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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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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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prove there exists bijective map

I am learning proof of bijective. I have a question... $V$ is a vector space and $W$ is a subspace of $V$. Also, $V/W$ is a quotient space. Operations $\oplus : V/W×V/W→V/W \otimes$ : F×V/W→V/W are ...
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1answer
25 views

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
35 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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1answer
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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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1answer
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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
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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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1answer
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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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3answers
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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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1answer
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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 ...
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Notation for quotient space obtained by collapsing a subset to a point?

If $X$ is a topological space and $A$ is a (closed, usually) subset of $X$, then the quotient space obtained by "collapsing $A$ to a point" is often denoted by $X / A$. Unfortunately, that notation ...
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The Metric in $\mathbb{R}/\mathbb{Z}$

When I read a math book, the book says $\mathbb{R}/\mathbb{Z}$ is the set of coset $\mathbb{Z}$ in $\mathbb{R}$ with quotient topology induced by the usual topology on $\mathbb{R}$. The ...
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2answers
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Adjunction space is homeomorphic to quotient: an easier proof?

All spaces are in $Top.$ It is trivially obvious geometrically that if $A\subseteq X$, and $f:A\to *$, where $*$ is a singleton, then $X/A\cong X\cup_f *.$ I have never seen a proof of this, however, ...
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How to define a Homeomorphism between these spaces?

Question Let $X$ be a set and let $(Y, \tau)$ be a topological space. Let $g:X \to Y$ be a given map. Define $$\tau'=\{U \subset X~|~ U=g^{-1}(V) ~\text{for some} ~V \in \tau\}.$$ Which of the ...
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$\mathbf{H}(3)$ is diffeomorphic to $\mathbf{SL}\left( 2,\mathbf{C}\right) \mathbf{/SU}\left( 2\right) $

I'm reading the book from Jensen's "Surfaces in Classical Geometries". Could anyone help me understand why $\mathbf{H}(3)$ is diffeomorphic to $\mathbf{SL}\left( 2,\mathbf{C}\right) \mathbf{% /SU}\...
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1answer
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$\|x+\mathcal{M}\|=0 \iff x\in \mathcal{M}$.

I recently asked this question. Existence of x∈X such that ∥x∥=1 and ∥x+M∥=1 for a closed subspace M And people said that when $\mathcal{X}$ is normed vector space, even if $\mathcal{M}$ is closed ...
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1answer
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Existence of $x\in \mathcal{X}$ such that $\|x\|=1$ and $\|x+\mathcal{M}\|=1$ for a closed subspace $\mathcal{M}$

I was proving a theorem stated below. Theorem. Suppose that $(\mathcal{X},\|\cdot\|)$ is a normed vector space and $\mathcal{M}\leq \mathcal{X}$ is a closed proper subspace. Then for any $\epsilon&...
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1answer
46 views

For a matrix A, find a subspace of $R^3$ such that the function represented by A satisfies given properties.

This is a continuation of this question: Find real number $a$ such that matrix $A$ is NOT diagonalisable For the matrix $A =$ \begin{bmatrix}2&5&-1\\0&2&1\\-1&8&-1\end{bmatrix}...
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Adjunction Spaces: Why does $A$ need to be a closed subset of $X$?

Let $X$ and $Y$ be topological spaces, and $f:A\rightarrow Y$ a continuous map from a subset $A$ of $X$ to $Y$. We can form the adjunction space $X \cup_f Y$ by appropriately quotienting the ...
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1answer
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Homogeneous space and quotient space for spin groups

We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space. These are in some sense spheres. If we embed the spin group $Spin(n)$ into $Spin(n+1)$, ...
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A question regarding the property of a quotient map.

Let $X$ and $Y$ be two topological spaces. Suppose $f:X \longrightarrow Y$ be a quotient map. Consider the equivalence class of $X$ formed by the disjoint non-empty fibres of $f$. Let $X/{{\sim}_f}$ ...
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Homogeneous space and quotient space for real projective spaces

We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space. Also $$ PO(n+1)/PO(n) \simeq P^n, $$ $P^n$ is the projective space. These are in some ...
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Quotient of $[-1,1]$ homeomorphic to unit circle

Let $I:=[-1,1]$ and $Q$ the quotient space where we identified $-1$ and $1$. Show hat $Q$ is homeomorphic to the unit circle $S^1\subseteq \mathbb{R}^2$. My first idea was to define $g\colon I\to S^...
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1answer
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Homogeneous space and quotient space for complex projective spaces

We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space. Also $$ PO(n+1)/PO(n) \simeq P^n, $$ $P^n$ is the projective space. These are in some ...
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1answer
60 views

Homogeneous space and nice manifolds

We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space. Also $$ PO(n+1)/PO(n) \simeq P^n, $$ $P^n$ is the projective space. Do we have similar ...
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1answer
56 views

Show that $\mathbb{R}^2/∼$ is homeomorphic to $\mathbb{R}$

Question: Let $∼$ be the equivalence relation on $\mathbb{R}^2$ given by ($x_1,x_2)∼ (y_1,y_2)$ if and only if $x_2 = y_2$. Show that the identification space $\mathbb{R}^2/∼$ is homeomorphic to $\...
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1answer
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If $A= \mathbb{Z}[X]/(X^n+1)$, is it true that $A/mA \cong \mathbb{Z}_m[X]/(X^n+1)$?

