Topological spaces Definition and 48 Threads

A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of...

Given a topological manifold, this may or may not admit a ##C^1## atlas (i.e. starting from its maximal atlas it is or it isn't possible to rip charts from it to get an atlas of ##C^1## compatible charts). A theorem due to Whitney states that from such a topological manifold ##C^1##-atlas (if...

##SO(3)## is a Lie group of dimension 3. It is the set of 3x3 matrices ##R## with the following properties: $$RR^T = R^TR=I, \text{det}(R)=+1$$ There exists a parametrization of ##SO(3)## that maps it on the sphere in ##\mathbb R^3## of radius ##\pi## where the antipodal points are identified...

I've a doubt regarding Lemma 1.35 (Smooth Manifold Chart Lemma) from J. Lee "Introduction to Smooth Manifolds" The proof claims that Hausdorff property follows from v). However v) includes the case where both ##p## and ##q## are included in the same ##U_{\alpha}##, i.e. their neighborhoods are...

Can you provide an example of homeomorphism onto the image φ:U→φ(U) where the image φ(U)⊂M is not open in M w.r.t. its assigned topology ? Thanks.

Consider a non-injective map ##\pi## from a set ##M## to a set ##N##. ##N## is equipped with a topological manifold structure (Hausdorff, second-countable, locally euclidean). Take the initial topology on ##M## given from ##\pi## (i.e. a set in ##M## is open iff it is the preimage under ##\pi##...

How do I find the metric tensor on ##S^1## x ##S^2##?

Hi, I've a doubt about the following example in "Introduction to Manifold" by L. Tu. My understanding is that if one assumes the subspace topology from ##\mathbb R^2## for the "cross", then one can show that the topological space one gets is Hausdorff, second countable but non locally...

Hi, in the books I looked at, the 2-sphere manifold is introduced/defined using its embedding in Euclidean space ##\mathbb R^3##. On the other hand, Mobius strip and Klein bottle are defined "intrinsically" using quotient topologies and atlas charts. I believe the same view might also be...

Hello. So, the question is do you know any applications of topological spaces which are not homeomorphic to R^n in physics? Motivation for the question i am making: as i think if a topological space is homeomorphic to R^n then differential calculus is allowed on it. Modern physics uses i think...

I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 4, Section 4.1: Sequences ... I need help in order to fully understand Example 4.1.1 ...Example 4.1.1 reads as follows: In the above example from Singh we read the following: "...

I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ... I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ... I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows: In the...

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand an aspect of Example 3.34 (c) on page 102 ... ... Examples 3.34 (plus some relevant definitions ...)...

Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8 ##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)## As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in...

I have a hypothetical universe where the distance between two points in spacetime is defined as: $$ds^2 =−(\phi^2 t^2)dt^2+dx^2+dy^2+dz^2$$Where ##\phi## has units of ##km s^{-2}##. The space in this universe grows quadratically with time (and, as I understand it, probably isn’t Minkowski...

I am newbie to topology and trying to understand covering maps and quotient maps. At first sight it seems the two are closely related. For example SO(3) is double covered by SU(2) and is also the quotient SU(2)/ℤ2 so the 2 maps appear to be equivalent. Likewise, for ℝ and S1. However, I...

<Moderator's note: Moved from General Math to Differential Geometry.> Let p:E→ B be a covering space with a group of Deck transformations Δ(p). Let b2 ∈ B be a basic point. Suppose that the action of Δ(p) on p-1(b0) is transitive. Show that for all b ∈ B the action of Δ(p)on p-1(b) is also...

Is it possible to have a topological space in which three 2-spheres A, B, C are such that B is in some sense nested inside A, C is nested inside B, but A is again nested in C. What about for three 2-tori in a similar manner?

Homework Statement We define ##X=\mathbb{N}^2\cup\{(0,0)\}## and ##\tau## ( the family of open sets) like this ##U\in\tau\iff(0,0)\notin U\lor \exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}## ##a)## Show that ##\tau## satisfies that axioms for...

Homework Statement show that the two topological spaces are homeomorphic. Homework Equations Two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse between them The Attempt at a Solution I have tried proving that these two spaces are homeomorphic...

https://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf I'm reading the paper linked above (page 10) and have a simple question about notation and another that's more of a sanity check. Given a space ##Y## and a spacetime ##X## the author talks about the associated Quantum Hilbert Spaces...

Homework Statement I'm trying to understand the intuition behind path-connectedness and simple-connectedness in finite topological spaces. Is there a general methodology or algorithm for finding out whether a given finite topological space is path-connected? Homework Equations how can I...

Homework Statement Hello All, I am experiencing Adventures in Topology. So far, so good, but I have an issue here. In the topological space (Real #s, U), show that 1 is not an element of Cl((2,3]).Homework Equations The closed subsets of our topological space are the converses of the given...

Homework Statement . If ##(x_{\alpha})_{\alpha \in \Lambda}## is a net, we say that ##x \in X## is an accumulation point of the net if and only if for evey ##A \in \mathcal F_x##, the set ##\{\alpha \in \Lambda : x_{\alpha} \in A\}## is cofinal in ##Lambda##. Prove that ##x## is an accumulation...

Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations The Attempt at a Solution I really don't know how to do this. Wikipedia entry for 'base sets' redirects to Pokemon...

Hi all! My question is the following. Suppose we have two normal topological spaces X and Y and we have a continuous map from a closed subset A of X to Y. Then we can construct another topological space by "glueing together" X and Y at A and f(A). By taking the quotient space of the disjoint...

Homework Statement Let ℝ be set of real numbers. Which of the following collection of subsets of ℝ defines a topology in ℝ. a) The empty set and all sets which contain closed interval [0,1] as a subset. b)R and all subsets of closed interval [0,1]. c)The empty set, ℝ and all sets...

Dear All, It sounds a strange question, we know that the measure theory is the modern theory while the topological spaces is the classical analysis (roughly speaking). And measure theory solves some problems in the classical analysis. My first question is that right? Second, Is every...

This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult...

The Markov chain, as you know, is a sequence of random variables with the property that any two terms of the sequence X and Y are conditionally independent given any other random variable Z that is between them. This sequence (which is in fact a family, indexed by the naturals) can and has been...

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...

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...

"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...

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 :)

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...

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...

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 ...

Not really homework, but a typical exercise question, so I figured it's appropriate to post it here. Homework Statement X,Y topological spaces f:X→Y x is a point in X Prove that the following two statements are equivalent: (i) f^{-1}(E) is open for every open E that contains f(x)...

Which would fit this description and with answers?

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...

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...

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...

"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...

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.

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...