Topological Definition and 250 Threads

Homework Statement Let X be a topological space, a subset S of X is said to be locally closed if S is the intersection of an open set and a closed set, i.e S= O intersection C where O is an open set in X and C is a closed set in X Prove that if M,N are locally closed subsets then M...

Hello. Please, help me with this exercise: Let X be a topological space and let Y be a metric space. Let f_n: X \rightarrow Y be a sequence of continuos functions. Let x_n be a sequence of points of X converging to x. Show that if the sequence (f_n) converges uniformly to f then...

Hey everyone, I'm an applied math undergrad whose research is in applications of topology to fluids and mechanics. I cannot find any grad school that has faculty research in topological physics. Can anybody point me anywhere? Thanks in advance...

Homework Statement http://img219.imageshack.us/img219/2512/60637341vi6.png Homework Equations I think this is relevant: http://img505.imageshack.us/img505/336/51636887dc4.png The Attempt at a Solution A topological isomorphism implies that T and T-1 are bounded and given is that all cauchy...

1. Any metric space can be converted into a topological space such that an open ball in a metric space corresponds to a basis in the corresponding topology (metric spaces as a specialization of topological spaces ). 2. Any topological space can be converted into a metric space only if there is a...

Are singular points necessarily mapped to singular points under topological transformations? A specific example would a 2-space deformation of a triangle to any closed string with no cross over points. Would the three singular points of the triangle be necessarily mapped to three singular points...

1. Homework Statement [/b] U is T1 open iff \Re/U is countable, U=\oslash, or U=\Re. What is the bd(0,1), the cl(rationals), and the int(rationals). Homework Equations The Attempt at a Solution Could somebody please explain this problem to me? I feel like I could try it once I...

A metric dp on the topological space X×Y, with dX(x,y) and dY(x1,y1) being metrics on X and Y respectively, is defined as dp((x,y),(x1,y1))=((dX(x,y))p+(dY(x1,y1))p)1/p What does each dp((x,y),(x1,y1)) mean (geometrically or visually)? as p\rightarrow\infty...

Let {\mathbb I} = {\mathbb R} \setminus {\mathbb Q} the set of the irrational numbers of the real line. What is the topological dimension of {\mathbb R}^2 \setminus {\mathbb I} \times {\mathbb I} ?

what is the meaning of topological black holes? thanks

Hi all, I do realize that my previous thread on CW complexes was unanswered, so perhaps I am posting my questions to wrong section of this forum. If so, please direct me to the right forum. Otherwise, I am having some problems understanding the smash product of two topological spaces. If anyone...

Perturbative Super-Yang-Mills from the Topological AdS_5xS^5 Sigma Model A proof of the AdS/CFT correspondence Is this serious ?

[SOLVED] Topological Properties of Closed Sets in the Complex Plane Homework Statement 1. Show that the boundary of any set D is itself a closed set. 2. Show that if D is a set and E is a closed set containing D, then E must contain the boundary of D. 3. Let C be a bounded closed convex set...

Hello, I know that the CS Lagrangian is a topological invariant, in the sense that it does not depend on the connection we choose. OK, but a TFT is a field theory whose Lagrangian and all other observables do not depend on the metric, a connection in general is not uniquely defined by a metric...

Homework Statement Prove: a topological group is discrete if the singleton containing the identity is an open set. The statement is in here http://en.wikipedia.org/wiki/Discrete_group The Attempt at a Solution Is that because if you multiply the identity with any element in the group, you get...

In Hartshorne's book definiton of a dimension is given as follows: İf X is a t.s. , dim(X) is the supremum of the integers n s.t. there exist a chain Z_0 \subsetneq Z_1...\subsetneq Z_n of distinct irreducible closed subsets of X My question is: Can we conclude directly that any...

1.suppose that f:X->Y is continuous. if x is a limit point of the subset A of X, is it necessarily true that f(x) is a limit point of f(A)? 2. suppose that f:R->R is continuous from the right, show that f is continuous when considered as a function from R_l to R, where R_l is R in the lower...

J. Scott Carter from The University of Southern Alabama is a great new QG-TQFT blues performer See and listen here: http://scienceblogs.com/pontiff/2007/12/quantum_gravity_topological_qu.php Thanks to Dave Bacon! Here is the song text: The Quantum Gravity Topological Quantum Field Theory Blues...

Homework Statement If U is open set in a topological space such that U = A union B, where A and B are disjoint, do both A and B have to both be open? I think that they do not, but I cannot think of a counterexample...perhaps (-1,0) union [0,-1). OK. That's a counterexample. So, now the...

I'm wondering if someone can furnish me with either an example of a topological space that is countable (cardinality) but not second countable or a proof that countable implies second countable. Thanks.

I'm really stuck on this simple problem: Let X be a topological vector space and U, V are open sets in X. Prove that U+V is open. It should be a direct consequence of the continuity of addition in topological vector spaces. But continuity states that the f^{-1}(V) is open whenever V is open...

Homework Statement Is it true that every point in a topological space is closed? In a metric space? Homework Equations The Attempt at a Solution

I've just read Quantum field theory of many-body systems Xiao-Gang Wen His web page http://dao.mit.edu/~wen/ I thought that his book might be easier than his papers. hehehe It's a textbook. I did get to learn a few things. Here is what wiki says about the subject. In physics...