Let $R= \mathbb{Z}[X]/(X^n+1)$ for some sufficiently large $n$. For $q \geq 2$, I want to show that $R/qR \cong \mathbb{Z}_q[X]/(X^n+1)$. I've tried to prove it, but I dont know the construction of $...
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Isomorphism theorem like theorem for cosets

I would like to a get a result similar to the third isomorphism theorem, but I want to deal with sets of cosets, rather than a quotient subgroup. Let me formalize it. Say $G$ is a group, with ...
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2answers
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How do we define functions on quotient sets?

Let $X$ be a set and $\sim$ an equivalence relation on $X$. I know how to define the quotient set $$ X/{\sim} := \{ [x] \in \mathcal{P}(X) \mid x \in X \}, $$ but I'm a bit confused about how ...
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1answer
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Identification Space in the plane

Let $X=\mathbb R^2$. Describe (visualize) the space $X/ $~ if ~ is the smallest equivalence relation satisfying the following conditions. a) $(x,y)$ ~$(x',y')$ if and only if $x = x' - 1$. b) $(x,y)$...
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1answer
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Composition of an Affine Map and the Quotient Map is Injective

In page 2 of the book Differential geometry: Cartan's generalization of Klein's Erlangen program by Sharpe, it stated the example of the projective plane $P(\mathbb{R}^{n+1})$. The book said if $a:\...
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1answer
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Real projective space is Hausdorff: is this proof correct?

Let $\mathbb{RP}^n=\frac{\mathbb{R}^{n+1}\setminus\{0\}}{\sim}$ be the real projective space of dimension $n$, where $\sim$ is the proportionality relation. Then $\mathbb{RP}^n$ has the quotient ...
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1answer
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Equivalence relation closed in product, but not closed?

Exercise 2.4.C in Engelking's Topology book asks for an example of an equivalence relation $E$ on a space $X$ such that: $E$ is a closed subset of $X^2$, but $E$ is not a closed equivalence relation....
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Why do we not have any subgroup of ${\mathbb Z}/{4 \mathbb Z}$ which maps isomorphically to the quotient group?

I have recently started watching the video lectures of the Harvard University Extension School regarding abstract algebra taught by Benedict Gross. I am now in lecture 9 where various results of ...
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183 views

Are $\mathbb{R}[X] / (X^2 +1)$ and $\mathbb{C}$ homeomorphic?

The fields $\mathbb{R}[x] / (x^2 +1)$ and $\mathbb{C}$ are isomorphic as fields, but I am trying to see if they are homeomorphic as well. $\mathbb{C}$ is given its standard topology, and we can ...
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1answer
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Prove that the Kolmogorov Quotient is $T_0$

I am trying to prove the following (more on the background below) Let $(X, \tau)$ be a space, $KQ(X) := X / \sim$ be the quotient set and $\pi : X \rightarrow X / \sim$ the projection which maps ...
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1answer
145 views

Does identifying opposite points in Euclidean space result in a smooth manifold?

Taking Euclidean space $\mathbb{R}^n$ and identifying all pairs of points $\{\mathbf{x}, -\mathbf{x}\}$ results in a topological quotient space $\mathbb{R}^n/\mathbb{Z}_2$. Is this quotient space a ...
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1answer
45 views

What is the explicit equivalence relation for an adjunction space $X\cup_f Y$

What is the explicit equivalence relation for an adjunction space $X\cup_f Y$ ? Intuitively, the construction of $X\cup_f Y$ is pretty straightforward: we begin with the data $(X,Y,B\subseteq Y,\, f:...
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2answers
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Universal property of quotient group to get epimorphism

I know there are already answers to the question of the universal property of quotient groups. For example here: Universal property of quotient group. My question is now: If I have the homomorphism: $...
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1answer
21 views

Understanding geometric realisation of a simplicial set

I am trying to understand what is geometric realisation of a simplicial set is. I am not able to understand what exactly do they mean in notes I found online. Sometimes, starting something fresh ...
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1answer
35 views

Consider $\mathbb{R}^3/E$ with $E = \{(x,y,z) \in \mathbb{R}^2 \vert x^2 + y^2 + z^2 \leq 1\}$

Consider $\mathbb{R}^3/E_i$ with $E_1 = \{(x,y,z) \in \mathbb{R}^2 \vert x^2 + y^2 + z^2 \leq 1\}$ and $E_2 = \{(x,y,z) \in \mathbb{R}^2 \vert x^2 + y^2 + z^2 < 1\}$. I want to study the quotient ...
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1answer
75 views

Path connected quotient implies path connected space

The following is a problem I questioned myself yesterday: Let $X$ a topological space and define the equivalence relation $\sim$ given by: $$x\sim y \Leftrightarrow \textrm{ $x$ and $y$ can ...
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1answer
49 views

The continuous image of a First Countable Space need not be First Countable (Willard 16.B.1)

In Stephen Willard's General Topology appears the following exercise: A quotient of a second countable space need not be second countable (for each $n\in \mathbb{N}$, let $I_n$ be a copy of $[0,1]$ ...