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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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Finding an explicit isomorphism between $\mathbb R^4 / \ker \ T$ and $\mathbb R^2$

I'm wondering if I have a valid answer to this. It is exactly (e) of the following: I first state that the two vector spaces are isomorphic because they have equal dimension. I then define a ...
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Proving a set of vectors is a basis for the quotient map between two vector spaces

I want to see if my work is justifiable. I am tasked with the following: I will neglect to prove (a), as the work for this is fairly straight forward. I will center my attention on (b). $$\text{...
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Real Grassmann manifold and orthonormal groups

I'm trying to prove that the Grassmann manifold $$G_k(\mathbb{R}^n) = \{E = {\rm {\it k} - dimensional\ subspace\ of\ } \mathbb{R}^n\}$$ is equivalent to: $$G_k(\mathbb{R}^n) = \frac{O(n)}{O(k)\...
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norm of element in equivalent class in quotient space

If we have a quotient space $E\backslash L_0$ where $E$ is a linear normed space and $L_0$ it's subspace the norm of an element $L$ in $E\backslash L_0$ is defined as $$\lVert L\rVert = \inf_{x \in L}{...
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Homotopic maps and attaching spaces

Let $f, g : \mathbb{S}^{n-1} \to X$ be two continuous maps from the sphere into a compact and Hausdorff space $X$. I want to show that if $f$ and $g$ are homotopic, then attaching an $n$-cell to $X$ ...
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The quotient linear transformation

I'm very confused by the quotient linear transformation, and I'll try and illustrate my confusions in the following example: I have no problem showing (a), however I have serious conceptual ...
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Is the quotient space finite-dimensional linear space?

We have space $V=\{(x_1,x_2,x_3..):(x_1,2x_2, 3x_3,...)\in l_{\infty}\}$ Is the factor space (quotient-space) $l_{\infty}/V$ finite-dimensional linear space? First of all in this exercise (I think)we ...
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Constructing a linear map from annihilator of a subspace to dual of the quotient space

If W is a subspace of V, let V/W denote the quotient of V by W and let (V/W)* denote its linear dual. Construct a non zero linear map from Ann(W) to (V/W)* (1)From Basis (2) Canonical I am weak in ...
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Three space property

I want to show that Finite dimensionality is a three space property. Let $X$ be a normed linear space and let $Y$ be a closed subspace of $X$. If $Y$ and $X/Y$ are finite dimensional spaces, then I ...
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Understanding the definition of the projective sphere and quotient spaces

Munkres Topology: Corollary 60.4 says that $p^{-1}(y)$ is a 2 point set. Why is this? I understand $P^2$ to consist of equivalence classes of $S^2$ which are 2 point sets of antipodal points $\{x,-x\}...
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Elements of a $T_{0}$ quotient space are closed in $X$.

Define an equivalence relation on a topological space $(X,\tau)$ as $x\sim y$ iff $\overline{\{x\}}=\overline{\{y\}}$. I want to show that each equivalence class $[x]$ is closed in $X$. My attempt:...
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Quotient map from topological group onto left cosets is open

I've been trying to prove the following exercise from "James Munkers'-a first course in topology": Let $G$ be a topological group, and let $H\leq G$. Induce the left cosets, $G/H$, with the quotient ...
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Open sets in quotient space defined by equivalence relation

Given the following exercise: Consider on $\mathbb{R}^2$ the subsets: Now the exercise asks one to give a $C^{\infty}$ atlas on S. To define open sets on a quotient space obtained from an ...
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The symmetry group / isometry group of the complex projective space

question: For the complex projective space of $n$-complex dimensions, $$\mathbb{P}^n,$$ what is the symmetry group / isometry group of this complex projective space $\mathbb{P}^n$? Attempt: ...
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CW-complex structure on the quotient

Let $X$ be an $n$-dimensional CW-complex and let $A \subseteq X$ be a subcomplex. I want to show that the quotient space $X/A$ admits a structure of a CW-complex with skeletons $(X/A)^j := \pi(X^j \...
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Is Converse True, i.e Quotient space of any space is locally connected then space is locally connected?

$X$ is a locally connected space, and $f: X \to Y$ is onto where $Y$ has the quotient topology then $Y$ is locally connected. ($X$ is locally connected and $f: X \to Y$ is onto where $Y$ has the ...
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Is the quotient space path-connected?

Let $\mathbb C^n = \mathbb C \times \mathbb C \times \dots \times \mathbb C$ be the complex space. We define the quotient space $\mathbb C^n_{sym}$ by $\mathbb C^n / \sim$ where $x \sim y$ if there ...
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~$_P$ is an equivalence relation in A.

