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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How does the wedge product appear in the quotient approach to constructing an exterior algebra?
Let $V$ be a finite dimensional vector space and let $T(V)$ denote the tensor algebra
$$T(V) = \bigoplus_{k=0}^\infty T^k(V)$$
where
$$T^k(V) = \underbrace{V \otimes \cdots \otimes V}_k.$$
Let $I(V)$ ...
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Closed curve in a torus diffeomorphic to a circle?
In Example 15.9 of Tu's "An Introduction to Manifolds: Second Edition", it is written
Let G be the torus $R^2/Z^2$ and L a line through the origin in R2. The torus can also be represented ...
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G-invariant Tietze Extension Theorem
The Tietze Extension Theorem states that if $V$ is a closed subset of a metric space $X$ and $f$ is a continuous function on $V$ then there exists $F$ continuous on $X$ such that $F|_{V} = f$.
Suppose ...
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X/A with respect to the quotient topology is Hausdorff?
Consider $X = [0,1]$ and $A = (a,b)$ (where $0 < a < b < 1$).
I want to prove $X/A$ is connected and compact.
Here's my approach :
If I take an open cover of $X/A$, the corresponding cover in ...
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Can every "noodly" space be decomposed into one or more manifolds?
The Premise:
Let $X$ be a topological space. For any pair of points $x_0,x_1\in X$, let $\mathrm N_X(x_0,x_1)$ be the set of all subspaces $S$ of $X$ such that $x_0,x_1\in S$ and $S\cong I$ for some ...
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Set of positive definite Hermitian matrices as quotient
Let
$$\mathcal{H}_n^{++} := \{ A \in \mathrm{Mat}_n(\mathbb{C}) \vert A = A^{\dagger} \text{ and } A > 0 \}$$
denote the space of positive definite Hermitian matrices. I have read in some ...
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Topological Quotient vs Group Quotient [closed]
I was wondering about if topological quotients and group quotients always agree for a topological group. By this, I mean if G is a topological group, G/~ is a topological quotient space, G/N is a ...
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When can we "switch" isomorphic things
I'm finishing a course on basic abstract algebra, which covers groups, rings, modules, and finite group representations. However, up to this point, I am not very sure about the concept of isomorphism.
...
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How to identify the number of degrees of freedom in a system of particles using the Principle of Relativity
So consider the following laws of motion in an inertial system :
$$ m_k \partial_{tt} x_k = - \partial_{x_k} V(x_1, .., x_k)$$ where $x_i \in \mathbb{R}^3$ is a particle of mass $m_i$. By principle of ...
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Is this induced quotient space Hausdorf?
Let Y be the quotient space obtained from $\mathbb{R}_{K}$ by identifying the set K to a point.
Let p : $\mathbb{R}_{K} → Y$ be the quotient map.
Is this quotient space hausdorf?
K is Closed in $\...
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How to compute $\pi_1 \left (S^1 \times S^1 \setminus \{(1,0,1,0)\}, (-1,0,-1,0) \right ).$
Compute $\pi_1 \left (S^1 \times S^1 \setminus \{(1,0,1,0)\}, (-1,0,-1,0) \right ).$
I know that it's torus minus a point. The idea is to use the polygonal representation of a torus i.e. a square ...
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Does cutting and pasting preserve continuity in the quotient space.
I have a hexagon with interior with the opposite sides being identified. In order find the identification space I did the following cutting and pasting.
My question is : Is it correct way to do the ...
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$\Bbb R/\Bbb Z$ is an angry electron, right?
I keep seeing $\Bbb R/\Bbb Z\cong S^1$. Shouldn't this instead be the quotient of $\Bbb Z$-many copies of $S^1$ by some fixed point? Isn't the correct quotient $S^1\cong\Bbb R/\{\{x,x+1\}:x\in\Bbb R\}\...
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Definition of Universal enveloping algebra of a lie algebra
I am a physics student, and I am trying to understand the Casimir operator from a formal perspective; therefore, I come to learn what's Universal enveloping algebra.
Two definitions of Universal ...
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When is a quotient a differentiable manifold?
The two most popular ways to generate a topological manifold from others are picking a subspace of a topological space and computing the quotient space of a topological space. In differential geometry,...
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Does the quotient induced topology coincide with the "corresponding" topology on the quotient space?
Let $ X $ be a Banach space, and $Y$ a closed subspace, and let the quotient map be $ q : X \to X/Y$.
Now, if $X$ and $X/Y$ are given the norm topology (where the norm on $X/Y$ is $\|x+Y\| = d(x,Y)$), ...
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Show that a quotient function is continuous
Let $R_1$ and $R_2$ be equivalence relations on the spaces $X$ and $Y$, respectively. Given a continuous function $f$ such that $xR_1y$ implies $f(x)R_2f(y)$, prove that the induced function $\smash[t]...
