Quotient space is a space where the underlying space is set of equivalence classes of some equivalence relation. As this concept appears in various areas, include also a tag specifying subject matter, such as (topology), (vector-spaces), (normed-spaces), etc.

learn more… | top users | synonyms

2
votes
0answers
31 views

How do I interpret “K2 mod K1”?

I've been doing a bit of work these past couple of nights on computing cyclic subspace decompositions, finding cyclic bases, and then computing the Jordan canonical form of matrices. My question is: ...
5
votes
2answers
51 views

How to recover the cohomology of a torus from its description of a quotient

Note: here, "cohomology" means "De Rham cohomology". I know how to compute the De Rham Cohomology of a torus $T=\left(S^1\right)^n$ using Kunneth formula. But a torus can also be obtain as a ...
4
votes
1answer
48 views

Showing that $\mathbb S^1$ is a deformation retract of the Mobius strip, rigorously.

Intuitively, I can see why this is. I've found a few threads about this, but they only provide, for example, a deformation retraction of $I \times I$ to its diagonal $D = \{ (x,x) \in I \times I \}$, ...
5
votes
4answers
101 views

An instance of quotient Space $X/M$

Hi Guys i can't find a simple example (with analytic description i mean) that helps me to understand the meaning of quotient space. I've understood the definition ($X$ normed linear space, $M$ closed ...
0
votes
0answers
8 views

Literature on Quadratic Residues in Polynomial Quotient ring

I have a literature-question. Given a field $\mathbb{F}$, finite or infinite, and an element $f$ in the polynomial ring $\mathbb{F}[t]$. I am searching for results of any kind about quadratic residues ...
1
vote
6answers
72 views

Constructing $\mathbb{C}$ from $\mathbb{R}$

I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$. As far as I understand, $\mathbb{R}[t]$ is the set of all ...
3
votes
0answers
31 views

A non-singular quotient of $\mathbb{A}^n$ by a cyclic group is isomorphic to $\mathbb{A}^n$

Let $G$ be a cyclic group acting linearly on $X := \mathbb{A}^n$. If we assume that the quotient $Y:=X/G$ is non-singular, does it follow that $Y \simeq \mathbb{A}^n$? If so, is it necessary to assume ...
0
votes
2answers
27 views

Interpreting a Quotient Group ($D_8/\langle r^2\rangle$) in 2 Distinct Ways

I seem to have a misunderstanding about quotient groups. Let $D_{2n}$ denote the group of symmetries of an $n$-gon and let $V_4$ denote the Klein-4 group. On one hand, if we identify $r^2$ with $1$, ...
3
votes
1answer
19 views

Homology of a pair of simplicial complexes

Let $K=\{a, b, c, ab, bc\}$ (in graph-theoretic terms, a path $a,ab,b,bc,c$, where $a, b,$ and $c$ are vertices and $ab$ and $bc$ are edges) be a simplicial complex and let $L=\{a, b, c\}$ be a ...
2
votes
3answers
75 views

What does the quotient group $(A+B)/B$ actually mean?

I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$. When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements ...
5
votes
1answer
56 views

Universal property of quotient group

Let $H\le G$ be a normal subgroup of $G$. Then we can think a natural projection map $\pi:G \to G/H$. Then this map has the following universal property: Let $\phi:G \to G'$ be a homomorphism. If $H ...
0
votes
1answer
41 views

Question on proving quotient space is homeomorphic to circle [duplicate]

I am new to quotient spaces and was given this in class. I really have no idea how to solve. I tried one approach that my teacher said to be incorrect so I'd really appreciate the help on this. I am ...
2
votes
1answer
31 views

Lifting idempotents from a quotient of a Banach algebra

In a quotient of a Banach algebra $A$, if an invertible element is connected to the identity by a continuous path of invertibles, then it can be lifted to an invertible element in $A$. Is there an ...
2
votes
2answers
36 views

Dimension and basis of a quotient space

I'm having some problems understanding this: $$V = \mathbb R^3\text{ and }W = \{(x,y,z) \mid x+y+z=0\}$$ So I want $V/W$ and a basis to it. $$\dim V = 3$$ $$\dim W = 2$$ $$\dim V/W = 1$$ But a ...
3
votes
1answer
56 views

Homeomorphism between $S^2$ and $CP^1$ via uniqueness of quotient

I am trying to show that $S^2$ and $\mathbb{C}P^1$ are homeomorphic making use of the following result - see e.g. Jack Lee Introduction to topological manifolds. Let $Y \xrightarrow{\pi_1} X_1 $, $Y ...
2
votes
0answers
40 views

Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = ...
0
votes
1answer
14 views

Is the orbit map for a group action closed in this case?

