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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Understanding Quotienting by Relations vs Quotienting by Generators

I understand the idea of a quotient algebra $A / I$ where $A$ is a $K$-algebra and $I$ is a two-sided ideal, i.e. I understand the projection map as an algebra morphism. However, I'm unsure about how ...
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Something intuitive in common with different quotients?

Quotient groups in group theory, quotient ideals in algebraic geometry, quetient in calculus equals the result after division -- do they have something in common and what is an intuitive motivation, ...
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'Multiplying' by a subgroup

Is the following true for abelian groups $G \leq N$ and $H$: $$ G/N \cong H \implies G \cong H \oplus N $$ A reference would be helpful. Thanks!
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Show that $Im T$ and $U/Nuc T$ are isomorphic for a linear transformation $T: U \longrightarrow V$

Show that $Im$ $T$ and $U/Nuc$ $T$ are isomorphic for a linear transformation $T: U \longrightarrow V$ Hi guys, I know how to show this for vectorial spaces with finite dimension, but I don't have ...
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Direct Product and Quotient of Groups

Quick (and basic) group theory question: Say G, H, K some (Lie) groups, does it in general hold that $$ (G \times H)/H = G $$ and that $$ H = K\times G \to K = H / G $$ And if so, does it then ...
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Is $U(1)$ a normal subgroup of $U(2)$ and does the question even make sense?

I have been wondering whether $U(1)$, defined as the group of complex phases (edit for clarity: I mean complex numbers of unit absolute value, such as $e^{i\alpha}$ with $\alpha \in \mathbb{R}$) with ...
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Does $[V(\lambda)/W(\lambda)] = [V/W](\lambda)$?

Let $T$ be a linear operator on a vector space $V$, and let $W$ be an invariant subspace. If $V(\lambda)$ denotes the $\lambda$-eigenspace of $V$ and $W(\lambda)$ the eigenspace of $T$ on $W$, then ...
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Coset Space as a Representation of a Lie Algebra

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some ...
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30 views

Does the forgetful functor from smooth manifolds to sets preserve colimits?

It is easily shown that the forgetful functor $F: \mathbf{Man} \to \mathbf{Set}$ preserves limits ($F$ is representable), but does it preserve colimits? It certainly preserves all examples of ...
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Quotient space of the reals by the rationals

Let $\mathbb{R}/{\sim}$ be the quotient space given by the equivalence relation $a \sim b$ if $a$ and $b$ are rational. I am trying to understand general properties of the quotient topology and this ...
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Trying to understand quotient ring.

From my understanding of the definition: $(G/U,+,\cdot)$ is quotient ring, where the set $G/U=\{a + U \mid a \in G\}\stackrel{?}{=}\{a+b \mid a \in G, b \in U\}$. For example: ...
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Finding a map from $X = (0,\infty) \times (0,\infty)$ to a cone

Determine the quotientspace $X / \Gamma$, where $\Gamma = <\phi>$, $\phi(x,y) = (x/2,2y)$ and $X = (0,\infty) \times (0,\infty)$. I think the quotient space has to be a cone, but I can't figure ...
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$[0,1]$ quotient space of $]0,1[$, but not the other way around.

I want to show that $[0,1]$ is a quotient space of $]0,1[$, but not the other way around. I found so far that $]0,1[$ is no quotient space of $[0,1]$. Assume it was, then there would be a surjective ...
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Compute the (multiplicative) inverse of $4x+3$ in the field $\frac {\Bbb F_{11}[x]}{\langle x^2+1 \rangle}$?

So I am finding a polynomial $px+q$ ($p,q \in \Bbb F_{11}$) which is multiplicative inverse of $4x+3$ in $\frac {\Bbb F_{11}[x]}{\langle x^2+1 \rangle}$. i.e. $[(4x+3)+\langle x^2+1 ...
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Identification space of square. Net, triangulation and surface classification

Space Z is made as an identification space of unit square $Q=${$(x,y) | 0\leq x, y \leq 1$} by making the following identifications: $ (0,y)$~$(1,y) $ for all $0\leq y\leq 1 $, $ ...
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Elements of tensor product

Let $R, S$ be rings such that $R$ is a subring of $S$ and $1_R = 1_S$. Let $N$ be a left $R$-module. Let the free $\mathbb{Z}$-module on $S \times N$ be $F_\mathbb{Z}(S \times N)$. Let $H$ be the ...
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Group works on topological space

I have to prove that if discrete and finite group G works on topological Hausdorff space X $ \varphi :G \times X \rightarrow X $ and $ \varphi $ is cotinuous function, then $ X / G $ is also a ...
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What is the relationship between a quotient space and annihalator?

