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.

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Quotient space is not second countable

I was searching for an easy example of a quotient space $X$/~ which is not a second countable space even though $X$ is a second countable topological space. I have found an example in the following ...
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Explicit Description for an Equivalence Relation

Given a set function $f : X \to X$ let $\sim$ be the equivalence relation $x \sim f(x)$. Contextually, I am working with the coequalizer of $f$ and $1_X$. I want to have as much information about the ...
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15 views

Dimension of unordered configuration space

I'm working on $X = \mathbb{R}^n$ and I'm considering the set of unordered sequence of points of $X$. Considering $F(p) = \lbrace (x_1, \dotsc, x_p) \in X^p ; i \neq j \implies x_i \neq x_j \rbrace$ ...
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1answer
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How to prove thar O(Ng) | O(g)

I have this exercize: $G$ is a group. $N\subset G$. Need to prove that: $$o\left(Ng\right)\mid o(g)$$ where $Ng\in G/N$. For now, without using the canonic homomorphism $\tau \left(g\right)=Ng$. ...
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Third isomorphism theorem, about quotients

I'm trying to understand the third isomorphism theorem statement, specifically the one in Wikipedia: https://en.wikipedia.org/wiki/Isomorphism_theorem#Third_isomorphism_theorem I'm stuck at the point ...
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Understanding Quotient spaces - Shrinking down

I am looking at Page 57 of Kreyszig's Functional Analysis, and I have been given an exercise: Let $X=\Bbb R^3$ and $Y=\{(\xi_1,0,0)| \xi_1 \in \Bbb R\}$ 1) Find $X/Y$: So $X/Y=\{[x]:x+Y,\forall ...
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Determining the value of A given $Z_4=Z_8\oplus Z_2/A$

Let Z define the integers and $Z_a$ define the integer group modulo a. I want to determine what A is. Given $Z_4\cong Z_8\oplus Z_2/A$, where $A\subset Z_8\oplus Z_2$, am I able to just say that ...
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Quotient Space $X^*$ is homeomorphic to the Subspace $S^2$ of $\mathbb R^3$

Let $X$ be the closed unit ball $\{ x^2 + y^2 \leq 1 \}$ in $\mathbb R^2$ and let $X^*$ be the partition of $X$ consisitingof all the one point set $\{ x \times y \}$ for which $x^2 + y^2 < ...
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An example of open closed continuous image of $T_2$-space that is not $T_2$

Engelking in his "General Topology" states that $T_2$ separation axiom is not preserved under open closed continuous surjections. In "General Topology" by Stephen Willard I have found two separate ...
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2answers
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An example of open closed continuous image of $T_0$-space that is not $T_0$

Engelking in his "General Topology" states that $T_0$ separation axiom is not preserved under open closed continuous maps. But I can't find any example of open closed continuous image of $T_0$-space ...
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51 views

The cone of a topological space is contractible (why is the homotopy well defined?)

If $X$ is a topological space, define ${\rm Cil}(X) = X \times I$ the cylinder over $X$, and the cone over $X$, $\operatorname{Con}(X) = \operatorname{Cil}(X)/{\sim}$ the quotient by saying that ...
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47 views

Finite normal subgroups of $SO(4)$

What are the finite normal subgroups of $SO(4)$? If these do not exist (or if they are trivial, e.g. from some projection to $SO(2)$), are there different finite normal subgroups of $O(4),$ $U(4)$, ...
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82 views

Is the sphere $S^2$ diffeomorphic to a quotient of the square?

If we take the square $[0,1]\times [0,1]$ and collapse the border, the resulting quotient space is homeomorphic to the sphere. The same holds if we take the square $[0,1]\times [0,1]$ with the ...
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53 views

The product of quotient spaces is a quotient space

Let $(X,\tau)$ be a topological space and let $\sim$ be an equivalence relation on $X$. Now define an equivalence relation $\approx$ on $X \times X $ by $ [(x,y)]_\approx = [x]_\sim \times[y]_\sim $ ...
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14 views

Quotient singularities types

I am reading on calculating the number of quotient singularity types, but I am not sure what the notation means, the paper that I am reading does not give any background on this. I was wondering if ...
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A Map From $S^n\to D^n/\sim$ is Continuous.

The following question is motivated by Thomas's answer here which can be used to prove that $\mathbf RP^n$ is same as the space obtained by identifying the antipodal points on the boundary circle of ...
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Real Projective Plane is Same as Identifying Antipodal Boundary Points of The $2$-Disc.

$\newcommand{\RP}{\mathbf RP}$ The real projective plane $\RP^2$ is defined as the quotient space $S^2/\sim$, where $\sim$ identifies the antipodal points of $S^2$. I want to show that $\RP^2$ is ...
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$\beta \mathbb{R}$ is a quotient space of $ \beta \mathbb{N} $

I know that a quotient space can be thought of as being an open continuous image of a space. Therefore, it would be enough to find some map from $ \beta \mathbb{N}$ open and continuous to $ \beta ...
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Quotient spaces - $\Bbb R^1\hookrightarrow \Bbb R^3$

I am trying to understand quotient spaces, and I constructed my own example to do this: $(\Bbb R=\{(a,0,0)|a\in \Bbb R^1\}) \hookrightarrow (\Bbb R^3=\{(\alpha,\beta,\gamma)|\alpha,\beta,\gamma \in ...
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Maximal ideals of R[x]/(f(x))

I have been studying for my Qualifying Exam and came across the following problem: Let $R\subseteq T$ be integral domains and suppose that $a\in T$ satisfies a monic polynomial of degree $d$ with ...
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38 views

Quotient set and inverse image.

(What i'm about to say is a subpart of a proof of a theorem). Let $M, N$ be a closed subspace and a finite dimensional subspace, respectively, of a normed linear space $X$. We define the natural map ...
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Questions about the Identification Topology and Equivalence Class from “Introduction to Topology” by Mendelson

I am currently reading Introduction to Topology by Bert Mendelson, and I have some questions regarding the topic on Identification Topology in his book. Let $(X,\tau)$ and $(Y,\gamma)$ be topological ...
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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: ...
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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 ...
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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 $D = \{ (x,x) \in I \times I \}$, ...
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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 ...
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12 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 ...
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78 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 ...
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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 ...
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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 $1$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 = ...
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1answer
19 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 ...
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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". ...
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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 ...
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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 ...
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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 ...
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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. $U = \pi^{-1} ...
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$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 ...
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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 = ...
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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.
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47 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 ...
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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 / ...
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36 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 ...