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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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A detail in the proof of Killing-Hopf theorem for Euclidean surface.

I am reading the book Geometry of Surfaces by Stillwell. In chapter $2$, he proves the following theorem: Theorem: (Killing-Hopf) Each complete, connected Euclidean surface is of the form $\mathbb{R}^...
Zoudelong's user avatar
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Issues with a quotient topological space [closed]

Let $X=S^1$ the unit circle and take $p\in X$ and define $A=X\setminus \{p\}$. Consider the quotient space $Y=X/A$, that means $x\sim y$ iff $x,y\in A$. Is $Y$ homotopic equivalent to $X$? My problem ...
Sigma Algebra's user avatar
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Suppose $M/G$ is a smooth quotient manifold and $N$ is a $G$-invariant submanifold of $M$. Then is $N/G$ a submanifold of $M/G$?

Let $M$ be a smooth manifold equipped with a, not necessarily proper, smooth Lie group action $G$. Suppose $M/G$ is a smooth quotient manifold. That is, there exists a smooth structure on the quotient ...
Spencer Kraisler's user avatar
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Linearly dependence of a set which contains cosets of a vector subspace. [duplicate]

Let $V(\mathbb{F})$ be a vector space and $W$ be a subspace of $V(\mathbb{F})$.Let $S=\{x_1,x_2,....,x_k\}$ be a linearly dependent subset of $V(\mathbb{F})$. Then $T=\{W+x_1,W+x_2,....,W+x_k\}$ is a ...
Mathew's user avatar
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Quotients of triangle groups

I want to find all quotients of ordinary hyperbolic triangle group (2,3,8). In general the "Triangle type" indicates the three positive integers p, q and r in the defining presentation ...
Zahid Malik's user avatar
1 vote
1 answer
40 views

Axioms of countability and quotient topology

I am trying to do the following exercise regarding axioms of countability and quotient topology: In $\mathbb{R}^2$ (with the euclidean topology) consider the equivalence relation: $(x,y) \sim (x',y') \...
user1255055's user avatar
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1 answer
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Definition of homology group as quotient in chain complex

I am working through some theory about abelian categories and complexes from "An Introduction to Homological Algebra" by Rotman. I don't understand one of the sections which I will explain ...
Flynn Fehre's user avatar
2 votes
3 answers
546 views

Why don't we simply say that the open sets in the quotient topology are projection of open sets of initial topology? [duplicate]

When we define the quotient topology, we say that an open set in it, are those sets which have pre image as an open set. But, why not just define it to be the image of open set of the initial set? ...
Cathartic Encephalopathy's user avatar
6 votes
3 answers
231 views

How do we understand intuitively how the quotient topology changes as we make the relation bigger or smaller?

Provided a relation on the set of a topological space, we can turn that relation into an generated equivalence relation, and hence induce a quotient topology on the quotient set. The open sets of the ...
Cathartic Encephalopathy's user avatar
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Feedback on and assistance with this proof about a particular quotient space of $\mathbb{C}P^1$

The goal here is to define the particular equivalence relation I'm attempting to describe, and then provide an equation (in this case, (2)) that can be used to determine whether or not two given ...
Simon M's user avatar
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Embedding the space to be attached into the adjunction

(Note that I am NOT asking about the space onto which glueing is done.) I am skeptical of the following statement: Let $A\subseteq X$ and $f\colon A\to Y$ be continuous and injective. Then $X$ embeds ...
Atom's user avatar
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Two dimensional cylinder

I am reading a paper referring a two dimensional cylinder, if i understand the definition is 2d cylinder= $\mathbb{R} \times \mathbb{R}/ \mathbb{Z}$ What's the intuition behind this? Does anyone have ...
user1174736's user avatar
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Axler Example 3.100 on quotient spaces

In Axler's linear algebra textbook, he gives an example of a quotient space as $\mathbb{R}^3/U$, where $U$ is a line in $\mathbb{R}^3$ containing the origin. The quotient, he claims, is the set of ...
Cardinality's user avatar
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Inequality relating quotient norm and norm

In an comment under an answer to this question How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? it is claimed that we have the ...
jcutler's user avatar
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What is a $T_2$ quotient?