If we have a vector space $V$ and subspace $W$, we have that $$\dim(V/W) = \dim V - \dim W.$$ Similarly for the annihilator $W^{\circ}$ we have that $$\dim W^{\circ} = \dim V - \dim W.$$ What is ...
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Topology given by a relation

I have a problem with creating an equivalence relation ~ in a set : $ S^{1}\times S^{2}$ so that $ (S^{1}\times S^{2})/$~ (a quotient space of the given relation) is homeomorphic to 3-dimensional ...
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Basic understanding of quotients of “things”?

My modern algebra needs some work. Am I right in thinking that $\mathbb{Z}/2\mathbb{Z}$ refers to the two sets $$\{\pm0, \pm2, \pm4, \pm6, \ldots\}$$ and $$\{\pm1, \pm3, \pm5, \pm7\}~~?$$ What about ...
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Defining the pre-image topology for $f:A\rightarrow B$

Let $f:A\rightarrow B$ be a function from a set $A$ to a topology $(B,\tau)$. Then I can define a subbase on $A$ simply by giving $f[A]\subset B$ the subspace topology inherited from $B$ ...
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Proving that $\operatorname{Im} f \cong V/\ker f$ for a linear map f.

In this proof, how do you show injectivity? In my notes, it says: For $f\colon V \to W$ linear operator and $V,W$ vector spaces let $\bar{f}\colon V/\ker f \to \operatorname{Im}f$ be defined by ...
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Does $\pi(\partial M)\subset \partial(M/\sim)$?

Let $M$ a set and $\sim$ an equivalence relation. Let $\pi: M\longrightarrow M/_\sim$ the projection. Do we have that $\pi(\partial M)\supset \partial (M/_\sim)$ ? (where $\partial A$ denote the ...
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Doubts about adjunction of elements to a ring

I've just solved the following exercise Suppose we adjoin an element $\alpha$ to $\mathbb{R}$ satisfying the relation $\alpha^2 = 1$. Prove that the resulting ring is isomorphic to the product ...
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Determining the structure of the quotient ring $\mathbb{Z}[x]/(x^2+3,p)$

I'm interested in the following problem from Artin's Algebra text: Determine the structure of the ring $\mathbb Z[x]/(x^2 + 3,p)$, where (a) p = 3, (b) p = 5. I know that by the isomorphism ...
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What is the quotient space $\mathbb{C}[x,y,z]/(x^2+y^2)$?

What is the quotient space $\mathbb{C}[x,y,z]/(x^2+y^2)$ and more generally how do I determine such spaces (if possible include some reference). Such quotients appear a lot in algebraic geometry and I ...
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Group actions by semi-direct products of groups

I have trouble to understand the second part of the following example which I hope someone can explain to me. First let me explain the initial situation which I feel comfortable with: Consider the ...
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Let $X$ be a Banach space, and $E$ a closed subspace, then $X/E$ is a Banach space [duplicate]

I am trying to prove that the following is true: Let $X$ be a Banach space, and $E$ a closed subspace, then $X/E$ is a Banach space So the quotient norm is $\|[y]\|_{X/E} = ...
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Using the Fundamental Isomorphism Theorem for groups to prove an isomorphism

Let $G$ be the group of all real matrices of the form $ \begin{bmatrix}a&b\\0&c\end{bmatrix} $ with $ac \ne 0$, under matrix multiplication. Let $H$ be the subgroup consisting of all the ...
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Fundamental Group of a Torus glued to a Hausdorff space

Question: Via analogy to the following Munkres Theorem: I am asked to explain the relation between the fundamental groups of a Hausdorff space $Y$ and a space $X$, which is obtained from gluing ...
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Is Quotient Spaces of Hausdorff also Hausdorff where the Quotient Spaces is T1?

Let X be Hausdorff space and X/~ be "~ Quotient Spaces", Given that X/~ is T1, Does it satisfy for X/~ to be Hausdorff?
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Normed Quotient Space

If $X$ is a normed vector space and $M$ is a proper closed subspace, I want to show that for any $\epsilon>0$ there exists an $x\in X$ such that $\|x\|=1$ and $\|x+M\|\geq 1-\epsilon$. Is there ...
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If $X/F$ is separable, must $X/F$ isomorphic to $F$?

