Topological Definition and 265 Threads

As well known, for any topological space (X,T), there is a smallest measurable space (X,M) such that T\subset M. We say that (X,M) is generated by (X,T). Right now, I was wondering whether the "reverse" is true: for any measurable space (X,M), there exists a finest topological space (X,T) such...

I'm used to the notation f : X --> Y for a map, where X and Y are sets. I recently came across this notation for a map between topological spaces, where the second item of each pair is a topology on the first: f : (X,{t}a) --> (Y,{tb}) Is the notation to be read "f maps each element of X...

Homework Statement Let (A,S) and (B,T) be topological spaces and let f : A -> B be a continuous function. Suppose that D is dense in A, and that (B,T) is a Hausdorff space. Show that if f is constant on D, then f is constant on A. Homework Equations D is a dense subset of (A,S) iff the...

Can someone explain the difference between the two? 2 topo spaces are isometric if they have the same metric properties and homeometric if they have the same topological properties. If a space is homeo it is iso, but not vice verse. Which begs the conclusion that every topological...

Can someone please recommend any topological QFT text .I searched amazon and found nothing

I am confused as to exactly what a local base at zero (l.b.z.) tells us about a topology. The definition given in Rudin is the following: "An l.b.z. is a collection G of open sets containing zero such that if O is any open set containing zero, there is an element of G contained in O". Ok, great...

Dear all, a homomorphism is a continuous 1-1 function between two topological spaces, that is invertible with continuous inverse. My question is as follows. Let's take the topologies of two topological spaces. Is there a 1-1 function between the two collections of open sets defining the...

I had been reading several articles on topological insulators (TI) including the Kane and Hasan's 2010 RMP. I am not very much clear about the Z_2 invariant TI. I mean, the even-odd argument proposed by Kane and Male (also argued by S. C. Zhang's group and Joel Moore's group in a different way)...

Definition. A complex K,when regarded as a topological space,is called a polyhedron and written |K|. I think it is easy to understand the definition,but there are some theorem and problems involving it confused me. 1.Let K be a simplicial complex in E^n,if we take the simplexes of K...

"Cauchy" Sequences in General Topological Spaces Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every...

Homework Statement Prove that Hausdorff is a topological property. Homework Equations The Attempt at a Solution For showing that a quality transfers to another space given a homeomorphism, we must show that given a Hausdorff space (X,T) and a topological space (Y,U), that (Y,U)...

Prove that the intersection of any collection of closed sets in a topological space X is closed. Homework Statement Homework Equations The Attempt at a Solution

Homework Statement Find a topological space which does not have a countable basis. Homework Equations Definition of basis : A collection of subsets which satisfy: (i) union of every set equals the whole set (ii) any element from an intersection of two subsets is contained in another...

On the plane R^{2} let, B= {(a,b) x (c,d) \subset R^{2} | a < b, c < d } a.) Show that B is a basis for a topology on R^{2}. This means I have to show that every x in R^{2} is contained in a basis element, and that every point in the intersection of two basis elements is contained in...

when we say "a topological action", do we only mean that the action is metric free? or is there some other meaning for this expression? What does the word topological mean exactly? Thanks!

Hello, I am learning about manifolds but I am not understanding part of the definition. This is what I'm looking at for defining the n-manifold M. (i) M is Hausdorff (ii) M is locally Euclidean of dimension n, and (iii) M has a countable basis of open sets I have a problem with (ii)...

Homework Statement Let Y be a topological space. Let CY denote the cone on Y. (a) Show that any 2 continuous functions f, g : X --> CY are homotopic. (b) Find (pi)1 (CY, p). Homework Equations I have no idea. The professor said one problem would be way out in left, to see who could make the...

Hi all, I'am starting a Phd In Theoretical Condensed Matter Physics, and I would like to produce a thesis on the Topological Insulators topic. Unfortunately I don't have a background in Consensed Matter Physics (in my curriculum there are exams about General Relativity, Quantum Field Theory...

