Topology Definition and 800 Threads

A topological space is a set X together with T, a collection of subsets of X, satisfying the following axioms: 1.The empty set and X are in T. 2.The union of any collection of sets in T is also in T. 3.The intersection of any finite collection of sets in T is also in T. The sets in T are...

Hi, I am currently a sophomore and a math major with thinking of adding computer science as either minor or second major. I get to register for my classes for Fall Quarter in a week, and I am thinking of taking 2 math classes: One will be numerical analysis, and the other is not yet...

Hello all! So, I'll be taking first-semester quantum mechanics and partial differential equations this fall, and would like to get a little bit of a head start by reading/working some problems on my own this summer. After some initial browsing, I've heard mixed-to-poor reviews concerning...

Homework Statement Show that the set S ⊆ C[0, 1] consisting of continuous functions which map Q to Q is dense, where the metric on C[0, 1] is defined by d(f,g) = max |f(x)−g(x)|. All else I need to know is what the question doesn't mention - what the set is dense in? I assume it doesn't...

In the proof of Proposition 3A.5 in Hatcher p.265 (http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf), at the bottow of the page, he writes, "Since the squares commute, there is induced a map Tor(A,B) -->Tor(B,A), [...]" How does this follow? The map Tor(A,B)-->A\otimes F_1 is the connecting...

In Hatcher, p. 262 (http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf), he writes, just before Lemma 3A.1, "the next lemma shows that this cokernel is just H_n(C)\otimes G. I can't say that I see how this follows. Thanks!

Hi everyone! I would like to solve some questions: Classify up to isomorphism the four-sheeted normal coverings of a wedge of circles. describe them. i tried to to this and it is my understanding that such four sheeted normal coverings have four vertices and there are loops at each of...

I am trying to show that the space Cone(L(X,x)) is homeomorphic to P(X,x) where L(X,x) = {loops in X base point x} and P(X,x) = {paths in X base point x} I firstly considered (L(X,x) x I) and tried to find a surjective map to P(X,x) that would quotient out right but i couldn't seem to find...

This is a very simple question... Because I'm not very good at these... notations... I feel like I need a clarification on what this means.. if X is a set, a basis for a topology on X is a collection B of subsets of X (called basis elements) satisfying the following properties. 1. For...

Let F: X x Y -> Z. We say that F is continuous in each variable separately if for each y0 in Y, the map h: X-> Z defined by h(X)= F( x x y0) is continuous, and for each x0 in X, the map k: Y-> Z defined by k(y) =F(x0 x y) is continuous. Show that if F is continuous, then F is continuous in...

x0 \inX and y0\inY, f:X\rightarrowX x Y and g: Y\rightarrowX x Y defined by f(x)= x x y0 and g(y)=x0 x y are embeddings This is all I have... f(x): {(x,y): x\inX and y\inY} g(y): {(x,y): x\inX and y\inY} right? soo... embeddings are... one instance of some mathematical structure contained...

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. b. Show that i is a homeomorphism <=> T'=T This is all I've got. According to the first statement... X \subset T and...

Topology of open set(newbie, I am stuck help!) Homework Statement Hi just found this found and have some basic questions about topology. If let say exist a metrix space (M,s) and two points x \neq y in M. Then show that there exists open sets V_1,V_2 \in \mathcal{T}_s such that x \in...

Hey, can anyone help me with this please. I am doing algebraic topology and am particularly stuck on exact sequences. I "understand" the idea of the definition for example: 0\rightarrow A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0 in this short exact...

On the wiki page on coherent topology, and more precisely, topological union (aka topology generated by a collection of spaces) (http://en.wikipedia.org/wiki/Coherent_topology#Topological_union), it is said that if the generating spaces {X_i} satisfy the compatibility condition that for each...

I admit I hardly know anything about topology, but I have the feeling it is a heavy, abstract part of mathematics. Yet, I heard that it can be important for physics. In which areas concepts of topology are crucial so that results cannot be guessed by common sense alone? Where are these...

Does anyone know where can I get a Topology for dummy? I'm learning Topology Spaces and Interior, Closure and boundary in the first two chapters of the textbook. I've had difficult time working on my homework assignments. Just wonder if some of you already had this course before willing to help...

