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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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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20 views

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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79 views

Quotient of direct sum of $l$ copies of a PID by an ideal.

Let $R$ be a PID, and $\psi:R^k\to R^l$ an homomorphism. I would like to know under what circumstances it's true that: $$ R^l/\operatorname{Im}\psi \cong R/I_1\times \cdots \times R/I_l, $$ for some ...
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26 views

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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47 views

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 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 uniformly ...
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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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25 views

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 ...
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If $X$ and $Y$ are g-equivariant homeomorphic then $X/G$ and $Y/G$ are homeomorphic

Let $X$ and $Y$ be $G$-sets (That is the group $G$ acts on $X$ and $Y$). We say that the function $f: X\to Y$ is G-equivariant if $f(g.x) = g.f(x)$ for all $x\in X$ and all $g\in G$. Prove that if ...
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If $R=K[X]/(X^n)$, can represent any element as polynomial with degree $<n$

Let $K$ be a field and $R=K[X]/(X^n)$ where $n \in \mathbb{Z}_{n\geq1}$ and $(X^n)$ is the ideal generated by $X^n$. We denote $x:=X+(X^n) \in R$, any equivalence class $r$ in $R$ has a representing ...
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how to define the map between the isomophism $(⊕Mα)/I(⊕Mα)\simeq ⊕(Mα/IMα)$?

May I ask how to define the map between the isomorphism $(⊕M_α)/I(⊕M_α)\simeq ⊕(M_α/IM_α)$?. where ${M_α}$ is a collection of $R$-modules, and $I$ is an ideal of $R$. we know $(⊕M_α)/I(⊕M_α)\simeq ...
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A subset $\tilde{V}$ in $X/\mathcal{R}$ endowed with the quotient topology is closed if and only if $p^{−1} (\tilde{V} )$ is closed in $X$.

For a topological space $X$ and a equivalence relation $\mathcal{R}$ on $X$, and $p:X\to X/\mathcal{R},\ x\mapsto[x]$, my notes has the proposition: A subset $\tilde{V}$ in $X/\mathcal{R}$ endowed ...
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Question about repeated quotient groups

I was wondering if there was a simpler representation for the quotient group $(K[x,y]/\langle xy\rangle)/\langle x, y-1\rangle$. Where $K[x,y]$ is the polynomial ring with variables $x,y$ over the ...
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Eigenvalues of an operator induced in a quotient space

Give an example of a vector space $V$, an operator $T \in \mathcal L(V)$ and a $T$-$\space$invariant subspace $U$ of $V$ such that $T/U$ has an eigenvalue that is not an eigenvalue of $T$. Attempt: I ...
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62 views

Vector Spaces: difference between Equivalence Classes and Quotient Spaces

I'm reading Herstein's Topics in Algebra and Halmos's Finite Dimensional Vector Spaces. I think I understand that if $V_F$ is a vector space over $F$ and $W$ a subspace of $V$, then there is an ...
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57 views

Tangent space and differential forms of quotient groups

I have difficulty understanding the following argument because it seems that many details are swept under the rug and I am looking for a detailed rigorous exposition on this (I'll try to make clear ...
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75 views

A statement in tom Dieck's Algebraic Topology

In the cone construction (section 9.3.1) of tom Dieck's book Algebraic Topology, the author defines the following map $$q:\Delta ^{n-1}\times I\rightarrow \Delta ^n,\;\;\;((\mu_0,\dots ...
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An action of $\mathbb{Z}_2$ on $\mathbb{R}^2$

Let $M$ be the manifold $\mathbb{R}^2$ and $G$ be $\mathbb{Z}_2 = \{1,-1\}$, the cyclic group of order 2. Let $G$ act on $M$ by $(x,y) \mapsto (-x,-y)$. I need to show that the orbit space $M\vert G$ ...
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If the fibers of a quotient map are all discrete, is this map a covering map?

If $p:\tilde{X}\rightarrow X$ is a covering projection then I know that for every point $x \in X$ the fibre above $x$, i.e $p^{-1}(x)$, has the discrete topology. Here $p$ being a covering map means ...
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44 views

Strange Quotient space $X / \mathbb{Z}$

For a practice-exam exercise I am trying to understand why $X/ \mathbb{Z}$ is homeomorphic to $S^1$. Here, $X = (-1,\infty)$, and $\mathbb{Z}$ is acting as an additive group on $X$ with the action: ...
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Matrix representation induced by quotient space

someone can help me with this question, I know how to solve ker(A) but I don't know how to develop matrix representation. Thanks!!!!!
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If $p:A\to B$ and $q:C\to D$ are quotient maps, $B$ and $C$ locally compact, separable spaces, is $p\times q$ a quotient map?