Suppose a compact Lie group $G$ acts on a manifold $M$ and let $\pi : M \rightarrow M/G$ be the orbit map. Can I say that $\pi$ is closed map? If $C \subseteq M$ is a closed set in $M$ then I only ...
2
votes
2answers
39 views

Characterizing the A-module M/S

I've been working through this for a little while, and I'm not 100% sure I understand what I'm supposed to be doing here, or maybe I'm not grasping correctly what they mean by "Characterize". ...
1
vote
2answers
39 views

Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
0
votes
0answers
20 views

Matrices of Quotient Representation

I've been working through some exercises in Sagan's book on the representation theory of the symmetric group and I came across the following exercise that's been giving me some trouble: Let $X$ be a ...
2
votes
1answer
79 views

Isometry between $X^\ast/M^\perp$ and $M^\ast$.

Let $X$ be a normed linear space, $M \subset X$ be a subspace, $M^\perp = \{x^\ast \in X^\ast \mid x^\ast\big|_M = 0\}$ be the annihilator of $M$, $X^\ast$ the topological dual of $X$, and let's ...
0
votes
0answers
29 views

Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed

Problem is: Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed. ($U$ is saturated $\iff$ $\exists V \in Y$ s.t. $U = \pi^{-1} ...
2
votes
0answers
16 views

$U$ takes the same value on $\pi$ then $U$ is saturated

Let $\pi : X \to Y$ be any map, and $U$ be a subset of $X$. The problem is: "$\forall x\in U$, $ \pi (x) = \pi(x') $, then $x' \in U$" then $U$ is saturated. (U is saturated $\iff$ $\exists V ...
1
vote
1answer
31 views

The topology on $X / G$ where $G$ acts on $X$

The elements are orbits, but how do we find the neighbourhoods? In particular, let $G= ( \mathbb R^+ , \cdot )$ and $X=[0, \infty )$. Let $G$ act on $X$ via the usual multiplication. Then $X/G = ...
3
votes
1answer
47 views

Is a compact subset of a topological group G/N closed if G is hausdorff?

I just used this hypothesis when I was proving a theorem.But I was not sure if this hypothesis is correct.
1
vote
0answers
37 views

Differences between a quotient map and a continuous function in topology

Def. for a continuous function: Let $X$ and $Y$ be topological spaces. A function $f : X \rightarrow Y$ is continuous if $f^{-1} (Y)$ is open in $X$ for every open set $V$ in $Y$. Def. for ...
4
votes
1answer
30 views

Dimension of a certain $L^p$ quotient space.

Define $L^p_0 := \{ f \in L^p : \int f = 0 \}$. I am trying to calculate the dimension of the cokernel of the inclusion operator $i:L^p_0 \to L^p$. That is, I am trying to calculate $$\dim ( L^p / ...
2
votes
1answer
33 views

Path components of quotient space

Let $X$ be a topological space and $A \subseteq X$ a subspace. Is it true that $X/A$ is path-connected if and only if $A$ meets every path component of $X$? Intuitively this seems reasonable but I'm ...
1
vote
1answer
35 views

Is this quotient of a disk Hausdorff?

Let $D$ be the disk in the complex plane, $D = \{ z : |z|<1\}$. Let $\sim$ be an equivalence relation on $D$ defined by $z_1\sim z_2$ iff $|z_1| = |z_2|$. Is the quotient space $D/{\sim}$ is ...
2
votes
0answers
33 views

Is a canonical projection always continuous in the quotient topological space

I'm solving some exercises on quotient topology, and I wonder if a canonical surjection is always continuous. Let $\pi: (X,\tau_X)\to (X/_\sim,\tau_\sim)$ be a canonical surjection. I think ...
2
votes
1answer
46 views

Difficulty understanding why a set is saturated with respect to a map

The following is the beginning of Example 7 on page 143 of Munkres' Topology. Example 7. The product of two quotient maps need not be a quotient map. Let $X = \mathbb{R}$ and let $X^*$ be the ...
8
votes
3answers
124 views

Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff

$\newcommand{\R}{\mathbf R}$ Let $V$ be an $n$-dimensional vector space and $k$ be an integer less than $n$. A $k$-frame in $V$ is an injective linear map $T:\R^k\to V$. Let the set of all the ...
2
votes
1answer
58 views