Homework Statement I have seen two definitions for the topology generated by a basis set: one here http://books.google.com/books?id=9cT2wI-Qrk4C&pg=PA23&dq=basis+for+a+topology&ei=wu3nRua6HYqKoQLlj81v&sig=nNFNyWXorlwCQ8WCwQBDTuI9HUc#PPA22,M1 and the other one is a set U is open if for every...

hello, I have just started to study topological field theories but my problem is that I do not find of books really specialized in the subject, can somebody inform me what I must do?

Homework Statement The nested set property: "If {F_k} is a decreasing sequence of non empty compact sets in a metric space (M,d), then their intersection is non empty." First I cooked up a proof of my own and then I read the one provided by the book. It seemed to me that they were perfectly...

Homework Statement Consider a topological space X Show that every point of X is contained in a unique path component, which can be defined as the largest path connected subset of X containing this point. The Attempt at a Solution What happens if we take X=Q? There are no path connected subsets...

Specifically, can they be determined (up to isomorphism of ordered fields) as the smallest connected ordered field?

Can someone please explain to me what the following notation/objects are: (Here X,Y are topological spaces) colim(X-->Y<--X) where the first arrow is a map f, the second is a map g. colim(X==>Y), where there are 2 maps f,g from X to Y (indicated by double lines, but couldn't draw 2 arrow...

I am trying to show that the connected sum of two topological surfaces does not depend on the open discs removed. Any hints?

one of the last Scientific American issues described a way to make a QC. I wasn't sure if I understood correctly. Could someone correct me if I'm wrong: you start with pairs of anyon particles -input- and then swamp ones that are next to each other either CW or CCW. As they pass through time...

If X and Y are two connected topological spaces then so is X \otimes Y. I want to understand the proof of this theorem but I am having some difficulties. Even though we went over it in class, it is still unclear to me. The professor constructed this continuous function: f:X\otimes Y...

I posted this on another forum, but had no response. Maybe because it's too stupid the bother with? Anyway... Say I have a set X and a topology T on X so that T = {X {} A} i.e A is an open subset of T. Then the complement of A is Ac = X - A, which is closed. Now the interior of A, int(A) is...

"visualizing" topological spaces I am taking my first topology course right now. My professor spends most of the time in class proving theorems that all sound like "if a space has property X then it must have property Y." Now this is fine, but my trouble comes in finding an example of a...

Can quanta of unlimited genus exist in theory?

I'm reading this book (Hartshorne), and it uses a funny definition of topological dimension, which I'm having a hard time convincing myself is the usual one. The definition is as follows: dim X is the supremum of natural numbers such that there exists a chain Z_0\subset Z_1\subset \dotsb...

eddo's thread got me thinking: How can you tell if a specific topological space is compact? It seems like it would be hard to do just starting with the definition of compactness.

How can I show that F:X\times I\to I given by F(x,t)=(1-t)f(x)+tg(x) is continuous, given that f:X\to I and g:X\to I are continuous (here I is the unit interval [0,1]). It seems that F is continuous, but I want to show that explicitly. Any help appreciated! X is any topological space. (I...

http://lanl.arxiv.org/pdf/quant-ph/0501041 the pioneer anomaly is due to topological phase defect of light

I don't know if this paper was commented on here while I was away at Christmas time, but I was just pointed to it by a paper on today's arxiv and I think it desrves notice. http://arxiv.org/abs/hep-th/0411073 Robert Dijkgraaf, Sergei Gukov, Andrew Neitzke, and Cumrun Vafa; Toplogical...

The critical density determines the Universe evolution along time. So, by measuring this density, we could know about the finite or infinite age of the Universe. But we don't know from General Relativity if our Universe is spatially finite or infinite. As far as I know, such question...

This is a chess/math puzzle I invented: Consider a fully set up chess board (in starting position). Invent a condition (new rule) in which black wins without either side making any moves. Note: this problem has a really abstract and *topological* approach. :smile: Please email me...

I was wondering if knotted topological strings can be asigned definite quantum charges apart from mass and angular mumentum. That would really be useful to glueball people who try to find candidate in the hadronic spectrum : they use only mass and angular momentum. If they had a mean to assign...

Is it true that a perfect generalized ordered space can be embedded in a perfect linearly ordered space? It is true that a perfect generalized ordered space can be embedded as a closed subset in a perfect linearly ordered space.

What is topological defects in cosmology? Is this a current field of research and what are their progresses?

I have printed a notes about differential geometry, and it says: -A Coo differentiable structure on a locally Euclidean, Hausdorff topological space M of dimension m is a collection of coordinate systems F Then it specifies the conditions that F must satisfy, but I'm a little lazy and won't...

I'm a noob starting out studying differential geometry and topology. Really probably somewhere in the multivariate calculus level, but I've been trying to understand the plethora of terminology I'm encountering with this higher math. I've been reading a lot on Wikipedia.org and PlanetMath.org...

I believe that the Axioms for TQFT were set out by Atiyah in 1990 and that one of the equivalent definitions of a TQFT is in category terms: a TQFT is a functor from the category of n-dimensional cobordisms to the category of Hilbert spaces, satisfying certain conditions. Is anyone familiar...

From the abstract: "An action principle is described which unifies general relativity and topological field theory. An additional degree of freedom is introduced, and depending on the value it takes the theory has solutions that reduce it to (1) general relativity in the Palatini form, (2)...

Chern-Simons theory is a 3d-TQFT and Crane-Yetter model is a 4d-TQFT There exist another Topological Quantum Field Theory apart of these two models?