R=2^\alepha 0 vs Continuum hypothesis! A result in "a taste of topology" A year ago or so I read a proof in A Taste Of Topology, Runde that the cardinality of the continuum equals the cardinality of the powerset of the natural numbers. But a few hours ago I found Hurkyl making that statement...

Let LG be the base point preserving loops (it's a Hilbert manifold). So LG = { f : S^1 -> G s.t. f(0)=1 } where G is a connected, simply connected Lie group. LG is embedded into the (vector space) Hilbert space L^2[0, 2pi] given by f |--> g(t) = f '(t)f(t)^-1 Is LG a closed subset of...

Hey there, I'm looking for popular theories concerning the topology of space-time. Can anyone point me in the direction of a link? Or perhaps even just give me the name of a theory or two? -Tim

Lemma 1 of page 35 says that the index at an isolated zero z of vector field v on an open set U of R^m is the same as the index at f(z) of the pushfoward f_*v = df o v o f^-1 of v by a diffeomorphism f:U-->U'. For the proof, he first reduces the problem to the case where z=f(z)=0 and is U a...

Homework Statement Is it true that if U is an open set, then U = Int(closure(U))? The Attempt at a Solution I feel like this may be true; I found counter-examples to the general form, Int(U)=(Int(closure(U)), but they all seem to hinge on U being not open (A subset of rationals in the...

Homework Statement Let ( X, \tau_x) (Y, \tau_y) topological spaces, (x_n) an inheritance that converges at x \in X, and let f_*:X\rightarrow Y[/itex]. Then, [tex]f[/itex] is continuos, if given (x_n) that converges at [tex]x \in X , then [tex]f((x_n))[/itex] converges at...

Show that any basic open set about a point on the "top edge," that is, a point of form (a, 1), where a < 1, must intersect the "bottom edge." Background: Definition- The lexicographic square is the set X = [0,1] \times [0,1] with the dictionary, or lexicographic, order. That is (a, b) <...

For each n \in \omega, let X_n be the set \{0, 1\}, and let \tau_n be the discrete topology on X_n. For each of the following subsets of \prod_{n \in \omega} X_n, say whether it is open or closed (or neither or both) in the product topology. (a) \{f \in \prod_{n \in \omega} X_n | f(10) = 0 \}...

Which would fit this description and with answers?

Homework Statement Let X be a topological space, Y a Hausdorff space, and let f:X -> Y and g:X -> Y be continuous. Show that {x \in X : f(x) = g(x)} is closed. Hence if f(x) = g(x) for all x in a dense subset of X, then f = g. Homework Equations Y is Hausdorff => for every x, y in Y with...

Consider the set of rational numbers, under the usual metric d(x,y)=|x-y| I am pretty sure that this space is totally disconnected, but I can't convince myself that the set {x} U {y} is a disconnected set. It seems obvious, but I can't find two non-empty disjoint open sets U,V such that U...

I'm not very good at topology but am reviewing it for the GRE Subject Test. Here's a question that I think I know, but would like to check with you guys. We define: Ek = B(0, k) - B(0, k-1), where B(0,k) is an open ball around the origin with diameter k. Now suppose that Tk is a subset of Ek...

Hello, I'm wondering, is it possible to study topology without having taken a course in multivariable calculus? I'm very eager to learn and my college don't offer too many math courses this spring (I'm moving to a bigger next fall though), so I'm thinking if I should take topology. I can...

Homework Statement Let X be a space. A\subseteqX and U, V, W \in topolgy(X). If W\subseteq U\cup V and U\cap V\neq emptyset, Prove bd(W) = bd(W\capU) \cup bd (W\cap V) Homework Equations bd(W) is the boundary of W... I think I have the "\supseteq" part, but I am having trouble with...

Homework Statement If A is a discrete subset of the reals, prove that A'=cl_x A \backslash A is a closed set. Homework Equations A' = the derived set of A x is a derived pt of A if U \cap (A \backslash \{x\}) \neq \emptyset for every open U such that x is in U. Thrm1. A...

