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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If $p:X\to Y$ is quotient map and $Z$ is a locally compact Hausdorff space, then $p\times I_z$ is a quotient map

The problem from my topology book is as follows. If $p:X\to Y$ is quotient map and $Z$ is a locally compact Hausdorff space, then $p\times I_z$ : $X\times Z \to Y\times Z$ is a quotient map where ...
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deformation retraction as mapping cylinder

In Hatcher's Algebraic Topology, the mapping cylinder is defined as the quotient space of the disjoint union $(X\times I)\sqcup Y$ (where $I$ is the unit interval) of a continuous $f:X\to Y$, where ...
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Show that $C/B \approx T/A$

Let $T=S^1 \times S^1$ be the torus with meridian $A=S^1 \times \{1\} \subset T$ Let $C=S^1 \times [-1, 1]$ be the cylinder with base circles $B=S^1 \times \{-1, 1\}$. Show that $C/B \approx ...
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What is $S^3/S^1$?

I have been given this space in a question but I am unsure what it means I know that $S^3=\{(z_1, z_2) \in \mathbb{C^2}\mid |z_{1}|^2 + |z_{2}|^2=1 \}$ Could you help me understand what set of ...
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Show that a smooth manifold modulo diffeomorphism group is a smooth manifold

Would like help in starting this exercise: Suppose $\Gamma$ is a group of diffeomorphisms of a manifold $\left( {X,C_X^\infty } \right)$. Suppose that the action of $\Gamma$ is ...
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Identifying the antipodal points of one boundary of the cylinder gives the Möbius band.

In Example 1.35, Hatcher writes in his book Algebraic Topology, the following (not paraphrased): Let $X=S^1\times I$, and let $A$ be the quotient space obtained by defining the relation $(z, ...
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quotient groups and SLOCC

I have a math-physics question, which is based on an interest in SLOCC systems for black hole entanglement. The Cartan decomposition of a group $G$ such that $H = G/K$ is such that the derivation or ...
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How to show that if $X$ is Hausdorff and $ \big\{ (x, y) : x \sim y \big\} \subseteq X \times X$ is closed then $Y$ is Hausdorff?

Let $\sim$ be an equivalence relation on a topological space $X$, and let $Y = X/\sim$ be equipped with the quotient topology. How to show that if $X$ is Hausdorff and the set $\big\{ (x, y) : x \sim ...
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$X$ hausdorff and $ \big\{ (x, y) : x ∼ y \big\} ⊆ X × X$ is closed implies quotient map is open.

Let $∼$ be an equivalence relation on a topological space X. $\ Y = X/∼ $ equipped with the quotient topology. How to show that if X is Hausdorff and the set $ \big\{ (x, y) : x ∼ y \big\} ⊆ X × X$ ...
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Difference between homeomorphism and quotient map

I am learning topology and would like to know the difference between Homeomorphism and Quotient maps. I will be grateful if anyone can help me in any way.
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Showing $S^2/\sim$ (real projective plane) is Hausdorff

Let $\pi:\;S^2\to S^2/\sim$ be the projection map where the relation on $S^2$ is $a\sim b\iff a =\pm b$. I am trying to show $S^2/\sim$ is Hausdorff. So take $\alpha,\beta\in S^2/\sim$ then ...
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Why is this function well definded?

I have to take a look at the relation of $ x \sim y: \Leftrightarrow \exists k \in \mathbb{Z} : x-y=5k$ in $\mathbb{Z}$. I have no idea how to show at $\mathbb{Z} / \sim$ that the addition is well ...
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Projection of measure with bowen - walters metric.

Given $X$ a compact metric space, $f:X\to X$ be a homeomorphism and consider the quotient space $Y^{1,f}=(X\times [0,1])/\sim$, where $(x,1)\sim(f(x),0)$ for all $x\in X$. Let $d^{1,f}$ be the ...
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A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n $

(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...
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Quotient topology on unit sphere

Let $\sim$ be the equivalence relation $$a\sim b\iff a=b\text{ or }a=-b,$$ for $a,b$ on the unit sphere $S^2$. Let $Q$ be the quotient space. How do I show that the quotient map is a covering ...
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Needing a clearer idea on “quotient topology”

So here's the definition of a quotient topology Let $(X,\tau)$ be a toplogical space and define some equivalence relation $\sim$ on the set $X$. There exists a natural surjection denoted $p:X \to ...
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circle actions on spheres

I'm considering the following action of $S^1$ on $S^3$: $$ e^{i\theta}.(z_1,z_2)=(e^{i\theta}z_1,e^{iq\theta}z_2) $$ It is clear that when $q=1$ the quotient space is $S^2$. Is there any description ...
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Proof verification: Munkres Theorem 22.1

