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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Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
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Matrices of Quotient Representation

I've been working through some exercises in Sagan's book on the representation theory of the symmetric group and I came across the following exercise that's been giving me some trouble: Let $X$ be a ...
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Isometry between $X^\ast/M^\perp$ and $M^\ast$.

Let $X$ be a normed linear space, $M \subset X$ be a subspace, $M^\perp = \{x^\ast \in X^\ast \mid x^\ast\big|_M = 0\}$ be the annihilator of $M$, $X^\ast$ the topological dual of $X$, and let's ...
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Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed

Problem is: Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed. ($U$ is saturated $\iff$ $\exists V \in Y$ s.t. $U = \pi^{-1} ...
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$U$ takes the same value on $\pi$ then $U$ is saturated

Let $\pi : X \to Y$ be any map, and $U$ be a subset of $X$. The problem is: "$\forall x\in U$, $ \pi (x) = \pi(x') $, then $x' \in U$" then $U$ is saturated. (U is saturated $\iff$ $\exists V ...
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The topology on $X / G$ where $G$ acts on $X$

The elements are orbits, but how do we find the neighbourhoods? In particular, let $G= ( \mathbb R^+ , \cdot )$ and $X=[0, \infty )$. Let $G$ act on $X$ via the usual multiplication. Then $X/G = ...
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Is a compact subset of a topological group G/N closed if G is hausdorff?

I just used this hypothesis when I was proving a theorem.But I was not sure if this hypothesis is correct.
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Differences between a quotient map and a continuous function in topology

Def. for a continuous function: Let $X$ and $Y$ be topological spaces. A function $f : X \rightarrow Y$ is continuous if $f^{-1} (Y)$ is open in $X$ for every open set $V$ in $Y$. Def. for ...
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Dimension of a certain $L^p$ quotient space.

Define $L^p_0 := \{ f \in L^p : \int f = 0 \}$. I am trying to calculate the dimension of the cokernel of the inclusion operator $i:L^p_0 \to L^p$. That is, I am trying to calculate $$\dim ( L^p / ...
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Path components of quotient space

Let $X$ be a topological space and $A \subseteq X$ a subspace. Is it true that $X/A$ is path-connected if and only if $A$ meets every path component of $X$? Intuitively this seems reasonable but I'm ...
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Is this quotient of a disk Hausdorff?

Let $D$ be the disk in the complex plane, $D = \{ z : |z|<1\}$. Let $\sim$ be an equivalence relation on $D$ defined by $z_1\sim z_2$ iff $|z_1| = |z_2|$. Is the quotient space $D/{\sim}$ is ...
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Is a canonical projection always continuous in the quotient topological space

I'm solving some exercises on quotient topology, and I wonder if a canonical surjection is always continuous. Let $\pi: (X,\tau_X)\to (X/_\sim,\tau_\sim)$ be a canonical surjection. I think ...
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Difficulty understanding why a set is saturated with respect to a map

The following is the beginning of Example 7 on page 143 of Munkres' Topology. Example 7. The product of two quotient maps need not be a quotient map. Let $X = \mathbb{R}$ and let $X^*$ be the ...
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Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff

$\newcommand{\R}{\mathbf R}$ Let $V$ be an $n$-dimensional vector space and $k$ be an integer less than $n$. A $k$-frame in $V$ is an injective linear map $T:\R^k\to V$. Let the set of all the ...
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Quotient group and adjoint matrix

The exercise 1211 in "Problems and Solutions in Mathematics" by Ta-Tsien: Let $M$ be an $n \times n$ matrix of integers. Suppose that $M$ is invertible when viewed as a matrix of rational numbers. ...
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A Continuous Choice of $k$-Subspaces of a Vector Space Gives a Continuous Choice of Bases

$\newcommand{\R}{\mathbf R}$ The Grassmannian $G_k(\R^n)$ as a topoplogical space is defined in the following way: Let $F_k(\R^n)$ be the collection of all the linearly independent lists of size $k$ ...
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Quotients vs equivalence relations

