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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Unique Haar measure on quotient is pushforward

Let $G$ be an abelian locally compact Hausdorff group with discrete subgroup $H$. Let $\mu$ be a Haar measure on $G$ and $\lambda$ the usual counting measure on $H$. Then we obtain a unique Haar ...
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fundamental group of quotient (singular) tori

What we know about fundamental group of quotient singular manifold? I'm particularly interesting in the case of quotient given by $X:=T/G$ where $T$ is a complex torus and $G$ is finite. Since $\pi_1$ ...
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Projective special linear group and projective space as homogeneous space

The real projective special linear group $PSL(n+1,\mathbb{R})$ acts transitively and effectively on the real projective space $\mathbb{R}P^n$. As a homogeneous space, it can be considered as a ...
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Quotient of affine space under negation

Suppose we are working over a field $k$. Consider the affine space $k^n$ and the $\mathbb{Z}/(2)$-action on it given by $(x_1,x_2,\dots,x_n) \mapsto (-x_1,-x_2,\dots,-x_n)$. I would like to compute ...
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Help finding mistake in proof involving the quotient map.

Consider the plane $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$ with the product topology which has basis consisting of all open squares of the form $$\tag{1} \left]a,b\right[ \times \left]c,d\right[ \...
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existence of non-zero proper T-invariant subspace

Let $V$ be a finite dimensional vector space and $T:V\rightarrow V$ be a linear map. Suppose $U$ is a T-invariant subspace. Define $\overline{T}:V/U \rightarrow V/U$ by $\overline{T}(v+U)=T(v)+U$. ...
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Exercise 3.E.14 on p.100 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. Is my solution ok? What is a typical solution to this exercise?

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. I solved Exercise 3.E.14 on p.100 as follows. I want to know if my solution is ok or not. I also want to know a typical ...
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Exercise 3.E.12 on p.99 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. Is my solution ok? What is a typical solution to this exercise?

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. I solved Exercise 3.E.12 on p.99 as follows. I want to know if my solution is ok or not. I also want to know a typical ...
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Computing singular homology groups of quotient space

I want to compute the homology groups of $X$, the quotient of $S^2 \times S^1$ by the relation $(x,z) \sim (-x,-z)$. I've already computed the homology groups of $S^2 \times S^1$ using Mayer-Vietoris (...
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Characterize open sets of this quotient topology.

Let $X$ the quotient space obtained from $\mathbb{R}\times\{0,1\}$ identifying $(x,0)\sim(x,1)$ if $|x|>1$. Which are the open sets of this quotient topology? First, I've made the next drawing to ...
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Quotient of $GL(\mathbb{R}^n)$ by $O(\mathbb{R}^n)$

I suppose the quotient $GL(\mathbb{R}^n)/O(\mathbb{R}^n)$ has manifold structure. Is there a name for this manifold? Google isn't helping find it.
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An example showing that a quotient space of an hausdorff space is not hausdorff.

Let be $\mathscr P$ the partition of $[0,1]^2$ defined through the position $$ \mathscr P:=\big\{\{x\}\times[0,1]:x\in\Bbb Q\big\}\cup\big\{\{(x,y)\}:(x,y)\in \mathbb R \setminus \mathbb Q\times[0,1]\...
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What are these quotient spaces of sphere products homeomorphic to?

For $A,B \subset \mathbb{R}^n$, consider the equivalence relation $(a,b) \sim(-a,-b)$ on $A \times B$. I'm trying to see what the quotient spaces of $S^2 \times S^1$ and $S^1 \times S^1$ by $\sim$ ...
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Question on the connectedness of the orthogonal group

I want to show that the quotient $O_2^- = O_2/SO_2$ is connected. My idea was as follows: It's easy to show that $SO_2$ is connected. $S0_2$ is a topological group (normal subgroup of a topological ...
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Can germs be defined as a quotient of vector spaces?

Summary: Let $M$ be a smooth manifold and $p\in M$. I know of two notions of germs of functions at $p$, the more restrictive of which can be written as a quotient vector space. I am curious whether ...
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Is this space with the trivial metric, a nontrivial quotient of a subspace of the Sorgenfrey topology?

I guess this is a question about how accurate my intuition is about the Sorgenfrey line being a richer topology than the standard metric space on $\Bbb Q$, with reference to a specific application. ...
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Prove that $(S^2/\{\pm 1\})/S^1 \cong S^2$

I'm searching an homeomorphism between $S^2$ and the space $A=\dfrac{S^2/\{\pm 1\}}{S^1}$, where the numerator is the quotient space given by the group action by multiplication of $\{\pm 1\}$ on $S^2$,...
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Subspace of a quotient space is Lagrangian

Let $(V, \omega)$ be a symplectic vector space. Let $W \subset V$ be coisotropic and let $U \subset V$ be Lagrangian. Show that the quotient space $ ((U \cap W)+W^{\perp})/W^{\perp} \subset W/W^{\perp}...
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Provee that $\mathbb{P}^2(\mathbb{R})/H \cong S^2$, where $H\subset \mathbb{P}^2(\mathbb{R})$ is a projective line

$\mathbb{P}^2(\mathbb{R})/H$ identifies the topological quotient space given by the relation $x \sim y \Leftrightarrow x=y$ or $x,y \in H$. I tried to build an identification $\mathbb{P}^2(\mathbb{R}) ...
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Partitioning the 2-sphere into compact homeomorphic subsets

This question is related to a previous one Covering the plane with compact sets, II (as yet unanswered). Can the 2-sphere be partitioned into an infinite family of disjoint compact subsets, all of ...
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If the Quotient Space contains $0+U$ as it's zero element, must we not then include it in the visualization?

