn",homolgy group is corresponding to "hole". 1.The question is whether "exact" and "exact sequence" have...

Homework Statement I saw the following statement in a proof that a second countable normal space is homeomorphic to the Hilbert cube: n = (B_i, B_j) where the = sign is replaced with an approximately equals which I do not know how to make in latex B_i and B_j are basis sets s.t. B_i...

Suppose I have some subset of R, not necessarily an interval, let it be denoted as A. I have some union (might be countable, might be finite, might be uncountable) of sets where each set is an open set of A and the union of the open sets is equal to A. Can I conclude that A is open? I am not...

[SOLVED]Finite-Compliment Topology and intersection of interior Homework Statement Given topological space (R^{1}, finite compliment topology), find counter example to show that Arbitary Intersection of (interior of subset of R^{1}) is not equal to Interior of (arbitary intersection of...

1) Prove rigorously that S={(x,y) | 1< x^2 + y^2 <4} in R^n is open using the following definition of an open set: A set S C R^n is "open" if for all x E S, there exists some r>0 s.t. all y E R^n satisfying |y-x|<r also belongs to S. [My attempt: Let x E S, r1 = 2 - |x|, r2= |x| - 1, r =...

I just discovered the following. But since half the things I find in topology turn out to be wrong, I feel I better check with you guys. What I convinced myself of this time is that if you have a function f:(X,T)-->(Y,S) btw topological spaces, and S' is a basis for S, then to show f is...

Dear all, I am interested in the connection between the smoothness of a planet and the gravitational acceleration at the surface. Specifically, what is the highest a mountain can be for different values of g? More pertitently, what % of the Earth's surface would be covered with water if the...

I'm trying to prove the following Theorem. Suppose T1 and T2 are topologies for X. The following are equivalent: 1. T1 is a subset of T2; 2. if F is closed in (X, T1), then F is closed in (X, T2); 3. if p is a limit point of A in (X, T2), then p is a limit point of A in (X, T1)...

This question pertains to physics, but has to do with the math. In order to find the flux of an electric field you can put a sphere around it and use that to find flux, since the amount leaving is the same at every point. My teacher said that if you put a cube around the field/charge, you would...

In Munkres' Topology he defines a Cartesian product AxB to be all (a,b) such that a is in A and b is in B. He says that this is a primative way of looking at things. And then defines it to be {{a},{a,b}} He says that if a = b then {a,b} will just be {a,a} = {a} and therefore will only be...

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...