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Doubt on universal property of quotient vector spaces
We have the following theorem (call it the universal mapping property for quotient spaces): $V,U$ are vector spaces over field $F$, $W⊂V$ a subspace. Then for every linear transformation $t:V→U$ so ...
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Prove that the composition of quotient functions a quotient function
Let $f: X \to Y$ and $g: Y \to Z$ be quotient maps. The idea is to show that the composite function $g \circ f: X \to Z$ is a quotient map. To do this, we must proof that $V^{\text{open}} \subseteq Z$ ...
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Norm on $H^1/2(\partial\Omega)$ by first isomorphism theorem.
I want to show that $\operatorname{ran}(T)$ equipped with
\begin{equation*}
\lVert v\rVert:=\inf\{u\in H^1(\Omega):Tu=v\}
\end{equation*}
is a Hilbert-Space.
I have seen that the definition of the ...
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Real Projective Space has homeomorphic atlas
I am trying to show that $\mathbb{RP}^{n}$ is a smooth manifold with atlas $({U_{i},\phi_{i}})$ where $\phi_{i}:U_{i}\to \mathbb{R}^{n}$ is defined by $[x_{0},\dots,x_{n}]\mapsto (x_{0}/x_{i},\dots,x_{...
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What does it mean to 'attach a cone along a subspace'?
I am a little bit confused about the terminology used in Section 2.4 of this paper. We have a finite pointed CW complex $B$ and a subcomplex $A$. Then "let $C$ be obtained from $B$ by attaching a ...
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Upper bound the dimension of Quotient Space
Let $X$ be a Banach space and let $Y$ be the kernel of a bounded linear functional on $X$, say $M: X \rightarrow \mathbb{R}$. Is it possible to upper-bound the dimension of the quotient space $X/Y$. I ...
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Sheaf cohomology on quotient stacks
Suppose I have a scheme $X$ over $\mathbb C$, acted on by a finite group $G$. Let $\mathcal F$ be a $G$-equivariant coherent sheaf of $\mathcal O_X$-modules.
Then I can form the stack quotient $[X/G]$,...
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Is $0\in \{ux+vy+wz=0\}\subseteq \mathbb C^6$ a quotient singularity?
I am trying to gain some intuition for telling when a variety has quotient singularities.
The example I am focusing on here is the affine variety $X$ which is cut out by the equation $ux+vy+wz=0$ in $\...
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Quotient Ring example issue
I am currently learning about lattices and came across a Mr.Bill Buchanan video that explains the Ring-LWE problem. I understand how modulo works and quite understand the math behind it.
However, I ...
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References for Topology on Graphs
A graph $ \Gamma = \Gamma (V,E) $ consists
sets $ V , E, $
A function $E \to E : e \mapsto \bar e$ such that $\bar {\bar e} = e,$ and $\bar e \ne e$ for all $e \in E$
A function $i : E \to V$ which ...
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Isomorphism in quotient spaces of linear spaces
Let $E$ be a linear space and $V$ a linear subspace. No assumption is made on the dimension of $E$.
We write $G = \{g \in GL(E) \mid g(V) = V\}$ where $GL(E)$ is the group of invertible linear maps of ...
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Check whether the map is a homeomorphism.
Let $\mathbb R \times [-\pi, \pi]$ be given the topology induced from $\mathbb R^2 \simeq \mathbb C.$ Consider the map $\mathcal E : \mathbb R \times [-\pi, \pi] \longrightarrow \mathbb C^{\times}$ ...
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Is the projection map $\pi_1:\mathbb{R}^2 \to \mathbb{R}$ a quotient map?
Is the projection map $\pi_1:\mathbb{R}^2 \to \mathbb{R}$ a quotient map? It is definitely a surjective and continuous map. I think it is also open and most likely not a closed one. Surjective, ...
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How can I show that these two topologies are the same?
$G$ is a topological group and $H$ an open subgroup of $G$ I want to show that the quotient topology of $G/H$, $\tau_{G/H}$ is equal to the discrete topology $\tau_{\text{discrete}}$.
I mean clearly ...
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Is the quotient space of a homogeneous space induced by a free group action s.t. the quotient map has a right inverse homogeneous?
Given is a topological space $X$ and a group $G \leq$ Aut($X$) with the property: for $\lambda \in G$ and $x \in X$, $\lambda(x)=x \Rightarrow \lambda=id_X$, also satisfying that the quotient map $q:X ...
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Quotient of projective group scheme by a finite group action
Let $X$ be a projective group scheme, over some base $S$, and let $G$ be a finite group acting on $X$ by $S$-isomorphisms.
I would like to understand if/when the quotient $X/G$ is representable by a ...
