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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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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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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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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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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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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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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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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 ...
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Generalize exterior algebra: vectors are nilcube instead of nilsquare

The exterior product on a ($d$-dimensional) vector space $V$ is defined to be associative and bilinear, and to make any vector square to $0$, and is otherwise unrestricted. Formally, the exterior ...
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Let $\mathbb{F}=\mathbb{F}_3$ Find an irreducible polynomial of degree 2 and construct a field of 9 elements as a quotient.

Find an irreducible polynomial p of degree 2, and use it construct a field of 9 elements as a quotient. Describe the cosets in the quotient explicitly, and use them to construct the addition and ...
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Let $V=\mathbb{R}^3$ and $W=\{ (x,y,z): x+y+z=0\}$ Describe $V/W$ geometrically and contrsuct an explicit isomorphism $W^\perp \cong V/W$

For an isomorphism I let $\phi:W^\perp\to V/W$ be defined as $\phi(x)=[x]$ To show $\phi$ is injective suppose $x\neq y$ for $x,y \in W^\perp$. Since $W=\{ (x,y,z): x+y+z=0\}$, $(1,1,1)$ is the ...
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Homeomorphism between [0,1]/~ and the Hawaiian Earring

Let $X$ be the quotient space [0,1]/~ where 0 ~ 1 ~ 1/2 ~ $\cdots$ ~ 1/n ~ $\cdots$ Let $H$ (the Hawaiian Earring) be the subspace of $\mathbb{R}^2$ consisting of the union of circles of radius 1/n ...
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Closed subspace and quotient norm

In the Banach space $C[0,1]$ consider the subspace $M=\lbrace g \in C[0,1]: \int_{0}^{1}g(t)dt=0 \rbrace $ Show that $M$ is closed in $C[0,1]$ and calculate the quotient norm $(\|f+M \|)$ where $f(t)...
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Isomorphism between quotient fields of polynomial rings [closed]

Let $R$ be an integral domain and $F$ be its field of fractions. If $X$ is a nonempty set, prove that there is a ring monomorphism from $R[X]$ to $F[X]$ that extends to an isomorphism of their ...
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Is there a general way to calculate the fundamental group of a quotient space?

Suppose $X$ is a path-connected topological space, and $A$ is a path-connected subset of $X$. My question is, is there a way to calculate the fundamental group of the quotient space $X / A$ in terms ...
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Boundary of polygonal presentation homeomorphic to bouqet of circles

Suppose $P$ is a regular 2n-gon, with sides in pairs to give a surface. I want to show that the image under quotient topology of boundary of this polygon is homeomorphic to wedge of n circles. I want ...
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What does modular space $\mathbb{H}/ \mathrm{SL}_2(\mathbb{Z})$ mean?

Juts a quick question. In Freitag's Complex Analysis as an example for The Quotient Topology it comes: The "modular space" $\mathbb{H}/\mathrm{SL}_2(\mathbb{Z}).$ Every element in $\mathbb{H}$ can ...
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Number of elements in the ideal of Ring.

We have $$x^9+1 = (x+1)(x^2+x+1)(x^6+x^3+1)$$ is factorization of irreducible polynomials over $GF(2)$ (Galois field). Then we know that one of its ideal for the ring is $$R = GF(2)[x]/(x^9+1)$$ One ...
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Continuous map from function space to quotient space maps through projection?

Suppose $F$ is a function space $Y^X$ with $Y$ discrete (so it has the topology of pointwise convergence), and $F'$ is another function space $Y'^{X'}$ with $Y'$ discrete, and suppose we have an ...
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Set of functions defined up to a sign

Let $C^{\infty}(I)$ denote the vector space of smooth functions from an interval $I$ to $\mathbb{R}$. Let $\sim$ be the following equivalence relation on $C^{\infty}(I)$: $$ f \sim g \Leftrightarrow f ...
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Why is restriction $\varphi_R$ a homeomorphism in showing cross is not locally Euclidean in Tu Manifolds?

Tu Manifolds Section 5.1 Definition of locally Euclidean of dimension n. Example To show $\varphi_R: U \setminus p \to B \setminus 0$ is also a homeomorphism, I think: $\varphi_R = r \circ \varphi ...
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General concept of 'quotient'

Working with group theory I've found multiple times the idea of quotient group as $G/H = \{gH\ |\ g\in G, H < G\}$. Nevertheless, you can find similar things in vectorial spaces as $\mathbb{R}^2/L =...
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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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1answer
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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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1answer
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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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1answer
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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 $R^n/span(\{x_i\})$?