Consider the following equivalence relation: $x\sim y$ provided $\mathcal N(x)=\mathcal N(y)$, that is, $x$ and $y$ have the same neighborhoods. It follows that the quotient of $\sim$ results in a $...
Steven Clontz's user avatar
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1 answer
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If $X$ is regular and $A$ is a closed subset of $A$, then $X/A$ is Hausdorff [duplicate]

If $X$ is regular and $A$ is a closed subset of $A$, then $X/A$ is Hausdorff How can I use to canonical quotient map $q: X \to X/A$ to prove the result? I tried picking distinct elements $x, y \in X/...
pera erdir's user avatar
3 votes
0 answers
117 views

Is $\mathbb R/\mathbb N$ homeomorphic to $\mathbb R/\mathbb Z?$

Where $\mathbb R/\mathbb N$ is the quotient topological space, where $\mathbb N$ is collapsed onto a point. Same with $\mathbb R/\mathbb Z.$ I was thinking I could prove it wasn't homeomorphic using ...
Alejandro Hernando's user avatar
2 votes
1 answer
50 views

Is there a name for the arc $\mathbb{S}^1 / (x \sim360 - x)$

I was playing with some ideas in a vague way and I have encountered this structure that arises from taking the space of angles $\mathbb{S}^1$ and quotienting it by the relation $(x, 360-x)$ (here $360$...
Sidharth Ghoshal's user avatar
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How does the quotient affect the complex valued metric?

Take a probability distribution without the normalization factor $f_t(x)=e^{{tL}}$ for suffcient statistic $L(x)=\frac{1}{\log x}.$ Take a map from the open unit interval $\mathcal M: I \to \Bbb C$ ...
zeta space's user avatar
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1 answer
77 views

Coherent topology with subspaces

EXAMPLE 2. Let $ J $ be a discrete space, and let $ E = [0,1] \times J $. Then the quotient space $ X $ obtained from $ E $ by collapsing the set $ \{0\} \times J $ to a point $ p $ is a linear graph. ...
Davood Karimi's user avatar
2 votes
1 answer
45 views

When $S^d/X$ is a manifold $S^d-X$ is homology equivalent to a point?

Upon reviewing for an algebraic topology final, I have found the following question which has stumped me: Let $S^d$ be a sphere of dimension $d\geq 1$, and $X\subset S^d$ a proper subset of $S$ such ...
Chris's user avatar
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Question and Notation regarding Quotients of Tensor Products

I read that "One can interpret $(\wedge^nV)^*$ as a quotient of $V^{\otimes k}$," in my class notes. I know that the notation $V^{\otimes k}$ is just $V \otimes V \otimes \dots \otimes V$ $...
RD Healthcare's user avatar
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0 answers
44 views

Pair of pants with k-genus

Let $\mathbb{H}$ be the hyperbolic plane. An usual($0$-genus) pair of pants $P_0$ is a $2$ dimensional sphere with $3$ holes and constant curvature $-1$. The boundary of a pair of pants consists of ...
Quanta's user avatar
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2 answers
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$G = \mathbf{Z}\times\mathbf{Z}$ and its normal subgroup $\langle (1,0)\rangle$. How is $G/N$ isomorphic to $\mathbf{Z}$? [duplicate]

I am new to quotient spaces and I came across this problem and it confused me: Given the group $G=\mathbf{Z}\times\mathbf{Z}$ and some normal subgroup $\langle (1,0)\rangle$.. The quotient group $G/N$ ...
baslerbuenzli's user avatar
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1 answer
31 views

Proving that a quotient space is not Hausdorff

a number of examples were given already, like this one : Show that the quotient space is not Hausdorf but I can't quite conclude on the following: Let ~ defined on $\mathbb{R}$ by : x ~ y if x = y = 0 ...
Jhnaby's user avatar
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An injection from the set of normal subgroups to subsets of irreducible representations

