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Questions tagged [quotient-spaces]

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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Polynomial factorization in $\mathbb{R}$ and $\mathbb{Z}_{[n]}$

I've the following polynomial: $$ a(x) = x^6 + x^5 + 2x^3 - 3x^2 +x -2 \in \mathbb{K}[x] $$ Set $\mathbb{K} = \mathbb{R}$. A factorization of $a(x)$ is: $$ a(x) = (x^2 + 1)^2(x-2)(x+1) $$ Now set ...
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Proof about length of quotient modules

Let $M$ be an $R$-module and let $x, y \in R$ such that $y$ is not a zero divisor of $M$ and $M / xyM$ has finite length. Show that $l(M/xyM)=l(M/xM)+l(M/yM)$. In the above, $l$ denotes the lenght ...
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number of connected components of $E(n)$

Let $E(n) := \{f : \Bbb R^n → \Bbb R^n , ||f (x)|| = ||x||\}$ be the group of affine isometries of $\Bbb R^n$. Prove that $T(n)$ the set of translations $x→ x + y, y ∈ \Bbb R^n$ verifies $T(n) \...
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Quotient of semi-simple representation is semi-simple

Prove that Quotient of semi-simple representation is semi-simple. Take $V=\oplus_i V_i$ a semi-simple representation of finite dimension of a finite group. For a fixed j, we have $V/V_j=\oplus_i (...
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1answer
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Understanding a specific quotient ring

I have the ring $R = \frac{k[x,y]}{\langle x^2\rangle}$ where $k$ is a field. But I'm having some trouble understanding exactly what this means. So, as I understand it, in general the ring $R = \frac{...
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Let $X$ a non-Hausdorff topological space and let $\sim$ denote an equivalence relation on $X$. Is it possible that $X/\sim$ be Hausdorff?

Let $X$ a non-Hausdorff topological space and let $\sim$ denote an equivalence relation on $X$. Is it possible that $X/\sim$ be Hausdorff? Show an example. I would like to know also if there is a ...
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Behavior of points and compact subsets of Hausdorff spaces

It is quite straightforwad to see that many prpoperties are shared by points and compact subspaces of Hausdorff topologies, for example in terms of separation properties. I was wondering if there is ...
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1answer
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If $X$ is normal and $A \subset X$ is closed, then the quotient space $X/A$ is normal.

If $X$ is normal and $A \subset X$ is closed, then the quotient space $X/A$ is normal. I am trying to show this statement. The idea I have is using the fact that if $p:X \to Y$ is a closed continuous ...
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1answer
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Preimage of codimension one subspace of the quotient $g/D^1 g$

I am trying to prove the following. Proposition Let $g$ be a Lie algebra such that the first derived algebra $Dg$ is a proper ideal of $g$. Consider the quotient $g/D^1 g$, and show that the preimage ...
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Quotient algebras of nilpotent Lie algebra are nilpotent

For the following proposition I found a proof in some notes that I don't understand. Below definition 1 defines the terminology I'm using, and proof attempt 1 gives my attempt at the proposition. ...
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Real projective space and $n$-sphere

I know that $$\mathbb{R}^n\mathbb{P}\cong \mathbb{S}^n/{\pm 1}$$ Is there another equivalence relation apart from treating antipodal points as the same which we can quotient out from $\mathbb{S}^n$ to ...
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Determining the ideals of a quotient ring

Given an ideal $I = \langle x^3 - x\rangle \subseteq \Bbb{R}[x]$, determine the ideals in the quotient ring $\Bbb{R}[x]/I$. I understand that the quotient ring is of the form $k[x_1...x_n]/I$ where ...
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Verify that an equivalence relation is open.

On $\mathbb{R}^{n+1}-\{0_{\mathbb{R}^{n+1}}\}$ define the following equivalence relation: $$x\sim y\iff\quad y=tx\;\text{for same}\; t\in\mathbb{R^{\times}}.$$ Problem. The equivalence relation $\...
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Set of equivalence classes

I'm new to equivalence classes here, so I hope someone could help me out Mathworld (http://mathworld.wolfram.com/EquivalenceClass.html) defined equivalence classes as "a subset of the form$ (x \in X:...
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Phase, Isochrons, Isochrons map and Lift

at the moment i read the following paper: https://arxiv.org/pdf/1512.04436v1.pdf I have some questions about it and i hope someone can help me. On page 4/5 they introduce isochrons and the isochron ...
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Definition of mapping telescope

In Kochman's stable homotopy theory, pg 121 prop 4.24 We let $X$ be a based CW complex. Let $X^n$ be an increasing sequence of subcomplex whose union equals $X$. We define $$TX = \bigcup_{n \ge ...
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1answer
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Quotient to make $X$ a $T_1$ space

Let $X$ be a topological space. We define a relation on $X$: $$x \approx y : \quad \Leftrightarrow \quad x \in \overline{\{y\}}.$$ In general $\approx$ is no equivalence relation since it lacks ...
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1answer
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Compute the homology groups $H_∗(X, X−p)$ for points $p∈X$ where $X = S(\Bbb CP^2)$

Let $X = S(\Bbb CP^2)$ be the suspension of $\Bbb CP^2$ . Compute the homology groups $H_∗(X, X−p)$ for points $p∈X$ . My attempt : For any topological space $Y$ , $S(Y)-pt \simeq C(Y)$ where $C(Y)...
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How to rigorously show that elementary operations on polygonal presentations yields homeomorphic spaces?