In Step 2 of Theorem 22.1 from Munkres' Topology: Let $p:X\to Y$ be a quotient map; let $A$ be a subspace of $X$ that is saturated with respect to $p$; let $q:A\to p(A)$ be the map obtained by ...
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$(X \times X) /{\sim'}\cong (X/{\sim}) \times (X/{\sim})$

The full description of this problem is: Let $X$ be a topological space. Let $\sim'$ be the equivalence relation on $X\times X$ defined by $(x,y)\sim'(x',y')$ iff $x \sim x'$ and $y \sim y'$ ...
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Restriction of diagonalizable endomorphism to an invariant subspace is diagonalizable - another approach

There are some questions discussing the diagonalizability of a restriction of a diagonalizable endomorphism to an invariant subspace, however, I have a question regarding a certain approach, which ...
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Is there some geometric intuition for the quotient $G/Z(G)$, where $G=GL_n(\mathbb{R})$?

Let $G=GL_n(\mathbb{R})$ be the $n$th general linear group. Its center $Z(G)$ is given by all scalar matrices $aI$ with nonzero determinant. How can I get an intuitive picture of $G/Z(G)$? I know that ...
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Exterior algebra subspace of all grade-n wedge products of a vector

Let $V$ be a finite-dimensional vector space, and let $\Lambda(V)$ be its exterior algebra. Then if $S_k = \text{span}(k_1,k_2,...,k_n)$ and $\hat k = k_1 \wedge k_2 \wedge ... \wedge k_n$, there is ...
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How to describe the vector space $\left( \frac{Z}{Im(T)} \right)^*$?

Just as a preface: I am not looking for an answer I just want help describing what the vector space looks like so that I can work on solving the problem. We have a linear transformation from $T: V ...
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Quotient topology, topological spaces

This is a practice final exam. My questions: 1) When the question defines X/~ as quotient topology. Does that mean I can write: $q: X \rightarrow X/\sim $ 2) Specifically, I am trying to prove ...
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Invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$

If $A$ is a ring with unit element $1 \ne 0$ let $A^{\times}=\{a \in A: a$ invertible$\}$. Find the invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$. ...
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Construct a homemorphism $\phi : T^2/A \rightarrow X/B $

Construct a homemorphism $\phi : T^2/A \rightarrow X/B $ $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by $A=S^1 \times\{1\}$. $X=S^1 \times [-1, 1]$ and $B = S^1 \times\{-1, 1\}$. ...
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Examples of a quotient map not closed and quotient space not Hausdorff

Is there any example of a closed relation $\sim$ on a Hausdorff space $X$ such that $X/\sim$ is not Hausdorff? Also, is there any example of a closed relation ~ on a Hausdorff space $X$ such that a ...
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Order in a quotient space of $\mathbb{R}^\mathbb{N}$ ($\mathbb{R}^\omega $)

Let $\mathcal{F}$ be a filter in $\mathbb{N}$ finer than Fréchet filter. In $\mathbb{R}^\mathbb{N}$ we define the equivalente relation : $(a_n) \equiv (b_n)$ iff $\{n | a_n = b_n\} \in \mathcal{F}$. ...
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Quotient of a scheme under infinite group action

Let $X=\mathbb{A}^2\setminus\{(x,y)\}$ be the affine scheme minus the origin (say over a field $k$). Consider the action of the group $G=k^*$ given by $(x,y)\mapsto (\alpha x,\alpha^{-1}y)$ for any ...
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Understanding quotient groups

Admittedly this will be probably be a naive question, but here it goes: Is it possible to flesh out in simple terms, for someone with little background in group theory, what it means to take the ...
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Value of polynomials in quotient ring

Let $K$ be a field. If we have a polynomial ring, $K[X_1,...,X_n]$, and an ideal $I$, we can form the quotient ring, $$K[X_1,...,X_n]/I.$$ For a given ideal, $I$, if we take an element of this ...
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Subspace as a representative system of the quotient space

Can someone help me with the following problem from Linear Algebra: Let $\Bbb K$ be a field and $V$ a vectorspace over $\Bbb K$ and $U$,$W$ two subspaces of $V$. Now I want to show: $$W \text{ is a ...
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Determinate the quotient topology

I was trying to find the quotient topolgy for the next example: Let R be the real numbers with the usual topology ($\tau$) and define the relationship $\mathcal{R}$ over R as follows, a ...
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Universal cover of a CW complex corresponding to an identification space

I am looking at a past exam paper for my introductory algebraic topology course, and am asked, for each of the following identification spaces, to find a CW complex homeomorphic to the space, draw the ...
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Book mainly discuss quotient topology

I am studying general topology, but I can't grasp the 'feeling' of quotient topology. I know the 'gluing' metaphor, but I always have no idea when I am asked to determine whether two quotient ...
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Homeomorphism $\phi : T^2/A \to X/B$. What are $ T^2/A$ and $X/B$?