It is a true or false question from an old test. At first I tried some counterexamples, using the line with two origins or taking $B$ as a quotient space of the real line by some not-open subset, ...
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The quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open.

I'd like to show that the quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open, where I'm considering $\mathbb P^n$ as the quotient space of $\mathbb R^{n+1} \setminus \{0\}$ ...
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54 views

Proving that S/I is a vector space

I'm given a polynomial ring $S=K[x_1,...,x_n]$ and $I$ is an ideal of $S$. I'm working on proving that the quotient ring $S/I$ is a vector spake over $K$. Since S is a ring, we already have some of ...
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Show that $F[x] / (x^2-1) \cong F \times F$ iff $char(F) \neq 2$

Let $F$ be a field. Show that $F[x] / (x^2-1) \cong F \times F$ iff $char(F) \neq 2$. Here is my work this far: If $char(F) = 2$, then $x^2-1 = (x-1)^2$, and hence $F[x]/(x^2-1)$ has a non-zero ...
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Projection of glueing identification is open map?

Let $(X_\alpha)_{\alpha \in A}$ be a family of topological spaces and let there be given for every $\alpha,\beta \in A$ open subsets $X_{\alpha\beta} \subseteq X_\alpha$ and homeomorphism $$ ...
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Error in claiming $N_0$ = $G_0$ when $N/N_0 \leq G/G_0$?

Let $N \leq G, N_0 \unlhd N, G_0 \unlhd G$ such that $N/N_0 \leq G/G_0$. Since the identity elements in $N/N_0$ and $G/G_0$ are same, $N_0$ = $G_0$. What is the error in this claim? It would be ...
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Which quotient spaces is Hausdorff in Klaus's book?

In Chapter3.3 "Properties of Quotient Spaces" of the book "TOPOLOGY" written by Klaus Janich and translated by Silvio Levy. I can't figure out which one is Hausdorff. Any one can explain it? Update: ...
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65 views

Obtaining the Möbius strip as a quotient of $S^1\times[-1,1]$

I am trying to obtain the Möbius strip as a quotient of $S^1\times[-1,1]$, where $S^1$ is, of course, the circle. My definition of Möbius strip is the quotient of the square $[0,1]\times[0,1]$ by the ...
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Some quotient of $[0,1]$ is homeomorphic to $[0,1] \times [0,1]$

I have been unable to solve this problem. I imagine you would start with some bijection $$f:[0,1] \rightarrow [0,1] \times [0,1]$$ But I don't get how any topological argument could be made here with ...
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42 views

Showing homeomorphism between interval's quotient spaces

We have two spaces: $[0,1]/C$ and $[0,1]/A$, where $C$ denotes the Cantor set and $A=\{0,1,\frac{1}{2},\frac{1}{3},...\}$. One needs to show they are homeomorphic. What I thought about is showing ...
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Weird isomorphisms of infinite groups

According to my interpretation to one of the answers in Splitting in Short exact sequence, $$\Bbb R \cong \Bbb Q \oplus \Bbb R / \Bbb Q$$ also, according to What is known about the quotient group ...
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How do I show that $S^1$ is the suspension of $S^0$?

How do I show that $S^1$ is the suspension of $S^0$? I have all the definitions here, I'm just bad at applying them. The suspension of a topological space $X$ is the quotient $CX / (X × ${$1$}$)$, ...
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Is the suspension space contractible?

Let $X$ be a topological space. The suspension of $X$, denoted $ΣX$, is the quotient $CX / (X × ${$1$}$)$, where $CX$ is the cone on $X$, the quotient space $(X × [0, 1]) / (X × ${$0$}$)$. Is $ΣX$ ...
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136 views

How to prove the Cone is contractible?

Let $\sim$ be an equivalence relation on a topological space $X$ with a subset $A ⊂ X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then define $X/A$ ...
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131 views

What space is this Homeomorphic to?

Let $X$ be a topological space and let $A ⊂ X$. Let $\sim$ be an equivalence relation on $X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then ...
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30 views

$\mathbb R$ ~ $[0,1) \cap (\mathbb R\ $\ $\mathbb Q)$

$\mathbb R$ is the set of the real numbers. $\mathbb Q$ the set of the rational numbers. So how I can prove this? $\mathbb R$ ~ $[0,1) \cap (\mathbb R\ $\ $\mathbb Q)$ I am also not sure what means ...
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Quotient morphisms in topological subconstructs

Let $(\mathscr{A},U)$ be a topological construct (that is, a concrete category $U\colon\mathscr{A}\to Set$ where every structured sink has a unique final lift) and let $\mathscr{B}\subset\mathscr{A}$ ...