Quotient group and adjoint matrix

The exercise 1211 in "Problems and Solutions in Mathematics" by Ta-Tsien: Let $M$ be an $n \times n$ matrix of integers. Suppose that $M$ is invertible when viewed as a matrix of rational numbers. ...
3
votes
1answer
16 views

A Continuous Choice of $k$-Subspaces of a Vector Space Gives a Continuous Choice of Bases

$\newcommand{\R}{\mathbf R}$ The Grassmannian $G_k(\R^n)$ as a topoplogical space is defined in the following way: Let $F_k(\R^n)$ be the collection of all the linearly independent lists of size $k$ ...
0
votes
0answers
50 views

Quotients vs equivalence relations

In a recent preprint I defined the category of quasi-frames (qframes for short) as follows: a qframe is a modular and upper continuous complete lattice; a morphism of qframes $f:L_1\to L_2$ is a map ...
4
votes
1answer
49 views

question about direct sum of vector fields and preservation under quotient spaces

Hello all I was given this question in linear algebra it is two parts and asks to prove or give a counterexample. We are given a vector space V and a subspace of it W and the quotient map $ \pi : V ...
6
votes
1answer
95 views

Unifying Connection Between Topological Embeddings and Quotient Maps

In a book on topology I'm reading the following theorem seemed striking to me, not for its proof, which I believe I understand, but because there's some nice symmetry going on that I'd perhaps like ...
0
votes
2answers
26 views

Quotient rings, polynomials are reducibility

I am trying to follow this solution. I am struggling to understand why 'If g is a member of R, then g divides the content of f'. Why is this true?
2
votes
0answers
47 views

Residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$

What are the residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$? After finding the maximal ideals of $\mathbb{R}[X,Y]$, which are of the form: $\langle X-a,Y-b \rangle$ with $a,b ...
1
vote
1answer
33 views

Clarification about Quotient Spaces

The question given to me is: Does there exist a vector space $V$ which has a nonzero subspace $U$ such that $V /U \cong V$ ? Provide an example or a proof that no such $V /U$ exists. Intuitively, I ...
3
votes
1answer
94 views

Are $S^1$ and $\mathbb{R}/{\sim}$ the “same thing”?

I am reading quotient space of topology and I am a little bit confused. I am looking at the relationship between $\mathbb{R},S^1$ and the quotient space $\mathbb{R}/{\sim}$, where the relation $\sim$ ...
4
votes
2answers
116 views

Compute Quotient Space

I have been struggling with this computation for a while now. I thought I was almost there, but it now results I still have nothing. So here is the initial problem: Let $c=\left\{ (x_j)_j \subset ...
1
vote
1answer
33 views

Quotient space of infinite dimensional vector space

On an exam today I used that if $X=\mathcal{C}[a,b]$ and $Y=\{f\in X : f(a)=f(b)\}$, then the projection $\pi: X\rightarrow X/Y$ has the property $\ker(\pi)=Y$. This led me to the following: Suppose ...
1
vote
1answer
28 views

Globally defined exponential in a particular homogeneous space

I'm currently working in a particular conformal compactification/completion of the Minkowski space-time, but I'm stuck at showing that the exponential of every vector field in it is globally defined. ...
1
vote
0answers
53 views

How do I show that this function is continuous? [duplicate]

I'm interested in showing that $CX=\frac{I\times X}{\{1\}\times X}$ is contractible. I defined the d.r $F(s,[t,x])=[(1-s)t+s,x]$ and the only missing part for me is to show that it is continuous. How ...
2
votes
1answer
49 views

About direct sum of abelian groups and quotient

I'm trying to understand properly the relations between quotient and direct sum. The first thing I wanted to know, and couldn't find online, is whether my guess is true or not: Assume $G_\alpha$ are ...
2
votes
0answers
94 views

Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
1
vote
1answer
52 views

“Quotient” as a verb

People here do use "quotient" as a verb: I searched for "quotienting" and got 12,890 results. [Edit: It's not as bad as I thought. Apparently I didn't understand how the search function works. When I ...
1
vote
1answer
43 views

Projection Mappings are Quotient Mappings?

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X=X_1\times X_2$ is a product space, then the first coordinate projection is a ...
0
votes
0answers
32 views

Induced map between quotient vector spaces well defined and linear?

Suppose we have vector spaces $V,W$ and subspaces $$ V\supset V'\supset V''\\ W\supset W'\supset W'' $$ suppose also that we have a linear map $A:V\to W$. What does it take for this map to induce a ...