In Rudin's book Functional Analysis, he makes the following claim about the interior A^\circ of a subset A of a topological vector space X: If 0 < |\alpha| \leq 1 for \alpha \in \mathbb C, it follows that \alpha A^\circ = (\alpha A)^\circ, since scalar multiplicaiton (the mapping f_\alpha: X...

Is this a legitimate definition for an "absorbing set" in a topological vector space? A set A\subset X is absorbing if X = \bigcup_{n\in \mathbb N} nA. This is the definition the way it was presented to us in my functional analysis class, but I'm looking at other sources, and it seems everyone...

I'm trying to simplify an action that has the term: levicivita_[a,b,c,d]*levicivita^[mu,nu,rho,sigma]*R^[a,b]_[mu,nu]*R^[c,d]_[rho,sigma] where a,b,c, and d are flat indices and mu nu rho sigma are curved indices I got the term: 4*e^mu_a*e^nu_b*e^rho_c*e^sigma_d*R^a,b_mu,nu*R^c,d_rho,sigma...

I'm trying to simplify an action that has the term: levicivita_[a,b,c,d]*levicivita^[mu,nu,rho,sigma]*R^[a,b]_[mu,nu]*R^[c,d]_[rho,sigma] where a,b,c, and d are flat indices and mu nu rho sigma are curved indices I got the term: 4*e^mu_a*e^nu_b*e^rho_c*e^sigma_d*R^a,b_mu,nu*R^c,d_rho,sigma...

Homework Statement Suppose X = A\cupB where A and B are closed sets. Suppose f : (X, TX) \rightarrow (Y, TY ) is a map such that f|A and f|B are continuous (where A and B have their subspace topologies). Show that f is continuous. What happens if A and B are open? What happens if A or B is...

I am having trouble proving this statement. Please help as I am trying to study for my exam, which is tomorrow Prove that the standard topology on RxR is equivalent to the one generated by the basis consisting of open disks. Thanks :)

I've recently started learning about topological insulators. I've read a considerable amount of (review) papers on the subject, yet I still only have a phenomenological understanding of what a topological insulator is. I know for example, that the gapless surface states have to be there because...

Hi, these days I have been trying to understand the essentials of the so-called topological insulators (TBI), such as Bi2Te3, which seem very hot in current research. As i understand, these materials should possesses at the same time gapped bulk bands but gapless surface bands, and spin-orbit...

Homework Statement Let X be a non empty T1 space (i.e. such one that for every two distinct points each one of them has a neighborhood which doesn't contain the other one). One needs to show that every connected subset of X, containing more than one element, is infinite. The Attempt at a...

Here's another problem which I'd like to check with you guys. So, let X be a topological space which satisfies the second axiom of countability, i.e. there exist some basis B such that its cardinal number is less or equal to \aleph_{0}. One needs to show that such a space is Lindelöf and...

Homework Statement Let X be a topological space. If A is a subset of X, the the boundary of A is closure(A) intersect closure(X-A). a. prove that interior(A) and boundary(A) are disjoint and that closure(A)=interior(A) union boundary(A) b. prove that U is open iff Boundary(U)=closure(U)-U...

Homework Statement In a first countable compact topological space, every sequence has a convergent subsequence. Homework Equations N/A The Attempt at a Solution I'm self-studying topology, so I'm mostly trying to make sure that my argument is rigorous. I understanding intuitively...

Hello my friends! My textbook has the following statement in one of its chapters: Chapter 8:Topology of R^n If you want a more abstract introduction to the topology of Euclidean spaces, skip the rest of this chapter and the next, and begin Chapter 10 now. Chapter 10 covers topological...

Homework Statement Let K be a closed&bounded set in R^n which isn't empty. Prove that K isn't open. Homework Equations No topology! I can't use the fact that the only sets in R^n which are closed and open are the empty set and R^n... Only the definitions of open sets and closed sets...