Prove Proposition 2.23. Use the notations defined in Definition 2.21 and 2.22. Definition 2.21. Let A be a set and R be an equivalence relation in A. Let x $\in$ A be an arbitrary element in A. The ...
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Are quotients of these two homeomorphic spaces still homeomorphic?

Let $D$ denote the open unit disk in $\mathbb C$ and $H$ denote the open left half plane in $\mathbb C$. We know there is a homeomorphism $f : D \to H$. Now let $$g = (g^1, \dots, g^n) := (f, \dots, f)...
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Question about the role of Quotient Spaces as Tensor Producs

According to Steven Roman's Advanced Linear Algebra, the very first objects that is required to construct the Tensor Product lies on a free vector space $\mathfrak{F}_{(\mathfrak{U}\times\mathfrak{V})}...
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Algorithm to find a basis of a quotient space $R^n/R^m$.

I have a set of $m$ vectors $\{x_i\}$, $x_i \in R^n$. How can I obtain a basis for $R^n/span(\{x_i\})$?
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Predual of $W^*$-subalgebra

I've seen many references claiming that if $\mathcal{N}$ is a $\sigma$-weakly closed *-subalgebra of a von Neumann algebra $\mathcal{M}$, then by taking $\mathcal{N}_\bot:=\{\phi\in\mathcal{M}_*|a(\...
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Homogeneous space construction theorem and expected dimensions of quotient maps

I am reading the notes linked here. On page 3 we read at Theorem 2 that the left coset space $G/H$ is a topological manifold of dimension $dim(G)-dim(H)$. Here $G$ is a Lie group and $H$ a closed ...
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Is there a simple characterization of the functions for which the Fourier inversion theorem holds identically?

I was thinking a little more about my previous quesion about whether there is a natural section of canonical representatives of the quotient map from the space $\mathcal{L}^2(\mathbb{R})$ of square-...
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What are the polynomial quotients $R/x$ and $R/(R/x)$ for $R = (\mathbb{R}[x]/x^n)$?

Define the polynomial ring quotient $R = \mathbb{R}[x]/x^n$. Is my understanding correct that $$ R/x \cong \mathbb{R} $$ and accordingly, as scalars divide all polynomials, $$R/(R/x) \cong \{1\}$$ ...
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Are these spaces quotient spaces?

I'm having trouble with this question: Consider the following spaces: Two circles of radius $1$ that are centered at $(0,1)$ and $(0,-1)$ (so that they touch at the origin - forming a "...
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What is $SO(n+1)/O(n)$ as a topological space?

Consider $SO(n),O(n)$ as topological groups. Find out $SO(n+1)/O(n)$ as a topological space. My attempt: Observed the inclusion : $O(n) \hookrightarrow{} SO(n+1)$ by, $$A \mapsto\begin{bmatrix} det(...
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Obtaining two-holed torus as a quotient of $\Bbb C$

The torus $T^2$ arises as a quotient space $\Bbb{C}/\Gamma$ for some lattice $\Gamma=\Bbb{Z}\gamma_1+\Bbb{Z}\gamma_2$ for $\gamma_1,\gamma_2\in\Bbb C$. One could think of this as gluing a $4g$-gon ...
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Prove that $[0,1]^n / S_n \cong \Delta^n$

Prove that $[0,1]^n / S_n \cong \Delta^n$, ( $S_n$ denotes the permutation group on n symbols, $\Delta^n:=\{(x_0,\dots,x_n)\in\Bbb R^{n+1} : x_0,\dots,x_n \ge 0 , x_0+\dots+x_n=1\}$, and the action is ...
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The quotient space of U(n)/SO(m)

Is there some good way to understand a quotient space of $$ U(n)/SO(m)=? $$ say $n=16$ and $m=10$? Can it be some kind of more familiar manifold? What it is? The Lie algebra generators of $U(n)$ ...
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Definition of a quotient group in Dummit-Foote

I'm reading the section on quotient groups in Dummit and Foote, and they give somewhat non-standard definition of a quotient group. I was wondering whether there is an easy way to see right away for ...
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U(n) v.s. SO(n) and their quotient / homogeneous spaces

By counting the number of generators, it is easy to see that unitary group U(n) has much more generator than SO(n). So it would make sense to consider the mod out group or the quotient space of U(n)/...
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Associativity of balanced products for $G$-spaces

This is Proposition 3.1 pg 4 Let $X$ be a right $G$-space, $Y$ a $(G,H)$ space, $Z$ a left $H$ space, then there is a natural homeomorphism. $$(X \times _G Y) \times_H Z \cong X \times_G( Y \...
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Differentiability of maps with quotient space as target set

Let $U\subset R^d$ open, $T>0$ and $\phi: U \to \mathbb{R}/{T\mathbb{Z}}$. What is the definition of $\phi$ being $C^k$ ? And how i get the partial derivatives in general? Can you recommend me ...
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Relationship between the determinant and cosets of a vector quotient space

