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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Infinite direct sum of quotient modules

Let $\{(M_i, N_i)\}_{i\in I}$ be a collection of $R$-modules where $N_i$ is a submodule of $M_i$ for each $i$ in a finite indexing set $I$. Then we can show $$\bigoplus_{i\in I} \frac{M_i}{N_i} \...
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Proving that a map is a quotient map

I am trying assignments in topology and I got stuck on this question: Prove that the map $f:\Bbb R^2 \to \Bbb R$ defined by $$f(x, y) = y^3 + xy^2 + x + y$$ is a quotient map. I have done a ...
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Under which conditions the rings $\mathbb{Z}_p[x]/(x^n+1)$ and $\mathbb{Z}_p[x]/(x^n-1)$ are fields? (for $p$ prime)

I'm looking for the necessary and sufficient conditions for any prime $p$ and any positive integer $n$ to make the quotient rings $\mathbb{Z}_p[x]/(x^n+1)$ and $\mathbb{Z}_p[x]/(x^n-1)$ not only rings ...
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How do we denote the $0$ vector in a Quotient Space $V /W$?

Just $0$ or $0 + W$? Unfortunately my textbook has no mention of the notation for this, nor can I find clarification for this online.
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Concerning the topology of a construction with rectangles

I have a flat space with a rectangular grid put on it, such that the rectangle has width $2r$ and length $h$.The grid has the following property : Take any rectangle $R$. The two rectangles to it's ...
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Field of fractions over $k[X,Y]$

Let $k$ be a algebraicly closed field and $p \in k^2$. It is to show, that if $z \in k(k^2)=\mathrm{Frac}(k[X,Y])$ is defined in every point $q \in k^2\backslash \{p\}$, then it already holds ...
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Paracompactness of Adjunction Spaces

In this question I will understand the definition of paracompactness to include the Hausdorff separation axiom. Let $X$ be a paracompact space and $A\subseteq X$ a closed subspace. Let $f:A\...
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On a representation of the Moebius strip

Let $f:\, (x,y,z)\in C\mapsto ((x^2-y^2)(2+xz),2xy(2+xz),yz)\in\mathbb{R}^3$. I have to prove that the image of this function is homeomorphic to the Moebius strip. I would like to use the universal ...
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Induced mapping into mapping cylinder

Can anyone please tell what is the induced map from mapping cone of f into Z? Also please explain how H and g induced G. Thanks in advance
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Subgroups of $\mathbb{R}/\mathbb{Z}$ are either cyclic or dense [duplicate]

So I know that subgroups of $(\mathbb{R},+)$ are either isomorphic to $x \mathbb{Z}$ for $x\in \mathbb{R}$, or dense in $\mathbb{R}$. I don't see how (if?) this descends to the quotient, though. ...
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Dimension of the Annihilator and the dual of the quotient space.

Let $W\subset V$ a vector subspace(not necessarily finite). I've been tryin to prove that the $dim(Ann(W))=dim((V/W)^*)$. Actually, the two spaces are isomorphic and i prove that, but I had an idea ...
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$f:X\to Y$ extends to a map $Z\to Y$ iff $f_*[g] = 0$

Let $f \colon(X,x_0)\to (Y,y_0)$ and $g \colon(S^n,s_0)\to (X,x_0)$ be base point preserving maps. Let $Z$ be the space that arises from $X$ by attaching an $(n+1)$-disk via $g$. I want to prove ...
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Question about topological hourglass

Let $X:=[-1,1]\times \mathbb{S}^1$, where each space is taken with the usual euclidean topology. Let $S:=\{0\}\times \mathbb{S}^1 \subseteq X$ and call $H:=X/S$. Is $H$ normal? My only idea is the ...
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Proving that $\mathbb{Z}^{d}/n\mathbb{Z}^{d} = (\mathbb{Z}/n\mathbb{Z})^{d}$

This question is a continuation of my previous question on quotient groups. I know that: \begin{eqnarray} \mathbb{Z}/n\mathbb{Z} = \{[0],...,[n-1]\} \tag{1}\label{1} \end{eqnarray} and, from the ...
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Showing that these two quotient spaces are not homeomorphic

Let, $P$ be the quotient space obtainted from $S^{2}$ by identifying two distinct points and $Q$ be the quotient space obtained by identifying three mutually distinct points in $S^{2}$. Show that ...
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Show that $g:X\to Z$ where $g((x,n)) = ((x,nx))$ is not a quotient.

Suppose that $X$ is the set of all lines $$L_{n} = \mathbb R \times \{n\} \quad\textrm{ for $n\in\mathbb Z^+$}$$ and $Z$ is the set of all lines that cross the center of plane and its slope is ...
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$\frac{I}{[0 \sim 1]} $ is homeomorphic to $\mathbb{S}^1$.

Let $I=[0,1]$ and the equivalence relation $\sim$ that identifies points $0$ and $1$. I used this theorem : $\textbf{Theorem:}$ Let $g: X \rightarrow Z$ continuous and surjective. Consider the ...
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Why does the Gromoll-Meyer Sphere have dimension $7$?

