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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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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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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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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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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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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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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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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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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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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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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
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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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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 ...
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Quotient Space Metric with Nice Equivalence Classes

Quotient Space Metric: The quotient metric for arbitrary quotient spaces is defined as If $M$ is a metric space with metric $d$, and $\sim$ is an equivalence relation on $M$, then we can endow ...
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Is $(S^1)^3/{\sim}$ a manifold? $\sim$ is equivalent relation on all triple permutations $(a,b,c)\sim (a,c,b)\sim (c,a,b)\sim … $ on $(S^1)^3$

Let $\sim_1$ be an equivalent relation on $(S^1)^2$, $(a,b)\sim(b,a)$, then $M_2:=(S^1)^2/{\sim_1}$ is homeomorphic to Mobius band and is a $2$-manifold,. What can we say about $M_3=(S^1)^3/{\sim_2}$ ...
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Open Sets in the Wedge Sum and a Homeomorphism

I am presently working through example 1.21 in Hatcher's book on wedge sums of topological spaces. He makes a few claims which I am having trouble verifying. First, let me set-up some notation. Let $...
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1answer
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Help proving this map is closed

Define the following equivalence relation on $\mathbb{R}^2$: $(x,y)\sim(x',y')$ iff there is $n\in \mathbb{Z}:(x',y')=(x+n,(-1)^ny)$. Let be $E=\mathbb{R}^2/\sim$ the quotient space, and $q:\mathbb{R}^...
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Quotient space $[0,1]/C$ is homeomorphic to $[0,1]/\left\{0,1,\frac12,\frac13,…\right\}$, $C$ denotes Cantor set. [closed]

How to prove quotient space $[0,1]/C$, where $C$ denotes Cantor set, is homeomorphic to $[0,1]/\left\{0,1,\frac12,\frac13,...\right\}$?
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Odd maps from $S^1$ to $S^1$ viewed from the quotient $\mathbb{R}/\mathbb{Z}$ perspective

Reading a proof about degree oddity of odd continuous maps from $S^1$ to $S^1$, I have the following: Let $f : S^1 → S^1$ be an odd map : $\forall z ∈ S^1 ⊂ \mathbb{C}, f (−z) = −f (z)$ $\...
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Spaces homeomorphic to qoutient space $\frac{\mathbb{H^2}}{\sim}$

What spaces are homeomorphic to $X=\frac{\mathbb{H^2}}{\sim}$, where $\sim$ is relation between points $(r_1,\theta _1)\sim (r_2,\theta _2)\sim (r_3,\theta _3)$ in polar cordinate $\mathbb{R^2}$ and $...
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Quotient topology clarification, what happens if we glue together a point on a boundary with a point in the middle

When I first learned about the quotient topology $X/\sim$ on a topological space $X$ the quotient space was defined to be $X$ with all the points identified by $\sim$ glued together. So if $X = [0,2]$ ...
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Quotient of Second Countable Space

I'm looking for an example for a second countable topological space $T$ such that there exist a quotient structure $T/\sim$ which is not second countable. Does there exist an example where $T$ is a ...
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Prove that quotient space is Hausdorff

Let $A$ be a closed subset of the interval $[0,1]$. Show that the quotient space $$W=(-2,2)\big/A$$ is Hausdorff. I guess we need to explicitly find the disjoint open sets for each two $x\neq y$ from ...
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Map from circle to real projective plane

can you help me with these proofs? Let $f: S^1 \to \mathbb{R}P^2 $ be a map from circle to real projective plane and $\pi: S^2 \to S^2/\sim$ be the quotient map where $\sim$ identifies antipodal ...
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Intuition on norm of quotient space

Theorem. Let $(X,\| \cdot \|$) be a normed space. Then $$ p(x+U) = \inf_{z \in U} \|z-x \|$$ defines a semi-norm on $X/U$ with $p(x+U) \leq \|x \|$. a) If $U$ is closed, then $p$ is a norm. b) If $U$...
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Tensor Product of Quotient.