I ran into the following question during my research. It seems like something which should already have an answer so, unable to come up with a solution myself or find one on the internet, I was hoping ...
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Find group $G$ and action of $G$ on $\mathbb{R}^2$ such that $\mathbb{R}^2 / G \approx M \setminus \partial M$, open Mobius strip

I want to find a group $G$ and an action of $G$ on $\mathbb{R}^2$ such that $\mathbb{R}^2 / G \approx M \setminus \partial M$, where $M$ is the Mobious strip, and $\partial M$ is its boundary, a ...
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$K = \mathbb{Z} / 2 \mathbb{Z}$. How many subspaces does the $K$-vector space $K^2$ have?

Let $K = \mathbb{Z} / 2 \mathbb{Z}$, How many subspace does the $K$-vector space $K^2$ have? I (hope to) already know the following When diving a whole number by 2, we can only obtain 0 or 1, so $K ...
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Why do topological quotients “bend” lines?

Why do topological quotients "bend" lines? http://mathonline.wikidot.com/topological-quotients-in-euclidean-space I have no problem with the idea that one constructs a topology on the line from its ...
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1answer
39 views

Does Hausdorffication preserve finite limits?

Does the left adjoint to the inclusion of T$_2$ spaces into the category of topological spaces preserve equalizers and finite products? (Apologies if this question has already been posted, but I did'...
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Level sets and quotients on affine varieties

Let $X \subset \mathbb A^n$ be an affine variety, say the zero set of polynomials $P_1,\dots, P_m$. Now, for an arbitrary polynomial $Q$, we can introduce an equivalence relation on $X$: two points $...
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Verification required on the Proof of $U\times V/U\cong V$

Is the following argument correct? Suppose $U$ is a subspace of $V$ such that $V/U$ is finite dimensional. Prove that $V$ is isomorphic to $U\times (V/U)$. Proof. Let $v_1+U,v_2+U,\dots,...
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Poincaré dual of the generators of $H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)$

We know $H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)=\mathbb{Z}_4$. So there are two classes of $\mathbb{Z}_4$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$. Wha are the Poincaré dual $(5-d)$-...
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Finite dimensionality of $V_1\times V_2\times \dots\times V_m$ implies $V_j$ is finite dimensional $\forall j$.

Is my proof of the following proposition correct? Proposition. Given that $V_1,V_2,\dots,V_m$ are vector spaces such that $V_1\times V_2\times\cdots\times V_m$ is finite dimensional. Prove that $V_j$ ...
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1answer
38 views

Action by automorphism group of a ring $R$ on $\mathrm{Spec}(R)$

Let $R$ be a ring, and let $G\subset \mathrm{Aut}(R)$ be a subgroup of the automorphism group. Then $G$ also induces an action on the affine scheme $\mathrm{Spec}(R)$. Namely, for $g:R\to R$ an ...
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70 views

Projective Space v.s. Quotient Space v.s. Fibration

What are the more precise relations between (a) projective space, (b) quotient space and (c) the base manifold under certain fibration? (1) Can every projective space (e.g. $\mathbb{RP}^n$, $\...
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Poincaré dual of $H^1(M,\mathbb{Z}_2)$ for a $\frac{\mathbb{CP}^2\times S^1}{\tau}$

Given a $M=\frac{\mathbb{CP}^2\times S^1}{\tau}$, where $τ$ acts as $−1$ on the sphere $S^1$ and a complex conjugation on complex projective space $\mathbb{CP}^2$. See Dold, Albrecht (1956), "...
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Exercise on quotient topology and countability axioms

Let $\, X := \mathbb{R}^3\Big/_{\sim} \,$ where $\, \sim \,$ is defined as: $\,x \sim y \iff x = y \quad \lor \quad \lVert x\rVert = \lVert y \rVert > 4$. Say wheter the canonical map $\, \pi :\...
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$\pi_j(G_{cont}/G_{finite})$

For the quotient group $$G'=G_{cont}/G_{finite},$$ knowing the homotopy groups of $G_{cont}$ and $G_{finite}$, one can determine homotopy groups from the long exact sequence $$ ... \to \pi_n(G_{...
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53 views

Finest topology such that a map is continuous : isn't it the discrete topology?

Let $X$ a set and $\mathcal R$ a relation of equivalence of $X$. Set $$q: X\to X/\mathcal R,$$ the natural projection. The quotient topology, it the finest topology such that $q$ is continuous. Isn't ...
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Blow-up and gluing coordinates

I am reading the book "Algebraic Geometry and Statistical Learning Theory" by Sumio Watanabe and have a question regarding Remark 3.16 (1) on page 95. He defines the blow-up of $ \mathbb{R}^2$ with ...