In a recent preprint I defined the category of quasi-frames (qframes for short) as follows: a qframe is a modular and upper continuous complete lattice; a morphism of qframes $f:L_1\to L_2$ is a map ...
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48 views

question about direct sum of vector fields and preservation under quotient spaces

Hello all I was given this question in linear algebra it is two parts and asks to prove or give a counterexample. We are given a vector space V and a subspace of it W and the quotient map $ \pi : V ...
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Unifying Connection Between Topological Embeddings and Quotient Maps

In a book on topology I'm reading the following theorem seemed striking to me, not for its proof, which I believe I understand, but because there's some nice symmetry going on that I'd perhaps like ...
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Quotient rings, polynomials are reducibility

I am trying to follow this solution. I am struggling to understand why 'If g is a member of R, then g divides the content of f'. Why is this true?
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Residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$

What are the residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$? After finding the maximal ideals of $\mathbb{R}[X,Y]$, which are of the form: $\langle X-a,Y-b \rangle$ with $a,b ...
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Clarification about Quotient Spaces

The question given to me is: Does there exist a vector space $V$ which has a nonzero subspace $U$ such that $V /U \cong V$ ? Provide an example or a proof that no such $V /U$ exists. Intuitively, I ...
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Are $S^1$ and $\mathbb{R}/{\sim}$ the “same thing”?

I am reading quotient space of topology and I am a little bit confused. I am looking at the relationship between $\mathbb{R},S^1$ and the quotient space $\mathbb{R}/{\sim}$, where the relation $\sim$ ...
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Compute Quotient Space

I have been struggling with this computation for a while now. I thought I was almost there, but it now results I still have nothing. So here is the initial problem: Let $c=\left\{ (x_j)_j \subset ...
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Quotient space of infinite dimensional vector space

On an exam today I used that if $X=\mathcal{C}[a,b]$ and $Y=\{f\in X : f(a)=f(b)\}$, then the projection $\pi: X\rightarrow X/Y$ has the property $\ker(\pi)=Y$. This led me to the following: Suppose ...
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Globally defined exponential in a particular homogeneous space

I'm currently working in a particular conformal compactification/completion of the Minkowski space-time, but I'm stuck at showing that the exponential of every vector field in it is globally defined. ...
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How do I show that this function is continuous? [duplicate]

I'm interested in showing that $CX=\frac{I\times X}{\{1\}\times X}$ is contractible. I defined the d.r $F(s,[t,x])=[(1-s)t+s,x]$ and the only missing part for me is to show that it is continuous. How ...
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About direct sum of abelian groups and quotient

I'm trying to understand properly the relations between quotient and direct sum. The first thing I wanted to know, and couldn't find online, is whether my guess is true or not: Assume $G_\alpha$ are ...
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Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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“Quotient” as a verb

People here do use "quotient" as a verb: I searched for "quotienting" and got 12,890 results. [Edit: It's not as bad as I thought. Apparently I didn't understand how the search function works. When I ...
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Projection Mappings are Quotient Mappings?

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X=X_1\times X_2$ is a product space, then the first coordinate projection is a ...
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Induced map between quotient vector spaces well defined and linear?

Suppose we have vector spaces $V,W$ and subspaces $$ V\supset V'\supset V''\\ W\supset W'\supset W'' $$ suppose also that we have a linear map $A:V\to W$. What does it take for this map to induce a ...
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Fundamental group of the mapping cone of a loop

You can read in Wikipedia the following: Given a space X and a loop $\alpha\colon S^1 \to X$ representing an element of the fundamental group of $X$, we can form the mapping cone $C_α$. The effect of ...
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Finding the Fundamental Groups of Some Modular Spaces

I'm looking to compute the fundamental group of a couple of different quotients of the $n$-torus. The first of these I'm interested is the space $\mathbb{T}^n/S_n$ where the symmetric group $S_n$ ...
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Showing that a function is continuous

Let $A$ be a topological space, and let $B$ be a quotient space of it. I have defined a continuous function $F:A\times[0,1]\to B$ such that it factors into a function $G:B\times[0,1]\to B$, i.e. that ...
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Injection of the mapping cone of $z^2$

We define the mapping cone of $f:S^1\to S^1=:Y$, $f (z)=z^2$ as the quotient space of $S^1\times [0,1]\sqcup Y$ where $(z,0)$ and $(z',0)$ are identified and where $(z,1)$ and $f(z)$ are identified ...
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Showing that the center of a Möbius strip is homeomorphic to a circle?