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Example of a quotient map between locally compact Hausdorff spaces that's not preserved by a product
I'm looking for an example of a quotient map $p : X \to Y$, with $X$ and $Y$ both locally compact Hausdorff spaces, such that there is a Hausdorff space $A$ for which $p \times 1_A : X \times A \to Y \...
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Quotient by compact topological group has closed projection
Let $X$ a topological space and $G$ a compact topological group, why is the quotient map
$$\pi \colon X \to X/G$$
closed?
For every closed subset $C$ of $X$
$$\pi^{-1}(\pi(C)) = \bigcup_{g \in G}g\...
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Resolution for exterior power of a quotient
Let us assume that we have exact sequence of vector spaces:
$$0\to U\to V\to W\to 0.$$
We can think of $0\to U\to V$ as a resolution of $W$.
Can we construct some canonical resolution of $\Lambda^n W$...
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Definition and construction of a quotient space using a toy example
To make headway into Agebraic Topology I need to precisely understand the definition and construction of the quotient space. Definitions in my textbooks and online have felt handwavy to me, and I don'...
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Why are quotients by free group actions "well behaved"
Let $M$ be a smooth manifold and $G$ a Lie group, then if $G$ acts smoothly, freely, and properly on $M$ it is a well known result that the quotient $M/G$ is a smooth manifold. In the context of ...
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Elements of quotients by groups of homeomorphisms
Definition by Manetti (rephrased):$X$ is a topological space. $G \subseteq Homeo(X).$ Let $x,y \in X$. Then $x\sim y$ if $\exists g \in G$, such that $y=g(x)$. This defines an equivalence relation on ...
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On quotients of subset rings over infinite sets
$\newcommand{\S}{\mathcal{S}}$
For a set $X$ we definte the subset ring $\S_X$ as the set of subsets of $X$ equipped witht the two operations
$$\begin{align}
A + B &:= (A\setminus B) \cup (B \...
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Homeomorphisms of quotient spaces. (With pictures)
I have a question about homeomorhic spaces: $\mathbb{R}^2$; $\mathbb{R}^2 /I$; $\mathbb{R}^2/D^2$; $\mathbb{R}^2/I^2$.
My attempt for first pair... I try to understand deformation of rectangular ...
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Question about topological embedding followed by quotient also being an embedding
Let $f : L\rightarrow M$ be a homeomorphism onto its image (aka a topological embedding), and let $q : M \rightarrow N$ be a topological quotient map.
Suppose $(q \circ f) : L\rightarrow N$ is ...
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Quotient Spaces (from Viro, Harlamov et al)
dears! I have some troubles with quotient spaces.
Let $A\subset X$ and $X=\{b_1,b_2, \dots, b_n, \dots, a_1,\dots, a_n,\dots\}$ $A=\{a_1,a_2,\dots,a_n,\dots\}$. Is it correct to imagine, think about ...
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Quotients of non-commutative rings.
All of the ring theory I know comes from a commutative perspective. I'm learning some non-commutative algebra right now, and I'm wondering whether the following theorem holds.
Let $R$ be a commutative ...
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Is the orbit space of discrete group act continuously and properly on manifold with boundary is still a manifold with boundary?
This question was inspired by Lee's Introduction to Topological Manifold problem 12-22 in chapter 12. Here is the original problem:
Give an Example of a manifold $M$ and a discrete group $\varGamma$ ...
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What is $k$-Hausdorffification?
Recall that a relation is $R\subset X\times X$ and $a~b$ iff $(a,b)\in R$.
According to Charles Rezk, given a space $X$, we can construct its $k$-Hausdorffification as follows, for any relation ~$_{\...
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Restriction of quotient spaces.
Let $V_{1,2}$ be two $\mathbb{R}$-vector spaces and $U_{1,2}\subset V_{1,2}$ two linear subspaces. If $f:V_{1}\to V_{2}$ is a linear map such that $f(U_{1})\to U_{2}$, it induces a well-defined map on ...
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What are cells in CW complex?
I was studying Hatcher's Algebraic Topology regarding CW complex. I am having trouble in understanding definition of CW Complex.
(1) Start with a discrete set $X^0$, the $0$-cells of $X$.
(2) ...
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Characterization of torus $I/\mathbb{Z}^{d}$?
The quotient space $\mathbb{R}/\mathbb{Z}$ be defined by the set of all equivalence classes with respect to the equivalence relation $x \sim y \iff x-y \in \mathbb{Z}$. This quotient space is then ...
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Proving the following question: Z[3i] isn't PID by use of the quotient map Z/(3i) isn't a field.
I was wondering if this methodology is sound. I've checked that for any element in $\mathbb{Z}[3i]$ if we take the quotient map with ideal $(3i)$, its kernel consists of the elements $ \{ 9a+3bi \}$. ...