In Lee's book Introduction to Topological Manifolds, he discusses elementary operations on polygonal presentations. Before the question, here are the terminologies that I am going to be using: A ...
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1answer
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Relational properties preserved under quotients

Suppose R is a binary relation on a non-empty set S. Let E be an equivalence relation on S. Now form the obvious quotient structure: Let S' be the set of all E-equivalence classes [s] of members s of ...
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Quotient of quotients in finite dimensional vector spaces

Suppose we are given filtrations of finite dimensional vectors spaces: $$ B_d\subseteq Z_d\subseteq C_d$$ $$0 \subseteq C_{d,1} \subseteq C_d$$ $$0 \subseteq Z_{d,1} \subseteq Z_d$$ $$0 \subseteq B_{...
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Showing that the upper half plane has the same induced topology from $(\Bbb R^2,T_{st})$ in the quoitent and subspace toplogy.

Say we have the surjective map $f:\Bbb R^2 \rightarrow H_+$. f(x,y)=(x,|y|). Where $H_+:=\{(x,y) \in \Bbb R^2|y\geq0\}$. I want to show that the quotient topology and the subspace topology are the ...
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Derivative of the quotient map $\mathbb R \to \mathbb R / T\mathbb{Z}$

We consider the quotient space $\mathbb R / T\mathbb{Z}$ and the quotient map $\pi:\mathbb{R} \to \mathbb R / T\mathbb{Z}\ $ defined by $\pi(t):= t\bmod T:=t+T\mathbb{Z}$. In a journal i read that $\...
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1answer
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Quotient of $n$-th skeleton by smaller dimensional skeleton

Suppose $X$ be a CW complex. Then we know that, $$\frac{X^n}{X^{n-1}}=\underset{\text{$\sigma$ an $n$-cell}}{\large\lor} \Bbb S^n,$$ Where $X^n$ is $n$-th skeleton of $X$ and $X^{n-1}$ is $(n-1)$-th ...
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1answer
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Are the only Riemann surfaces which are quotients of $\mathbb{C}$the cylinder and the toruses? Why?

Consider the Riemann surfaces $\mathbb{C}^\times=\mathbb{C}\setminus\{0\}$ and $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice in the complex plane (i.e. a discrete additive subgroup of $\mathbb{C}$...
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How to think intuitively quotient of quotient set in equivalence relations

If $G$ and $H$ are arbitrary equivalence relations in $A$, prove that $$A/(G\circ H) ≈ (A/G)/(G\circ H/G).$$ How to think $(G\circ H)/G$ intuitively, especially if $(G\circ H)$ is empty set, and ...
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Quotient space in topology

In this thread: What are the epimorphisms in the category of Hausdorff spaces? the quotient space of a topological space $X$ by a closed set $A \subset X$ is mentioned : $X / A$. This confused me, ...
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Show that the restriction map $f: \ X \to Y$ is not a quotient map.

Let $X=[0,1] \cup (2,3]$ and $Y=[0,2]$. Define a map $f:X \to Y$, where the toplogy on $X$ and $Y$ are the subspace topology from Euclidean topology on real line which is defined by \begin{align*}f(...
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2answers
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How to Find limits and co-limits of diagrams over Vector?

I am having trouble understanding how to find limits and colimits of specific diagrams over the category of finite dimension vector spaces. I understand the definitions of cones, terminal objects, ...
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1answer
67 views

What all topological properties are preserved under attaching a 2-cell?

Let $Y$ be a topological space. Let $f:\mathbb{S}^1=\partial\mathbb{D}^2\rightarrow Y$ be a continuous map. By attaching $2$-cell to $Y$ we mean the space $Y\bigsqcup \mathbb{D}^2$ under the ...
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Qutient Topologies in Banach Spaces

I hope the title is not misleading. I am currently reading a paper, where the author uses the following argument: First, let $ f:A\rightarrow B $ be a linear and continuous map between Banach spaces $...
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1answer
34 views

Find a homeomorphism

Let $X=A\cup B \cup C$ where $A=\{(x,y) :(x+2)^2 +y^2 =1\}$ and $B=\{x^2+y^2 \leq 1\}$ and $C=\{(x,y) :(x-2)^2 +y^2 =1\}$. Find a homeomorphism between the quotient space $X/B$ and $E=\{(x,y) :(x-1)^...
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Show that $( X \sqcup_f Y)/Y $ is homeomorphic to $X/A$.