Some things I understand to be true: (1) A finite dimensional $\mathrm{C}^*$-algebra $A$ is of the form $$A\cong \bigoplus_{j=1}^NM_{n_j}(\mathbb{C})\qquad (n_j\in \mathbb{N}).$$ (2) With respect to ...
JP McCarthy's user avatar
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Is the factor theorem for modules true for rings

I have already known a result from factor theorem for modules: Theorem. For any $R$-module homomorphism $f: M \rightarrow N$, then $ M/\mathrm{ker} f \stackrel{\sim}{=} \mathrm{Im} f$. Proof. Show ...
nhhlqd's user avatar
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4 votes
1 answer
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Let $Y$ and $Z$ be subspaces of the finite dim vector spaces $V$ and $W$, respectively. Let $R=\{α\in L(V,W):α(Y)\subset Z\}$. What is $\dim R$? [closed]

How would I go about proving this? I tried by considering maps from $Y$ to $Z$ and the space of all such maps would have $\dim Y\times\dim Z$ but I have no idea how to extend it to the entire vector ...
Ran An's user avatar
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2 answers
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Factorisation of continuous maps

I'm studying general topology and a question has come to my mind. I am referring to the class of theorems that in algebra go by the name of "homomorphism theorems". In my topology course, we ...
Amanda Wealth's user avatar
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1 answer
55 views

CW complex - closure finiteness and weak topology

Well, this is embarassing. To be honest, during my PhD, I haven't really bothered too much regarding the topology of CW complexes. Back then, I understood the first few pages of Hatcher's book (the ...
May's user avatar
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1 answer
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Rings that appear as quotients $B/I$ of subrings$B \subseteq F$ of fields $F$ and for $I \subseteq B$ ideals

What are the rings $A$ that appear as quotients $B/I$ of subrings$B \subseteq F$ of fields $F$ and for $I \subseteq B$ ideals? For each $A$, give an explicit formula for a ring $B$ and a field $F$. ...
love and light's user avatar
2 votes
1 answer
69 views

Statement from Proposition 0.18 of Hatcher

On page $16$ of Hatcher's, following proof is given: but I can't see the induced deformation retract's existence. My attempt is to use "passing to the quotient", but somehow at the end I can'...
Mahammad Yusifov's user avatar
1 vote
0 answers
57 views

Preimage of a ball in a quotient metric space

Let $(X,d)$ be a metric space and define for each $x\in X$ and for $\varepsilon >0$ the ball $B_\varepsilon^X (x) := \{ y:y\in X \land d(x,y) < \varepsilon\}$. Now, consider a subset $A\subseteq ...
Andreas Compagnoni's user avatar
1 vote
2 answers
42 views

Topological Quotients: Understanding $X/\sim$ and $X/Y$ with Insights into the disk Structure.

I could use some assistance in clarifying a concept. In topology, when we have a space denoted as $X$, we can create a quotient space (a space of equivalence classes) denoted as $X/\sim$, where $\sim$ ...
Mousa hamieh's user avatar
1 vote
1 answer
106 views

Is the adjunction space of two Hausdorff spaces also Hausdorff?

I was reading the definition of CW-complex in terms of pushouts given by Lück's Algebraische Topologie: Homologie und Mannigfaltigkeiten (Chapter 3). It is stated (though not proven) that such a ...
Julius Maximus's user avatar
2 votes
0 answers
50 views

Isotropy subgroup of $\operatorname{GL}_{n+1}(\mathbb R)$ acting on $\mathbb R \mathbb P^n$

The general linear group $\operatorname{GL}_{n+1}(\mathbb R)$ (as a Lie group) acts smoothly on the real projective space $\mathbb R \mathbb P^n$, via $A \cdot [x] := [Ax]$. By $[x]$ here, I mean the $...
Joseph Kwong's user avatar
1 vote
0 answers
56 views

A particular quotient in the study of tensor products of $\mathfrak{sl}_2$-modules

I’m studying Verma modules of $\mathfrak{sl}_{2} = \mathfrak{sl}_{2} (\mathbb{C})$. Let’s introduce standard notation. Elements $h,e,f\in \mathfrak{sl}_{2} $ form the basis of $\mathfrak{sl}_{2}$, ...
Matthew Willow's user avatar
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1 answer
89 views

I showed $V$ is isomorphic to $U \times (V/U)$ without using the assumption that $V/U$ is finite-dimensional. What did I do wrong?

