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 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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Example of lattice of subgroups of quotient group [closed]

I've studied a theorem that explains what is the lattice of subgroups of a quotient group. The result is the following: Given a group G and a normal subgroup N if we denote by Sub(G) the lattice ...
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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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Intuition for universal quotient maps [migrated]

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman, the two characterizations of ...
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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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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. ...
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Continuous map from topological space $A\rightarrow B$ as composition of quotient map and unique continuous map from quotient space to $B$

Let $(A,T_A)$ be a topological space; $R$ an equivalence relation; $Q=A/R$ the quotient set with $i:A\rightarrow Q$ the quotient map and $T_Q=\{V\subset Q|i^{-1}(V)\text{ open in $X$}\}$ the quotient ...
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Show that $[a][\star][b] :=[a \star b]$ defines an operation on $G/\sim$.

Let $G$ be a set equipped with an operation $\star$ and an equivalence relation $\sim$. Suppose that $\sim$ is compatible with $\star$, i.e., for elements $a$, $a'$, $b$, $b'$ of $G$, $$\text{if}\ a ...
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Geometrical intuition of quotient spaces

Hi i have stumbled across this example of a quotient space but i am not quite sure i really understand what it means either algebraically or geometrically. Fixing a vector space $V$ and a subspace ...
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Induced map by quotient is open

Let $f:X\longrightarrow Y$ be an open, continuous map and $p:X\longrightarrow X/R$ be a quotient map. I think the map (assuming it is well-defined) $\hat{f}:X/R\longrightarrow Y$ is an open map. My ...
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$V$ is vector space of polynomials $\mathbb R[X]_{<4}$. Let $U = \{P \in V : P(3) = 0\}$. Find a basis of $V/U$.

$V$ is the vector space of polynomials $\mathbb R[X]_{<4}$. Let $U = \{P \in V : P(3) = 0\}$. 1) Find a basis of $V/U$. 2) Write down the matrix that represents the canonical ...
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A quotient topology exercise.

Define the quotient space: (1) The real line $\mathbb{R}$ with $[-1,1]$ collapsed to a point. (2) The real line $\mathbb{R}$ with $(-1,1)$ collapsed to a point. The first one I think the quotient ...
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Exercise involving the quotient topology

Define a partition of $X = \mathbb{R}^2 − \{0\}$ by taking each ray emanating from the origin as an element in the partition. Which topological space appears topologically equivalent to the quotient ...
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Covering Projections and Quotients

Let $Y$ be a covering space of $X$, where $p:Y\to X$ is a covering projection. For $x\in X$, define the fiber of $x$ as $p^{-1}(x)$. Set up an equivalence relation on $Y$ as $y_1\sim y_2$ if they are ...
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Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$ [duplicate]

I want to prove that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}.$ The first part of the question asks us to show that if $\phi$ is a nonzero ring ...
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More elegant way to make the argument that $\mathbb{Q}[x,y]/\langle x,y \rangle \cong \mathbb{Q}$

My argument goes as follows: Observe that $I = \langle x,y \rangle$ is exactly the set of polynomials with no constant term. Then for any $p(x) \in \mathbb{Q}[x,y]$, we can write $p(x) = p'(x) + ...
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Let $I$ be an ideal of a ring $R$. Then Show that $|(R/I)| = 1 $ if and only if $R = I$.

(i) Show that $|(R/I)| = 1 $ if and only if $R = I$. (ii) Show that if $R$ has an identity 1 then (if $I \neq R$) so does $R/I$, and if $R$ is commutative, then so is $R/I$. I know that the quotient ...
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set theoretic equivalence in quotient ring

If I am given a ring $R$ and a 2-sided ideal $K\subseteq R$, I know that I have a well-defined quotient ring $R/K$. My question is the following: We know that if we have $a,b\in R$, then ...