In Axler's book Linear Algebra Done Right 3.ed he defines the quotient space to be $V/U=\{v+U:v\in V\}$. As an example it is stated, that if $U=\{(x,2x)\in\mathbb{R}^2:x\in\mathbb{R}\}$ then $\mathbb{...
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Prove that this quotient space is normal

Consider the topological space $(\mathbb{R}^2,\tau _E)$, where $\tau _E$ is the Euclidean topology, and the subset $A=\mathbb{R} \times \{0\} \subset \mathbb{R}^2$. Then, if $X=\mathbb{R}^2 /A$ (the ...
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Existence of well defined map $R/J\to R/I$ implies $J\cong M\subseteq I$?

I'm currently trying to prove by myself some proposition related to Hopf-Galois theory (from the paper "Galois Correspondences for Hopf Bigalois Extensions", by Peter Schauenburg). The ...
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Compute the $\pi_{1} (S^{3} /G)$ and $H_{m} ( S^{3} / G)$ for the following $G$

I am studying for a qualying exam and there is one exercise from the previous years exams which I don't know how to approach. Let $n$ be a positive integer and $$ G = \{ g \in \Bbb C^{\times} | g^{n} =...
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How is a quotient simplicial complex by a group action defined?

Suppose we have a simplicial complex $K=(V, S)$ a simplicial left action of a group $G$ on $K$, i.e. an action of $G$ on the set $V$ of vertices of $K$ with the property that $\sigma \in S \implies g \...
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When will a quotient space has finite choices

Consider $S^1: x^2+y^2=1$ and two symmetry, inversion and reflection. Inversion let $x\to-x,y\to-y$ and $x\to-x,y\to y$. The equivalence relation imposed by inversion and reflection are denoted as $\...
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$P(a,b,c)=P(bc,ca,ab)$ weighted projective planes for pairwise coprime $a,b,c$

Let $a,b,c\geq 2$ be pairwise coprime integers. The (complex) weighted projective plane $P(a,b,c)$ is the quotient of $\Bbb C^3-\{0\}$ by the action of $\Bbb C^*=\Bbb C-\{0\}$ given by $t\cdot (x,y,z)=...
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Meaningful definition of angle on torus $\mathbb T^2 = \mathbb R^2 / \mathbb Z^2$

Does there exist a meaningful definition of the angle between two points on the torus? I am working on $\mathbb T^2 = \mathbb S^2 = \mathbb R^2 / \mathbb Z^2$. The representation I choose is $[-\...
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quotient topology on $A^{*}$ equals subspace topology on $\pi(A)$ for $A^{*}:= \pi^{-1}(\pi(A))$ open or closed.

Let $\pi:X \to X/\sim$ be a quotient map. Let $A \subset X$ and define $A^{*} := \pi^{-1}(\pi(A)) \subset X$. If $A^{*}$ is open or closed in X, then the subspace topology on $\pi(A)$ is the same as ...
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Dimension of quotient space $L^2(\mathbb{R}, m) / \{ f \in L^1\cap L^2 : \int_{\mathbb{R}} f dm = 0 \} $

The subspace $$ \left\{f \in L^1\cap L^2(\mathbb{R},m) : \int_\mathbb{R} f\ \mathrm{d}m=0 \right\} $$ is a norm-closed subspace of $L^2(\mathbb{R},m)$, so it is valid to consider the quotient space. ...
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Showing that $D^n/\partial D^n \approx S^n$, i.e. the quotient of $n$ dimensional disk with its boundary is homeomorphic to an $n$ dimensional sphere

Preamble: I am trying to show that $D^n/\partial D^n \approx S^n$, where $\partial D^n = S^{n-1}$. I've defined a mapping $f:D^n \to S^n$ as $f(x) = \begin{cases} s_1(s_2(x)) &: |x| < 1\\ N &...
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Reflexive coequalizers of pseudo-metric spaces

In Jiri Rosicky's recent paper Metric monads, he claims in the proof of Proposition 5.1 that reflexive coequalizers in the category $\mathsf{PMet}$ of (generalized/extended) pseudo-metric spaces have ...
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Do these topologies on spaces of subsets always embed into their larger counterparts?

