Topology Definition and 818 Threads

http://en.kioskea.net/contents/initiation/topologi.php3 What connects the cables coming from the computers to the "bus"? I can't quite get the idea of connecting these cables to another cable when connecting devices aren't mentioned.

Many interesting proofs in GR (regarding black holes and singularities, etc.) involve topological methods. However, I don't understand how a theory embodied in Einstein's equations, which appear to me to be local rules of evolution, can ever change the topology since a patch of spacetime...

Homework Statement (i) Let U be a topology on Z, the integers in which every infinite subset is open. Prove that U is the discrete topology. (ii) Use (i) to prove that if U is a topology on an infinite sex X in which every infinite subset is open, then U is the discrete topology on X...

how can I understand the difference between algebra, sigma algebra and topology If I take the set A that contains a,b,c,d,e,f the set C that contains A,phi,{a},{b,c,d,e,f} then C is algebra on A and C is sigma algebra on A and (A,C) is topological space is that true? what is the...

Hello, could you please check if the reasoning is correct. This is not a homework, just a part of an exercise in a book I'm reading at the moment. Suppose X is a set, \mathcal{B}:=\{S\subset{}X:\bigcup{}S=X\}, \\...

I am reading from my text, and was just wondering if someone could provide additional information on the following examples. 0.1 Examples. For any set X each of the following defines a topology for X. (1) T_{*} = {A \subseteq X|a \in A \Rightarrow X \subseteq A}, Indiscrete Topology. (2)...

Homework Statement Determine if the set of points (x,y) on y = |x-2| + 3 - x are bounded/unbounded, closed/open, connected/disconnect and what it's boundary consist of. Homework Equations The Attempt at a Solution I know that the set is closed, and then by definition of a closed set...

i have problem in understanding these notes:confused:if some body please help me out in understanding them.to mention,i have just started this topic and have no background knowledge as well. wat to do??:cry: :rolleyes: help me...

Given two topological spaces X and Y, there is a standard way to define a topology on X x Y called the product topology. Given a subset S of X, there is a standard way to define a topology on S called the subset topology. Given an equivalence relation ~ on X, there is a standard way to...

Hello, there is a basic lemma in topology, saying that: Let X be a topological space, and B is a collection of open subsets of X. If every open subset of X satisfies the basis criterion with respect to B (in the sense, that every element x of an open set O is in a basis open set S, contained...

I have what's certainly a totally "newbie" question, but it's something I've been wondering about.. Suppose we have a simple boundary value problem from electrostatics. For instance, suppose we have a conducting sphere held at some potential, \phi = \phi_0. Because the sphere is conducting...

How difficult is Spanier's Algebraic Topology text to understand? How about the exercises?

I can't seem to find out how to prove this theorem: A collection {fa | a in A} of continuous functions on a topological space X (to Xa) separates points from closed sets in X if and only if the sets fa-1(V), for a in A and V open in Xa, form a base for the topology on X. Could anyone help me...

I have no idea where else to ask this: I have the 2nd edition of the book and I noticed that on the spine of the book it says "Secon Edition" instead of "Second". I'm just wondering if this is on every copy of the second edition of the book? Or, is your copy like this?

Is there any topology where it is possible to trisect an angle using only straight lines and circles?

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...

the question is to prove Urysohn's metrization theorem. But there some steps I need to show first. Assuming X is normal, second countable. We show there is a homeomorphism of X onto a subspace of [0,1]^w (the Hilbert cube which is metrizable), so X is metrizable. We first show we can assume...

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?