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.
1,977 questions
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Fundamental Group of Annulus Quotient
I was working on a problem from Lee's Introduction to Topological Manifolds and I am a bit confused as to what this space is supposed to look like. Here is what I am given:
Let $Q$ be the following ...
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How to prove the direct product of identification map is not an identification map
How to prove the direct product of identification map is not an identification map? Let ${X}_{1} = \left\lbrack {0, 1}\right\rbrack \times$ $\mathbb{Z}$, on ${X}_{1}$ we have an equivalence ...
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Connection between definition of topological continuity and the quotient topology
Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ is continuous if $f^{-1}(U)$ is open for every open subset $U\subseteq Y$.
If we now require $Y$ to be merely a set and $f$ to be surjective, ...
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Why is $X= \mathbb{R}_{fc}$ homeomorphic to $X/A$?
I m just trying to understand the following fact. I read the corresponding section in Topology by Munkres a few times but simply cannot understand how:
Let $X= \mathbb{R}_{fc}$ be the finite ...
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dimension of GIT quotients
Let $X$ be a non irreducible affine variety and $G$ a linearly reductive algebraic group, acting freely on X. I have seen in several places that the dimension of the the orbit space $X/G$ (that ...
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If A is closed then $X/A$ is a hausdorff space [duplicate]
I m trying to proof the following statement:
Let $X$ be a topological space and $A\subset X$ a subset. We write $X/A$ as de quotient space of $X/\sim$ wherein $\forall x,y \in X: x \sim y \iff (x =y $ ...
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If $X/A$ is a hausdorff space, proof that $A$ is closed in $X$ [duplicate]
Let $X$ be a topologic space and $A \subset X$ a not-empty subset. We note with $X/A$ the quotient space of $X/ \sim$ where $\forall x,y \in X: x \sim y \iff (x = y $ or $x,y \in A)$. The topology ...
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Determine the nilradical of $K[X]/(X^n)$
Let $K$ be a field, define $A:=K[X]/(X^n)$, I want to show that the nilradical $$\mathfrak{N}_A=\{P \in A | \exists n : P^n=0\}=(X)$$
My Problem is that I don‘t seem to use $K$ being a field.
My proof ...
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Proving $X^{n-1} \cong (X^{n-1} \sqcup_\alpha \partial D^n_\alpha) / (x \sim \varphi_\alpha(x))$ in CW-complexes
In this answer, it is mentioned that $$X^{n-1} \cong (X^{n-1} \sqcup_\alpha \partial D^n_\alpha) / (x \sim \varphi_\alpha(x))$$ where $\varphi_\alpha: \partial D^n_\alpha \rightarrow X^{n-1}$ is the ...
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Projectivization of an affine variety with exactly $1$ fixed point under the action of $\mathbb G_m=k^\times$
Section 9.5 Weyman - An Introduction to Quiver Representations (2017)
Often one has to omit some points of the variety in order to construct a more interesting quotient. To illustrate this, suppose ...
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Proving the suspension of a continuous map is continuous
Given a continuous map map between topological spaces $f\colon X\to Y$ one can define the suspension of the map $f$ by
$$Sf\colon SX\to SY,\quad [x,t]\mapsto [f(x),t].$$
Where $SX$ is defined as the ...
user1457233
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Would an analyst be interested in quotient vector spaces?
I am trying to think of a justification of why one would be interested in quotient vector spaces from an analysis prespective. I know that from a geometric(/Topological) view, one maybe interested in ...
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Product of quotient spaces vs. quotient of product spaces
For an equivalence relation $\sim$ on a topological space $X$, form the natural $f^{\ast}$ map from the quotient
space $(X \times X)/\sim'$ to $(X/\sim) \times (X/\sim)$,
where $\sim'$ is the ...
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Non-metrizable quotient despite a closed equivalence relation on a separable metric space
Nik Weaver has the following exercise in Chapter 1 of his Measure Theory and Functional Analysis:
1.21. Find an example of a closed equivalence relation on a separable metric
space such that the ...
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Moment maps on Marsden–Weinstein–Meyer quotients
I'm learning a bit of symplectic geometry and more specifically about moment maps, so I'm currently playing around a bit with the concept. In particular, I'm curious about ways of constructing new ...
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Homology of the quotient space by group action and $C_*$ preserving homotopy pushouts
Let $G$ be a Lie group, $E$ is a principle $G$-bundle and $X$ a topological space with right action of $G$. We have pushout diagram in the category of topological spaces
\begin{array}
A X \times G \...
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Quotient map, normality of $X$ implies normality of $Y$
The problem is from Munkres' Topology, in the section on separation axioms. It goes as follows:
Let $p : X \to Y$ be a closed, continuous, surjective map. Show that if $X$ is normal, then so is $Y$.
[...
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Methods to show that a singularity in a variety is not a quotient singularity
Let $X$ be a variety (irreducible, normal) over an algebraically closed field $k$ of characteristic $0$, and let $p\in X$ be a singular point. To simplify things, I'll assume that $X$ is affine and ...
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The bundle structure on $E/G = \bigcup_{x\in M} E_x/G_x$
I'm learning about the bundle structure on $E/G$ where $E$ is a vector bundle over $M$, and $G$ is a subbundle. I've reproduced my understanding below and asked some questions. Thanks for your help!
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topology of unrdered product of $\mathbb{RP}^2$
Let $X$ be a topological space and $Y$ be the quotient space $X\times X/\sim$ where the equivalence relation is defined as follows: $(x_1,x_2)\sim (x_2,x_1)$ for all $x_1,x_2\in X$ and $(x,x)\sim(y,y)$...