Exercise. Suppose $U$ is a subspace of $V$ such that $V/U$ is finite-dimensional. Prove that $V$ is isomorphic to $U \times (V/U).$ Outline of my proof. We need to construct a linear bijection ...
Paul Ash's user avatar
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Continous function $f:Y\to Y$ between quotient spaces [closed]

I was thinking in The following situation. Let $X$ be a topological space and $Y=X/\sim$ with quotient map $q:X\to Y$. Let $f:Y\to Y$ be a function. Is there an easy way to prove that $f$ is continous ...
Nestor Bravo's user avatar
1 vote
1 answer
96 views

Example of a Riemann surface

Let $\Gamma = \omega_1 \mathbb{Z} + \omega_2 \mathbb{Z}$ with $\omega_1, \omega_2$ independent over $\mathbb{R}$. Let $E_{\Gamma} = \mathbb{C}{/ \Gamma}$. Show that $E_{\Gamma}$ is a Riemann surface. ...
Andreadel1988's user avatar
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1 answer
65 views

Quotient Space: $V/U$ where $V=\mathcal{P}_2(R) \times \mathbb{R}^2$ and $U=\mathbb{R}^2$

Is this quotient space even defined? If I add some $v \in V$ to some $u \in U$, I would be adding a polynomial and a constant for the first coordinate (possible) and an ordered pair and a constant for ...
DarthArtoo4's user avatar
2 votes
1 answer
144 views

Is this quotient space a manifold?

Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R^{\times}\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ ...
Eric Ley's user avatar
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1 vote
1 answer
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Is the induced map on the quotient space bounded?

Let $T: V \rightarrow W$ be a continuous/bounded linear map between Banach spaces. Let $J \subseteq \ker(T)$ be a closed subspace. Then I know that this induces a unique linear map on the quotient $\...
Henry T.'s user avatar
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1 answer
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Reference for quotient lattices and universal property?

Question: Are there any references explaining the definition (or definitions) of quotient objects in the category of lattices? In particular a characterization in terms of universal properties would ...
hasManyStupidQuestions's user avatar
2 votes
1 answer
103 views

Free Commutative Monoid Quotient by Relations?

Say I have a commutative monoid $M$ that is generated by three elements $A,B,C$, where I have that $A+C=2B$. I want to write this a free (does that even mean anything?) monoid $\mathbb N^3$ with ...
Chris's user avatar
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1 vote
0 answers
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Continuity of quotient map with respect to topology induced by metric

Let $C([0,T];\mathbb{R}^d)$ denote the space of continuous functions with the usual supremum norm and given a path $x\in C([0,T];\mathbb{R}^d)$ let $x^s$ denote the stopped path $x(t \land s)$ with $t\...
Oscar's user avatar
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Passing an involution to a quotient algebra

This question is inspired by the discussion under this MO answer. I hope I have captured correctly what is going on in the below. Let $A_0$ be a finitely generated universal unital complex algebra $...
JP McCarthy's user avatar
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Does $Spec(k[G/[P,P]])$ have worse than quotient singularities?

Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a connected reductive linear algebraic group over $k$, and let $P$ be a parabolic subgroup of $G$. So we have the flag ...
Dave's user avatar
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2 votes
0 answers
44 views

Different topologies for the real projective base as a manifold

I am stuck with the following problem. I am given the $d$-dimensional real projective space $\mathbb{R}P^d$ as the set of equivalence classes of lines in $\mathbb{R}^{d+1}$, i.e. \begin{equation} \...
dancingqueen's user avatar
2 votes
3 answers
108 views

The norm on $\mathbb{R}^2$ is a quotient map

I think this is a simple example of a quotient map: $$ f:\mathbb{R}^2 \rightarrow \mathbb{R}_{\geq 0} \text{ defined by } f\left(x\right) = \left|x\right|$$ where $\mathbb{R}^2$ and $\mathbb{R}_{\geq ...
stuz's user avatar
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