Let $X$ be a topological space. Let $\mathcal{P}^\ast(X) = \mathcal{P}(X) \setminus \{\emptyset\}$ denote the power set of $X$ without the empty set. For a non-zero cardinal number $\alpha$ (which we ...
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A quotient topological space that isn't Hausdorff

Consider the group action $G=\mathbb{Z}$ on $X=\mathbb{R}^2\setminus\{(0,0)\}$ given by $n \cdot (x,y)=(2^nx,2^{-n}y)$. How to prove that $X/G$ isn't a Hausdorff space? I tried by searching 2 points ...
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Trouble with understanding quotient maps

Show that the cone $c(S^n)$ is homeomorphic to $\overline{B}^{n+1}$. The cone is defined as $c(S^n) = (S^n \times I)/(S^n \times\{1\})$. The book I'm reading says that this quotient comes with a ...
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Sum of a set with an equivalence class

I was studing and I can't understand the meaning of a notation. Let $\mathcal{F}(I)$ be the real algebra of functions $I \rightarrow \mathbb{R}$, where $I$ is a non-empty and finite set. We denote the ...
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For a Lie-group G and embedded Lie-subgroups K < H < G, prove that $\pi:G/K \rightarrow G/H$ is a surjective smooth submersion

The question is already in the title. For a Lie-group G and embedded Lie-subgroups $K < H < G$, prove that $\pi:G/K \rightarrow G/H$ is a surjective smooth submersion. Where it is meant that $K$ ...
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For Lie Groups $ H < G$, where H is embedded. The projection $\pi :G \rightarrow G/H$ maps components onto components

Let $G$ be a Lie-group and $H < G$ an embedded Lie-subgroup. Show that: The natural projection $\pi:G \rightarrow G/H$ maps connected components onto connected components. We already showed in the ...
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2 answers
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Difficulty understanding the meaning of quotient rings with regards to polynomials.

I am having difficulty understanding quotient rings with regards to polynomials. For example, the quotient ring $\mathbb{F_2}[x]/(x^3 + 1)$ maps to the set of all remainders when a polynomial of $x$ (...
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Let $U$ be the subspace {$(x,y,z)∈ \Bbb R^3:x=0 , y=0$} of $\Bbb R^3$

Let $U$ be the subspace {$(x,y,z)∈ \Bbb R^3:x=0 , y=0$} of $\Bbb R^3$. Then show that if $ v_1,v_2∈R^3$ be vectors such that the set {$v_1+U,v_2+U$} span {$\Bbb R^3/U$}. Then {$v_1,v_2,v_3$} doesn't ...
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2 votes
2 answers
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Intuition for quotients through exercise on rings

Show that $\Bbb Z \times \{0 \}$ is an ideal of $\Bbb Z \times \Bbb Z$ and describe the elements of the quotient ring $(\Bbb Z \times \Bbb Z)/(\Bbb Z \times \{0 \})$. Pick $a,b \in \Bbb Z \times \{0 \...
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Covering space and the Klein bottle [closed]

Hello I need help solving this exercise, I just want a path not a solution. Consider the equivalence relation $\sim$ on $\mathbb{R}^2$ generated by $(x, y) \sim (x + 1, y) $ and $(x, y) \sim (-x, y ...
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4 votes
1 answer
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Is there a simple characterization of compact subspaces of the "unit sphere" in $\mathbb{R}^\omega$?

Let $\mathbb{R}^\omega$ be the countably infinite product of $\mathbb{R}$ with itself in the product topology. $\mathbb{R}^\omega$ is metrizable but the metric doesn't arise from a norm. A natural ...
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4 votes
1 answer
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Is the "unit sphere" in $\mathbb{R}^\omega$ metrizable?

Let $\mathbb{R}^\omega$ be the countably infinite product of $\mathbb{R}$ with itself in the product topology. $\mathbb{R}^\omega$ is metrizable but the metric doesn't arise from a norm, so a natural ...
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16 votes
5 answers
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Why is addition defined, and not implied, on quotient spaces?

Small question. In chapter 3, section E, page 96 of "Linear Algebra Done Right", addition in quotient vector spaces is defined this way: I understand why scalar multiplication has to be ...
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Show that the quotient space is not Hausdorf

This is problem 2.13.3 in the Gamelin and Greene topology book. 1. Theorem Define $ X = [0,1] \times [0,1] $ with equivalence relation $$ (s_0, t_0) \sim (s_1, t_1) \iff t_0 = t_1 > 0 $$ (1) ...
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Why and when is this invariant polynomial map proper?

Let $G$ be a compact Lie group acting on $\mathbb{R}^n$. In the book "C∞-Differentiable Spaces", they give the following proof (Lemma 11.13 in the image) that the map $p$ is proper, where $p(...
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What is the shape I get after moding out the sphere $S^2$ by $A$?

I am trying to answer the following question: $(a)$ Compute the homology groups $H_n(X, A)$ when $X$ is $S^2$ or $S^1 \times S^1$ and $A$ is a finite set of points in $X.$ And I am trying to use the ...
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Criterion for proving that a quotient space is finite dimensional

In the book of Introductory Real Analysis of Kolmogorov and Fomin (page 122), there is some theorem about the dimension of quotient spaces. Let $L$ be vector space and $L'$ be a vector subspace. It ...
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The Axiom of Choice and a definition of addition in a quotient space of a vector space

I am thinking about the Axiom of Choice and I am trying to understand the Axiom with some but a little progress. Some time ago I could not understand why the obvious "proof" of the Axiom of ...
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