Consider the Möbius strip as the quotient space obtained from $[0,1]\times[0,1]$ when identifying $(0,t)$ with $(1,1-t)$ for $0\leq t\leq1$. Now consider its center, that is, the image of the set ...
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Let $A$ be the annulus in $\mathbb{C}$, what is the space $A/{\sim}$ generated by identifying all points on the “inner circle” with each other?

The annulus is $A=\{z\in\mathbb{C}:1\leq |z| \leq 2\}$ and the "inner circle" here is the set of points $\{z\in\mathbb{C}:|z|=1\}$. If we identify all points of the inner circle together, then, ...
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How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Taken from Hatcher, I think the free product $G =\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ should have infinite elements taking the form of words containing alternate elements, i.e. $a, b, ab, ...
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Quotient topology by identifying the boundary of a circle as one point

The following is an example taken from Munkres topology book: I don't understand why does $X^{*}$is homeomorphic to $S^{2}$, is this a basic fact that I don't understand or is it an example of ...
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Find a quotient map $f:(0,1) \rightarrow [0,1]$ where the intervals $(0,1)$ and $[0,1]$ are in $\mathbb{R}$ and endowed with the subspace topology.

Find a quotient map $f:(0,1) \rightarrow [0,1]$ where the intervals $(0,1)$ and $[0,1]$ are in $\mathbb{R}$ and endowed with the subspace topology. I am really not to sure where to start. I know ...
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Quotient norm question

http://mathoverflow.net/questions/99860/upper-semicontinuity-in-cx-algebras In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x ...
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Equivalence relation, product and quotient spaces

I have a problem with the following: "Define a relation $\sim$ on $R^2$ by $(u,v) \sim (x,y)$ if and only if both $u-x$ and $v-y$ are integers. Show that for each point $(x,y) \in R^2$ there exists ...
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On units in subrings ( or ideal ) and quotient ring of ring with unity

Let $R$ be a finite ring with unity and $S$ be an ideal ( or subring ) , let $R^*$ be the group of units of $R$ and $S^*:=R^* \cap S$ , then does $|S^*|$ divide $R^*$ ? Moreover , if $S$ is an ideal ...
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Quotient space isometrically isomorphic to $c$

Can some one help me fill the details in the next proof? It will be much better if someone knows a simpler way to do it. The problem states Take $C[0,1]$, with the usual norm $\|\cdot\|_\infty$, ...
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Bijection between a Hilbert and a Banach space

I am working on a question where I have to show that for a Hilbert space $\mathscr{H}$, and closed linear subspace $Y \subset \mathscr{H}$, $\mathscr{H}/Y$ is a Hilbert space, isomorphic to ...
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Why is $D^n/\sim$ homeomorphic to $\mathbb{RP}^n$?

Let $\sim$ be the equivalence relation on $D^n$ (the $n$-dimensional unit disc) which identifies antipodal points on the boundary $\partial D^n = S^{n-1}$. Show that $D^n/\sim$ is homeomorphic to ...
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In the category of uniform spaces, is the completion of a quotient map also a quotient map?

Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous map $f: X \to Y$ is a quotient map if for every map $g$ from $Y$ to a uniform space $Z$ such that $g \circ f$ is ...
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Dimension of quotient normed linear space

Suppose $M$ is a normed linear space. $L$ and $N$ are two closed subspaces of $M$ such that $L \subseteq N$. Then $L$ is a closed subspace of $N$. Let $\text{dim}(M/N)=r$ and $\text{dim}(N/L)=s$. My ...
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CRC calculate quotient

As above, i have absolutely no idea how to calculate the quotient (10110110). Some mentioned that there is no need for it, but my exams required me to understand how to get the quotient. Please help ...