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Do Smooth Quotient Maps on Manifolds Always Locally Resemble Projections?
I'm investigating the relationship between smooth submersions and smooth quotient maps in the context of smooth manifolds, and I'm particularly interested in understanding if every smooth quotient map ...
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Is the graph quotient a colimit?
Let $\mathcal{Grph}$ be the category of undirected simple graphs with self-loops allowed. A morphism in $\mathcal{Grph}$ between two graphs $f:G \rightarrow H$ as a map on the vertex set that ...
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$T_0$ spaces and equivalence relation
In a topological space $X$, define $x\equiv x'$ to mean that $cl\{x\} = cl\{x'\}$. Let $\ Y $ be the set of all equivalence classes thus defined, let $\tau$ map each point of $\ X $ into its ...
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A quotient space which is not Hausdorff
I have been working on Problem 3-17 in Lee's Introduction to Topological Manifolds. Here is Problem 3-17
This problem shows that the conclusion of Proposition 3.57 need not be true if the quotient map ...
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Good, fast-implementation algorithms for quotienting a finite-dimensional vector space over the reals with its subspace?
After doing a bit of digging, I can't find any native method in Julia's linear algebra package that let's me quotient a vector by a subspace. The Wikipedia article seemed to mainly focus on the theory ...
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Continuous/surjective maps between quotient spaces of $\mathbb R$ and $[0,1]$
Problem: Define the equivalence relation $R$ on $\mathbb R$ by $xRy$ iff $x-y\in \mathbb Z$ and define the equivalence relation $Q$ on $[0,1]$ by $xQy$ iff $x=y$ or $\{x,y\}\subset \mathbb Q\cap [0,1]$...
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Quotient of $\mathbb R^2$ that is not Hausdorff
Problem: Give an example of an equivalence relation $\sim$ in $\mathbb R^2$, whose equivalence classes are closed sets, but the quotient space $\mathbb R^2/\sim$ is not Hausdorff.
So far: My thinking ...
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Fields from Euclidean Domains: When do operations modulo an irreducible in an Euclidean Domain $R$ generate a field?
Somehow searching for this question I only find "all fields are euclidean domains" and "euclidean domains with unique quotient-remainder are either $F$ or the polynomial ring $F[X]$ for ...
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Enforcing invariance under isometries on a manifold that approximates data
I have some data, say a finite set $\mathcal{S}$ of grayscale images of given dimension $m\times n$, seen as vectors in $\mathbb{R}^{m \times n}$, with each coordinate expressing the intensity of one ...
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Internal direct sum of quotient space and subspace
I'm trying to (thoroughly) understand quotient spaces in linear algebra as I'm looking into geometric control. I, however, seem to keep running into discrepancies in papers, which makes me question my ...
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Is $\mathbb{F}_2^n/ rs(H)$ the same as the kernel of $H$?
I am not very familiar with quotient spaces but I am trying to understand what the authors of this paper mean in Lemma 1. Specifically, for a binary parity check matrix $H$, they consider the object
$$...
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Reference for the notion of quotient over a join-semilattice [closed]
In this question, a very natural way of defining the quotient of a join-semilattice over an ideal is described (in the community-wiki answer). This construction seems to differ from what it is usually ...
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Geometric interpretation of an equivalence relation
Consider $X=[-1,1]\subset \mathbb{R} $ with the standard subspace topology. Define the equivalence relation $a \sim b \iff [(a=b) $ or $(a=-b$ and $|a|<1)]$. My Professor suggested this as an ...
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Does gluing two points prevent simple connectedness?
Let $X$ be any path-connected Hausdorff space and let $x,y\in X$ be distinct points. Let $X/(x\sim y)$ be the quotient space obtained by identifying $x$ and $y$. Is it possible for $X/(x\sim y)$ to be ...
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How is torsion on a subset of the circle defined modulo an irrational rotation? [closed]
Can you help me define torsion generated by modulo an irrational rotation?
Rational torsion
In a Prufer p-group the elements of $\Bbb Z[\frac1p]/\Bbb Z$ are arrived at by reducing $\Bbb Z[\frac1p]$ ...
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Motivating Coequalizers from Set Quotients?
I am trying to interpret the definition of coequalizers as a generalization of quotients by equivalence relations in $\mathsf{Set}$. For products this is rather natural. My notes present an argument ...
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Linear isotropy group at the origin of a homogeneous space
Let $G$ be a Lie group and $H$ a compact subgroup. I am trying to understand Example 1.3 in Kobayashi & Nomizu's "Foundations..." , in Chapter IV.1. They say we consider the linear ...
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Quotient Groups and the Zero Element
I'm trying to understand tensor products from a rigorous mathematical perspective, and while there are definitely other ways to understand it, understanding it using quotient spaces will also help me ...
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Continuous surjection $\mathbb R^m\to \mathbb R^n$ that is not a quotient map
(I have found examples 1,2 that answer my original questions, so the question here is refined)
This question has been asked many times in this site, but all examples I see are maps between some ...
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Why are non-compact Hermitian symmetric spaces Stein spaces?
I look for a proof or at least a reference that Hermitian symmetric spaces are Stein spaces.
Because an irreducible Hermitian symmetric space is biholomorphic to a symmetric bounded domain, my ...
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Details in the proof of quotient manifold theorem
The theorem is stated as: Let $G$ be a Lie group acting smoothly, freely and properly on a smooth manifold $M$. Then $M/G$ is a topological manifold of dimension dim$M$ - dim$G$, and has a unique ...
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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}^...
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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 ...
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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 ...
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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
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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') \...
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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 ...
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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?
...
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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 ...
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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 ...
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