Topological Definition and 265 Threads

Does every (continous and second countable) topological manifold have an Euclidean neighbourhood around each of its points whose closure equals the whole manifold?

The definition of Y being connected in a topological space (X, tau) is that you can't find two non-empty, open and disjoint sets whose union is Y. This doesn't quite make much intuitive sense to me. For example, consider R with the usual topology. Then clearly, Y= [0,1] union [2,3] is not...

If A\subseteq B are both subsets of a topological space (X,\tau), is it true that any closed subset of A is also a closed subset of B?

Hi. I'm trying to find the degree of the map of f(g,h)=g.h (i.e. multiplication in g) for fixed g. It is a map G-->G (if we fix g). We can assign a degree to this map for any topological group for which the last non-zero homology group is Z and proceed like we do for the degree of a map...

If A and B are subsets of G, let A*B denote the set of all points a*b for a in A and b in B. Let A^(-1) denote the set of all points a^(-1), for a in A. a)A neighborhood V of the identity element e is said to be symmetric if V = V^(-1) . If U is a neighborhood of e, show there is a...

The integers Z are a normal subgroup of (R,+). The quotient R/Z is a familiar topological group; what is it? okay... i attempted this problem... and I don't know if i did it right... but can you guys check it? Thanks~ R/Z is a familiar topological group and Z are a normal subgroup of...

Let H be a subspace of G. Show that if H is also a subgroup of G, then both H and\bar{H} are topological groups. So, this is what I've got... if H is a subgroup of G then H \subset G. Since H is a subspace of G then H is an open subset. But, i don't even know if that's right. How...

Is it reasonable to work with a linear space whose subspaces are considered as open subsets of the linear space when the linear space is considered as a topological Space? Actually, this linear space is spanned by a topological space with known topology.

I am reading a paper right now on lattice QCD that presents a "method that improves the cooling method and constructs an improved topological charge operator based on the product of link variables forming rectangular Wilson loops." Unfortunately I...

It is a fact that if X is a compact topoloical space then a closed subspace of X is compact. Is an open subspace G of X also compact? please consider the following and note if i am wrong; proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the...

I have been going through some papers on lattice QCD lately, and many of them mention "topological charge". I was wondering if someone could either explain what is meant by this term, or point me to a resource that has an explanation. Thanks

I had a quick question: Is the following proof of the theorem below correct? Theorem: If C is a convex subset of a Topological vector space X, and the origin 0 in X is contained in C, then the set tC is a subset of C for each 0<=t<=1. Proof: Since C is convex, then t*x + (1-t)*y...

Let C([0,1]) be the collection of all complex-valued continuous functions on [0,1]. Define d(f,g)=\int\limits_0^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx for all f,g \in C([0,1]) C([0,1]) is an invariant metric space. Prove that C([0,1]) is a topological vector space

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} ?

Basic premiseses: Any universe can be expressed using a suitable topology containing points. At least the following (coherent) definitions exist for the multiverse (there can be more/others): 1. Topological disconnected universes Universes which could be very much the same as ours, and could...

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 ?

1. Thread topological defects and domain walls. In which cases these defects involve new items in Hamiltonians of models? The connection between thread topological defects and first-order phase transitions (for instance in the Nielsen-Olesen model, helium II, in which Chalatnikov quantum...

[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?