In exercise 10.33 of the book "Matrix Groups for Undergraduates" the Gromoll-Meyer sphere is described by taking the quotient of a smooth left action of $Sp(1) \times Sp(1)$ on the manifold $Sp(2)$ ...
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Prove the torus homeomorphic to this quotient space [duplicate]

Let $D = \{(x,y)\in \mathbb R^2∣\ 0\leq x\leq 1, 0\leq y\leq 1\}$ and $\sim$ be the equivalence relation on $D$ defined as $(x,0)\sim(x,1)$ for all $0\leq x\leq 1$ and $(0,y)\sim(1,y)$ for all $0\leq ...
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Canonical projection on Quotient space maps open ball to open ball

I’m struggling to understand the last part of this proof where it says $\pi(x)=\pi(x-z)$ proves the claim. To prove the claim I suppose this must imply that $||x|| \leqslant ||x-z||$ so that $||x||<...
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Constructing a vector field with no zeros

Let, $\phi: X \rightarrow X$ be a diffeomorphism of a smooth compact manifold $X$ with no boundary. Let, $X_{\phi}$ be the quotient manifold $(X \times[0,1])/ \sim$ and $(x,1) \sim (\phi(x),0)$. How ...
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invariant subspace and quotient space

let $T \in\mathcal{L}(V)$ be a linear operator on $V$ and let $U$ be $T$-invariant subspace of $V$.Suppose that $v_{1},...,v_{k}$ are elements of $V$ such that $$Tv_{j}=\lambda_{j}v_{j},\space \space ...
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Do the same rules apply to polynomial rings as to ring extensions?

Let $p$ be a prime and $g(x)$ irreducible $mod \ p$ over $\mathbb{Z}$. And let $d\ne 1 \ mod \ 4$ be a squarefree integer with $\omega:=\sqrt{d}$. We know that $(p,g(\omega))$ is a prime ideal over $...
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Finite dimension quotient ring

Let $R=C[x_1,...,x_n]$ and $M$ be a maximal ideal of $R$ such that $R/M$ is a finite dimensional $C-$algebra. Can we deduce that $R/M^n$ for n>1 is also finite dimensional $C$-algebra? We know that $...
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Left actions of quotient maps

I want to show that $G \times G/H \to G/H$, $(g,xH) \to gxH$ is a left action for $G/H=\{gH: g \in H\}$ where $G$ is a Lie group. If I write, $e.gxh=gxh$ and $g(xh_1)(xh_2) \to g(xh_1xh_2)=(gxh_1)...
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When does taking the product $- \times X$ preserve reflexive coequalizers in $\mathsf{Top}$?

By $\mathsf{Top}$ I mean the category of topological spaces and continuous homomorphisms. I am aware that if $(X,\tau_X)$ is a topological space, then the functor $- \times (X,\tau_X) : \mathsf{Top} ...
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Quotient Topology, equivalence relation. I need help to prove if X is homeomorphic to X/~

Let $X = [0,3] \subset\mathbb{R}$ and consider the following equivalence relation: $$ x\sim y \Leftrightarrow x=y \vee x,y \in [1,2]$$ and call $Y = X/\sim$. (1). Establish if $Y$ is ...
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$T_0$-Identification of a topological space

In Willard's General Topology, section 13.2.c, for any topological space $X$ is defined a quotient space $X/\sim$ such that $x \sim y$ iff $cl(\{x\})=cl(\{y\})$ where $cl(.)$ is the topological ...
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Proving a map in a commutative diagram is continuous.

Suppose $G$ and $H$ are topological groups, with $H \subset G$. I have the following commutative diagram with $f$ being continuous and $p$ being an open surjection (canonical projection). Does this ...
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Question in proof of open mapping theorem

Why is the mapping $\tilde A: X/N \to Y,\ \tilde A(x+N)=A(x)$ one to one (injective) and onto (surjective)? $A$ is a continuous, linear map from a F-space $X$ to a topological vector space $Y$ and $A(...
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Which notation is best for $R/I$

When $R$ is a ring and $I$ is an ideal of $R$, I have seen a variety of notational uses for the cosets in $R/I$, and I'm not sure which one is best in which context. For $a\in R$, if $C_a\in R/I$ is ...
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Splitting Fields, roots and quotient rings

With a lot of free time on my hands, I've been looking at bits of abstract algebra I never studied as a student (25+ years ago). I think I'm getting to grips with the ideas of rings, ideals and ...
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ideals of Lie algebras

I have a question about ideals of Lie Algebras and the quotient Lie Algebra. Why are ideals of Lie algebras defined the way they are? I am assuming this has something to do with the quotient algebra, ...
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$p: \mathbb{R} \to \mathbb{R}/A$ Prove $p$ is an open map $\iff$ $A$ is open.