The definition of a homeomorphism between topological spaces X, Y, is that there exists a function Y=f(X) that is continuous and whose inverse X=f-1(Y) is also continuous. Can I assume that the function f is a bijection, since inverses only exist for bijections? Also, I thought that if a...

What do you think about the reason why topological objects are physically interesting?

why are topological insulators called TOPOLOGICAL insulators? what factor of topology apperas in the phenomenon

Matter as topological entropy (Fotini disordered locality) I have to go in 5 minutes or so, but will get back to this. Fotini M. has this scheme or picture of disordered locality. The root meaning of topology is locality. Disordered locality is disordered topology. Topological entropy. I...

Let X be an infinite set and p be a point in X, chosen once and for all. Let T be the collection of open subsets V of X for which either p is not a member of V, or p is a member of V and its complement ~V is finite. Now, let (a_n) be a sequence in X (that is, for all n in N, a_n in X) such...

Hi Guy's I am just starting out in topology and I was wondering if someone might know of a good link that may have venn diagrams of some important topological terms ie closure of A, int A, limit points etc. regards Brendan

I've come across this question during revision and don't really know what you would say? Any help? Regard a 2 x 2 matrix A as a topological space by considering 2x2 matrices as vectors (a,b,c,d) as a member of R4. Let GL2(R) c R4 be the subset of the 2x2 matrices A which are invertible, i.e...

In calculus, the definition of the arc length of some curve C is the limit of the sum of the lengths of finitely many line segments which approximate C. This is a perfectly valid approach to calculating arc length and obviously it will allow you calculate correctly the length of any...

Homework Statement For any a \in \left( -1,1 \right) construct a homeomorphism f_a: \left( -1,1 \right) \longrightarrow \left( -1,1 \right) such that f_a\left( a \right) = 0 . Deduce that \left( -1,1 \right) is homogeneous.Homework Equations Definition of a homogeneous topological...

I recently took a great interest in topological quantum computing - so great an interest I am even considering it as a thesis topic for grad school (though I am still a junior undergrad and have awhile to figure that out). What would be some useful courses to take to pursue theoretical research...

Topological string theory is a description devoid of metric and hence is background independent and everything emerges from pure topological considerations. This should put it at the front of all other candidate string theories, but that is not the case (it is certainly considered important, but...

I had the following thought/conjecture: Two topological spaces are homeomorphic iff the two topologies are isomorphic. When I say that the two topologies are isomorphic, I mean that they are both monoids (the operation is union) and there is a bijective mapping f such that f(A) U f(B) = f(A...

hi I was having difficulty with this problem in the book If (1/n) is a sequence in R which points (if any) will it converge (for every open set there is an integer N such that for all n>N 1/n is in that open set) to using the following topologies (a) Discrete (b) Indiscrete (c) { A in X ...

Is it an oversimplification to say a countably infinite set is an algebra while an uncountably infinite set is a topology?

Homework Statement Let boundary = bdy, ∩ = intersection and C = contained. Show that the bdy (A) ∩ bdy(B) C bdy (A ∩ B). Homework Equations The Attempt at a Solution I can draw a diagram of this idea and visualize it my mind, but I cannot formally show this (this is second proof...

Homework Statement X is a set and P(X) is the discrete topology on X, meaning that P(X) consists of all subsets of X. I want to prove that X is metrizable. Homework Equations My text says that a topological space X is metrizable if it arises from a metric space. This seems a little unclear to...

so first let's take RP^2. I have a little trouble grasping why we can put a subspace topology on it. So RP^2 is the set of all lines through the origin in R^3. So if we take some subset S of RP^2 and the if set of points in R^3 which is the union of these lines in S is open then we can say we...

Can someone explain how topological vector spaces are more general than normed ones. So this means that if I have a normed vector space, X, it would have to induce a topology. I was thinking the base for this topology would consist of sets of the form \{y\in X: ||x-y||<\epsilon, \textrm{for some...