Let $X$ be a topological space and let $A \subset X$ be a subset. We define the topological space $X/A$ to be the quotient space $X/\mathcal{R}$ where $\mathcal{R}$ is the equivalence relation defined ...
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Quotient space not the same as the original space

I am trying to understand why quotient space is not the same as the original space. Let $V$ be a vector space and $W$ be its vector subspace. If I define $a+W:=\{x=a+w; w\in W\}$ then the quotient ...
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Factor Rings over Finite Fields

Given a polynomial ring over a field $F[x]$, I can factor, for example, the ideal generated by an irreducible polynomial $ax^2 + bx + c$: $F[x]/\left<ax^2 + bx + c\right>$, and guarantee that ...
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Span and Independence of Quotient Vector Space

Let $V/U$ be a quotient vector space with the standard equivalence relation $v_1 \sim v_2$ iff $v_2 - v_1 \in U$. Let $(v_i \mid i \in I)$ be a collection of vectors in $V$ indexed by $I$. Then $([...
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1answer
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Rudin functional analysis, theorem 4.9 part (b)

Let $M$ a closed subspace of a Banach space $X$. b) Let $\pi:X \to X/M$ be the quotient map. Put $Y = X/M$, For each $y^* \in Y^*$, define $$ \tau y^* = y^* \pi $$ Then $\tau$ is an isometric ...
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1answer
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Is compactness (without Hausdorff) enough to get a closed quotient map?

This is some attempt to generalise the classical result for a compact Hausdorff space $D$, if $E\subseteq D\times D$ is closed, then the quotient map $p:D\to D/{E}$ is a closed equivalence relation. ...
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$H/K$ when $K$ is not a subgroup of $H$.

Let $G$ be a group and $H$, $K$ ($K$ is normal) two subgroups of $G$ but neither $H$ is subgroup of $K$ nor $K$ is subgroup of $H$. What would be the problem with $H/K$? If I define for $x,y \in H$ ...
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1answer
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Confused about quotient ring

For $I$ an ideal of a commutative ring $R$, I'm confused about the object $R/I$. My understanding is that this is defined by $R/I := \{rI: r \in R\} = \{\{ri: i \in I\}: r \in R\}$. However, by the ...
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Show that an element is irreducible but not prime.

Show that in the ring $\mathbb Q[x, y]/(x^3 -y^2)$ the element $x +(x^3 -y^2)$ is irreducible but not prime. Not sure how to show this. I know that $(x^3 -y^2)$ is a prime ideal but I cannot ...
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Quotient space homeomorphic to sphere.

If $\textbf{x} \sim \textbf{y}$ iff $\textbf{x}=\lambda \textbf{y}$ for some $\lambda \in \mathbb{R}-\{0\}$. If $\textbf{x} \sim_+ \textbf{y}$ iff $\textbf{x}=\lambda \textbf{y}$ for some $\lambda \...
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1answer
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Is there an explicit function from the quotient space of an annulus to a torus?

The annulus is ${1\leq x^2+y^2\leq4}$ I would like to show that the quotient space of the annulus given by the equivalence relation $(x,y)\sim(x,y)$ and $(x,y)\sim(2x,2y)$ if $x^2+y^2=1$ is ...
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1answer
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Relation between quotient map and quotient space

Given a surjective function $p$ that takes a space $X$ to it’s decomposition $X*$ (decomposition here means a partition into equivalence classes), is it always the case that the function $p$ is a ...
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37 views

Saturated subsets of quotient map.

From an example in Munkres Topology:$$\\$$ Consider the projection map $\pi_{1}: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ onto the first coordinate; it is continuous and surjective. It is also ...
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156 views

Proper actions in complex geometry

[skip to next blockquote for the actual question] Complex geometry books often treat quotients by 'properly discontinuous' actions, such as the action of a lattice $L \subseteq \mathbb{C}$ on the ...
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2answers
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Using the quotient manifold theorem to show real projective spaces are smooth manifolds

The goal of this post is, as stated in the title, Use the quotient manifold theorem to show real projective spaces are smooth manifolds. I know this is an overkill, but I am just curious about ...
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1answer
30 views

quotient module $\mathbb{Z}^3/(2,0,3)\mathbb{Z}$

Is $\mathbb{Z}^n/(m,a_2,...,a_2)\mathbb{Z}\cong\mathbb{Z}/m\mathbb{Z}\oplus\mathbb{Z}^{n-1}$ for $0<m<a_2,...,a_n\in\mathbb{Z}$ (as $\mathbb{Z}$-modules)? If not, how can you compute something ...
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0answers
24 views

The Rank of an Endomorphism over a Quotient space, which is generated by an invariant subspace

I have problems answering the second part of the following question: Let $U$ be a $K$-vectorspace with finite dimension, $W \subset U$ a subset of $U$ and $\varphi :U\to U$ an endomorphism. 1.) Show:...
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Is $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ free?

I'm trying to solve a question which asks me to determine whether the quotient $\mathbb{Z}$-module $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ is free. I'm then supposed to find some ...