$A$ is a subset of $\mathbb{R}$ with more than 1 point and $p: \mathbb{R} \to \mathbb{R}/A$ is the quotient map. Prove that $p$ is an open map $\iff$ $A$ is open. I know if $A$ is open then for each ...
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About quotient space problem

Quotient space: Let V be any vector space and $W \subset V$ a subspace. For any $v\in V$, let $v+W$ and denote the set: $v+W=\{v+w\mid w \in W\} \subset V $, called the coset of W containing v. Let $V/...
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Units in $\Bbb Z / 3 \Bbb Z[x] / I$ where $I=(x^3+2x^2)$

I'd like to know how to find all the units in the quotient ring $R/I$ where $I=(x^3+2x^2)$ and $R=\Bbb Z / 3 \Bbb Z[x]$. I know $$R/I = \{a_0+a_1x+a_2x^2 + I \; / \; a_0+a_1x+a_2x^2 \in R\},$$ so $R ...
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Homeomorphic to a product space

Consider $\mathbb{R^2}$ with usual topology. We define $\mathcal{S}$ as the equivalence relationship on $\mathbb{R^2}$ that identifies to one point all elements in $\mathbb{Q^2}$. I need to check if $...
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Question about prime ideals and Quotient rings

I`m struggling with the following solution I wanted to use: Let S := $\mathbb{C}[x,y,z] /(z^2-xy) \cong \mathbb{C}[x,y,\sqrt{xy}]$ Then $S/(x-y) \cong \mathbb{C}[x,x,\sqrt{x^2}] = \mathbb{C}[x]$, ...
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A problem about invariant and quotient space

Assume V is finite dimensional,that $T:V \rightarrow V$ is a linear operator, and W is a T-invariant subspace. Define $\bar{T}: V/W \rightarrow V/W$ by $\bar{T}(v+W)=T(v)+W$ for any $v+W \in V/W$. Let ...
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Proving homeomorphism in Möbius strip

Consider the Möbius strip defined by the following equivalence relation on the subspace $[0,1]\times]-a,a[$ of $\mathbb{R}^2$: $$(x,y)\sim (x',y')\implies (x,y)=(x',y')\vee|x'-x|=1\:\text{and}\:y'...
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Quotient Topology of Multiplication Map

We have $f(x,y)=xy$ where $f:\mathbb{R}_{[-)} \times \mathbb{R}_{[-)} \to \mathbb{R}$. What is the quotient topology? I know that the topological basis in $\mathbb{R}_{[-)} \times \mathbb{R}_{[-)}$ ...
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Understanding the 2-torus as a definition of equivalence classes

Note: My questions are marked in boldface. In Devaney's book "An Introduction to Chaotic Dynamical Systems", Westview Press (2003), on p. 190 he defines the $2$-torus by To describe the torus, let ...
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Dimension of quotient space and basis.

I'm quickly reviewing the concept of quotient space. Suppose $V$ is a vector space of dimension $m$ and $W$ is a vector subspace of $V$. Then the quotient vector space is defined as $V/W$, which is a ...
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Relative GIT quotient.

I´m studying Mumford's geometric invariant theory and I came to the following question. Let us suppose that $k$ is an algebraically closed field of characteristic 0, and $G$ is a reductive linear ...
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The maximal normal subgroup in U(n) and SU(n)

Let $U(n)$ be a the unitary group the quotient $$\frac{U(n)}{U(n-1)} \simeq S^{2n-1}$$ and I believe that $$\frac{SU(n)}{SU(n-1)} \simeq S^{2n-1}.$$ Obviously the $U(n-1)$ is not a normal subgroup ...
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The magic properties of the quotient space from $U(2^{l-1})/{\rm Spin}(2 l)$

Inspired by a previous post, Embed a Spin group to a special unitary group I am wondering what are the magic properties of the quotient space from $$U(2^{l-1})/{\rm Spin}(2 l)$$ that makes such an ...
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Quotient space (vector space) and linear maps

my professor give an exercise about quotient spaces envolving vector spaces and I have some questions about that. Consider $f:V \to V$ a linear map and $W \subset V$ a subspace. Take $\phi: V/W \to ...
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99 views

Quotient manifold of a finite group action

Let R be a (crystallographic) root system on an Euclidian space $(E,⟨−,−⟩)$ and $$W:=gen\{σ_r∣r∈R\}=gen\{σ_r∣r∈R^+\}$$ its associated reflection group. Taking $$M=E-\cup_{r\in R^+} H_r,$$ where $H_r$ ...
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56 views

Describe the elements of the quotient space and find a basis for it

Describe the elements of the quotient space of $V/S$ and also find a basis for it. $V = P_{10}$, the space of all real polynomials of degree at most 10, $$S=\operatorname{span}(\{{x,x^3,x^5,x^7,x^9}\}...
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15 views

Extending a right inverse for the quotient map

Let $X$ be a compact Hausdorff space with a faithful continuous $S^1$-action. Further, assume that the orbit space is $[0,1]\times[0,1]$, and denote the quotient map by $\pi:X\to[0,1]\times[0,1]$. Now,...

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