Let $K$ be a field and $L/K$ a field extension. Suppose $A$ is a $K$-algebra and $I$ is an ideal. I want to show that $$ (A/I\otimes_K L) \to (A\otimes_K L)/(I\otimes_K L)$$ So i define a map $$f: A\...
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Volume of a 3D torus as a form on a quotient space

I'm trying to calculate the volume form $dx\wedge dy\wedge dz$ on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$: $\int_{\mathbb{T}^3}dx\wedge dy\wedge dz$ I've been told that it's simply the volume of ...
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Show that the space $X$ is not a surface

I would like to show that the following space is not a surface: $X$ is made as an identification space of the unit square $Q=\{(x,y)\mid 0\leq x,y\leq1\}$ with the identifications: $(0,y)\sim(1,y)$ ...
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The action of the group $\Gamma=\mathbb{Z}$ on the manifold $\mathbb{C}^n-\{0\}$

Let $\Gamma=\mathbb{Z}$ be the additive group of integers and give it the discrete topology. Suppose $\Gamma$ acts continuously on the topological n-manifold $\mathbb{C}^n-\{0\}$ by the map $x \mapsto ...
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Sheaves on a GIT quotient

As stated in the title, my question regards sheaves on a GIT quotient. Let me fix the notation: $G$ is the group scheme acting on the scheme $X$ and both $X$ and $G$ are $k$-schemes. Searching online ...
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Quotient with divisors of zero

Is there a ring $R$ with divisors of zero which have an ideal $I$ (non-null neither equal to $R$) that, the quotient $R/I$ also has divisors of zero?
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Characterization of quotient maps

The following is quoted from https://en.wikipedia.org/wiki/Quotient_space_(topology) Quotient maps q : X → Y are characterized among surjective maps by the following property: if Z is any topological ...
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1answer
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Corollary of the Hahn Banach theorem

I want to prove the following corollary of the Hahn-Banach theorem. Let $X$ be a normed space. For every closed linear subspace $Y\subseteq X$ and $x\in X-Y$, there exists $x'\in X $ such that $x'|Y=...
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Unique complex structure on the modular curve $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$

Is the complex structure on the modular curve coming from the quotient $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$ unique? (Here $\mathbb{H}$ is the upper half plane in $\mathbb{C}$) According to ...
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1answer
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Computing Coordinate Rings of Varieites

I am a complex analyst who was been screwed over by fate and now has to work with elliptic curves for my doctoral dissertation. This entails learning about (non-category-theoretic) algebraic geometry. ...
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When is a space homeomorphic to a quotient space?

Is the following theorem true? It seems straightforward but I haven't seen it published anywhere, not even as a corollary, so I'm concerned I've missed something. Discussions that introduce quotient ...
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1answer
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Quotient maps and open maps

I was doing Exercise A.36 in Lee's Introduction to smooth manifolds which states the following: Let $q: X \rightarrow Y$ be an open quotient map. Then $Y$ is Hausdorff if and only if $R = \{(x_1, ...
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Uniqueness in the Universal Property of Quotient Maps

Here is Munkres' way of phrasing the universal property of quotient maps: Let $p : X \to Y$ be a quotient map. Let $Z$ be a space and let $g : X \to Z$ be a map that is constant on each set $p^{-1}...
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1answer
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Proving $f(x)=(\cos(2\pi x),\sin(2\pi x)))$ is not closed and is the quotient mapping

$S^1$ is the subspace of $\mathbb{R}^2$ consisting of all $y\in\mathbb{R}^2$ such that $d(y,0)=1$, where $d$ is the Eucledian metric. Let $f:\mathbb{R}\to S^{1}$ be given by $f(x)=(\cos(2\pi x),\sin(2\...
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The name “section” for the operation of selecting representatives of an equivalence class

This is a question about terminology and sources. While looking for a name for the operation of "picking a representative from an equivalence class", I came upon the Wikipedia article on equivalence ...
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What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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32 views

Prove or disprove if a quotient map from X to Y with Y Hausdorff, then X is Hausdorff. [closed]

For two open disjoint subsets U and V, I want to show their pre-images are disjoint open subsets of X or not. But I have no idea how to do it. Any help would be appreciated. Thanks in advance!
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1answer
81 views

Is every quotient by a finite group an orbifold?

It is required, in order to be an orbifold, to be locally like $\mathbb{R}^n/\Gamma$ where $\Gamma$ is a finite subgroup of $GL(n,\mathbb{R})$ and that the fixed points of the action of $\Gamma$ have ...