The question I am working on asks me to construct a homeomorphism $\phi : T^2/A \to X/B$ where $T^2$, $A$, $X$ and $B$ are given as follows: $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by ...
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Easy way to show $D^n/S^{n-1}\cong S^n$

I can work out an heuristic argument for $n=2$: (homeomorphically) turning the disc $D^2$ to something like a funnel (no pipe of course), then gradually contracting the open "mouth" of the funnel to a ...
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$\mathbb{R}^{N}/\Sigma_{n}$ as a topological space

Let $\Sigma_{n}$ denote the symmetric group on $n$ letters. $\Sigma_{n}$ acts on unordered pairs $\{i,j\}$ via $\sigma(i,j)=\{\sigma(i),\sigma(j)\}$. Let $e_{\{i,j\}}$ be a basis for $\mathbb{R}^{N}$ ...
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Suppose $\phi \in L(V,F)$ and $\phi$ is not the zero map. Prove that $\dim V / \ker(\phi) = 1$.

I'm working on a quotient space problem from Axler (3.E #15): Suppose $\phi \in L(V,F)$ and $\phi$ is not the zero map. Prove that $\dim V / \ker(\phi) = 1$. Note that $F$ is the ground field for ...
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Showing that a bounded linear function that is identity on a certain subspace is identity e'erywhere

Let $X$ be a normed space, $M$ a closed subspace, and a quotient of $X$ over $M$ defined as an ordered pair $(Q, \pi)$ such that $Q$ is a normed linear space $\pi$ is a bounded linear function from ...
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Definition of quotient of a topological space by a group action

I was going through the following lecture note on topology as I was trying to understand quotient topology . http://homepage.math.uiowa.edu/~jsimon/COURSES/M132Fall07/M132Fall07_QuotientSpaces.pdf ...
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Can the quotient by a nonabelian group yield an abelian singularity?

Let $V$ be a complex vector space with a faithful linear action of a finite group $G$. Viewing $V$ as affine space (with coordinate ring $\mathbb{C}[V]$), the quotient $V/G$ is the affine variety with ...
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Proof of the universal property of the quotient topology

In this question: universal property in quotient topology I saw the following theorem: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/{\sim}$ be the ...
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Quotient spaces homeomorphic to the reals

Let $Z$ be a topological space. Given a subspace $A$ of $Z$, define an equivalence relation $R_A$ such that its equivalence classes are $\{x\}$ for $x\in Z\setminus A$ and $A$. Let $Z/A$ be the ...
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The function that identifies the closed unit ball in $R^2$ to $S^2$

Let $X$ be the closed unit ball in $\mathbb{R^2}$, and let $X^*$ be the partition of $X$ consisting of all the one point sets for which $x^2+y^2 <1$, along with the set $S^1=\{(x,y)|x^2+y^2=1\}$. ...
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Proving that a specific map exists

Let $(X, T_X)$ be a topological space and $\sim$ an equivalence relation on $X$, let $Q = X / \sim$ be the quotient space and let $q : X \to Q$ be the quotient map. Let $Z$ be a topological space and ...
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The quotient group (space) of $\Bbb Z/H$

Find the quotient group(space) of $\Bbb Z/H$ if $H = 6\Bbb Z $ and it is also a subgroup of $\Bbb Z$. Do the same if $ H = \langle[4]\rangle $ in $ \Bbb Z_{12}$ The quotient group of $\Bbb Z/H$ $H ...
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Trying to understand the proof for complements of subspaces of finite codimension

I'm trying to read through the book "Introduction to Global Variational Geometry" by Demeter Krupka and I don't fully understand a proof regarding the existence of a topological complememnt for a ...
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50 views

Non-separated quotient of separated scheme

I am reading Mumford's GIT book. I found the following claim there. Let $X$ be an algebraic variety. Let $G$ be an algebraic group acting on $X$. Then the categorical quotient of $X$ by $G$ may be ...
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Quotient map for space of leaves of a foliation

I have been told the following fact: Given a foliation $F$ of a smooth manifold $M$, then the projection $\pi : M \to M /F$ onto the space of leaves (points on the same leaf are identified) is open. ...