Homework Statement (a) Prove that R, with the cocountable topology, is not first countable. (b) Find a subset A of R (with the cocountable topology), and a point z in the closure of A such that no sequence in A converges to z. Homework Equations (The cocountable topology on R has as...

If A is not open in a topological space, does it follow A is closed?

"Easiest" topology textbook/book I am having a terrible time learning topology. Abstract algebra comes easily, as does analysis but Topology is not making any sense whatsoever to me and I honostly try harder in it than my other classes and it gets me 1/10th the progress if not thousands less...

Homework Statement Let d be the usual metric on RxR and let p be the taxicab metric on RxR. Prove that the topology of d = the topology of p. Homework Equations The Attempt at a Solution I am trying to show that the open ball around point (x,y) with E/2 as the radius (in...

Hello all, I have a question I'm having a hard time with in an introductory Algebraic Topology course: Take two handlebodies of equal genus g in S^3 and identify their boundaries by the identity mapping. What is the fundamental group of the resulting space M? Now, I know you can glue two...

I've just started looking at Rudin (Real and complex analysis, the big one) in which he offers the following definition (1.2): a)A collection \tau of subsets of a set X is said to be a topology in X if \tau has the following three properties: i) \emptyset \in \tau and X \in \tau ii)...

As I understand it a topology on a set X is a collection of subset that satisfy three conditions 1) The collection contains X and the null set 2) It is closed under unions (perhaps a better way to say this is any union sets in this collection is again in the collection). 3) The intersection...

I am trying to show that if X is a topological space, ~ an equivalence relation on X and q:X-->X/~ the quotient map (i.e. q(x)=[x]), then the quotient topology on X/~ (U in X/~ open iff q^{-1}(U) open in X) is characterized by the following universal property: "If f:X-->Y is continuous and...

Hi...I'm new to the forum but I need help with the following question. I need to find a topology on N for which there are exactly k limit points. k is a positive integer. Tips I have received: find countable subsets in R...then a bijection will produce the needed topology on N? Any help is...

I understand that topo-physiologically, the human body is a donut. i.e. not only are we a donut physically, but we are a donut as an organism. Our skin is an interface between the outside world and the inside of our bodies. Bacteria and other microbes have to get through our skin defense...

I know the importance of Topology, but I need to know if not taking this course in my last year will make a big difference. I will be a senior math major and I may decide to go to graduate school. I will take a year off to try to find some work experience to decide exactly what I want to do...

i've texed up three proofs in from elementary topology. can someone please check them? actually i'll just retype them here for convenience 8.2.5 Let f: X_{\tau} \rightarrow Y_{\nu} be continuous and injective. Also let Y_{\nu} be Hausdorff. Prove : X_{\tau} is Hausdorff...

u is the usual topology, cf is the cofinite topology. yes proof: pick a and b in (R,cf) ((0,1),u) ~ ((a,b),u). then the identity on (a,b) is continuous is because (R,cf) \subset (R,u). map 0->a and 1->b. the fn is continuous at the end points because no subset of the image is open in...

Hi, does it make sense to posit a curved (hall of mirrors type) space without higher dimensions? In other words, if I say we live in a 3 dimensional torus shaped universe, does that statement necessarily entail there's at least a 4 dimensional overall hyperspace? Or can I have a torus curved...

So I'm trying to teach myself some topology, and the first thing I noticed, was that a metric space is a topological space under the topology of all open balls.. But then, consider the intersection of two open balls, can someone prove to me that the result is another open ball? Or do they...

I'm struggling with something that I suspect is very basic. How do I should that the closure of a connected set is connected? I think I need to somehow show that it is not disconnected, but that's where I'm stuck. Thanks

Hi, I have a conceptual question. In a project I am working on, we are dealing with \Re^{n} (with the usual topology), and I am working on characterizing some objects. In particular, I am dealing with the intersection of two simply connected open sets (that do not have any sort of...

[SOLVED] Topology of curved space Homework Statement The distance between a point (r, theta) and a nearby point (r + dr, theta + d\theta) on a positively curved sphere is given by ds^2 = dr^2 + R^2 \sin ^2 (r/R)d\theta ^2 NOTE: I mean that ds^2 = (ds^2). My question is - how do I...