Suppose $X$ is a Banach space and $F$ is a closed subspace of $X$. Clearly $X/F$ is a Banach space, equipped with quotient norm. Question: If $X/F$ is separable, must $X/F$ isomorphic to $F$? The ...
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Evaluation of a formal series at a point

Let $R$ be a ring. On the ring of polynomials $A[X]$ I can define an evaluation map in $r \in R$ (the morphism that sends $X$ to $r$, extended by linearity and multiplication). This allows to ...
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Strange endomorphism that translates in a functor one (?)

Let $C$ be a category composed by three pieces: $C_{-2}, C_0$ and $C_2$, where $C_{-2}=C_2=\mathbb{K}-$modules and $C_0 = \frac{\mathbb{K}[x]}{(x^2)}$-modules. Let $E$ be the functor that acts this ...
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Calculating quotient ring over polynomial ring without guessing

Suppose we have a quotient ring over a polynomial ring, i.e. we have an ideal $I$ and a ring, $K[X_1,...,X_n]$, then when we can we identify $K[X_1,...,X_n]/I$? What do I mean by this? Well, for ...
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Collapsing the Unit Ball in $\mathbb{R}^n$

For a homework question, I need to prove that $\mathbb{R}^n/\overline{B_1(0)}$ is homeomorphic to $\mathbb{R}^n$, where $\overline{B_1(0)}$ is the unit ball centered at the origin. Clearly, the ...
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Quotients with connected preimages of points

The Hahn-Mazurkiewicz says that every locally-connected continuum is a quotient of the unit interval. However, this quotient usually has disconnected fibers (preimages of points). For instance, the ...
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Quotient Space Notation

Quick question, mostly just for my knowledge, but I'm working on a problem: Determine whether the indicated set $A$ is an ideal in the indicated ring $R$: $$A = \{0,2,4,6,8\},~~R = ...
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Quotient ring by ideal generated by several elements in steps.

I came up with the following intuition about a question. Let $R$ be a commutative ring and let $a,b \in R$. I consider the ideal $I=(a,b)$ and I wonder if the ring $R/(a,b)R$ is isomorphic to the ring ...
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Some insights in quotient topologies

One of Munkres' exercises tasks us to consider the following quotient space $\mathbb{R}^2/\sim$ in $\mathbb{R}^2$ given by the following equivalence relationship: $$(x,y)\sim(a,b)\iff x^2 + y^2 = a^2 ...
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Quotient group of invertible matrices isomorphic to $\mathbb{R}\times$

I need to show that the quotient group $G/H$ is isomorphic to $\mathbb{R}×$; where $G = \{ A = (a b 0 d): \det A \neq 0 \}$, $H = \{B = (a b 0 a): \det B \neq 0\}$, $\mathbb{R}\times$ is the group of ...
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Identification Topology (Mendelson's 'Introduction to Topology', page 103)

Let $f: (X, \sigma) \rightarrow (Y, \tau)$ be continuous and let $a \sim b$ if $f(a) = f(b)$. Noting $\sim$ is an equivalence relation, let $\pi(x)$ map $x$ to its equivalence class. Noting $\pi: X ...
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What does it mean that in a factor/quotient group certain elements get “glued” together

In these these notes on the generalized quaternion group it is written that: [...] $Q_{2^n}$ is made by taking a cyclic group of order $2^{n-1}$ and a cyclic group of order $4$ and "glueing" them ...
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Find a polynomial $q(x)$ of minimal degree in $F_3[x]$ so that the following condition is true:

$q(x)\equiv x ~(\text{mod}(x^2+2))$ $q(x) \equiv 1 (\text{mod} ~x)$ $q(x) \equiv (x+1) (\text{mod} (x^2 + 2x + 2))$ I know that this is a question about the Chinese remainder theorem but I have no ...
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Let $f = \prod_{i=1}^n (X - z_i)^{r_i} \in \mathbb{C}[x]$. Prove that $\mathbb{C}[x]/(f) \simeq \prod_{i=1}^n \mathbb{C}[x]/(X-z_i)^{r_i}$

This exercise started asking to prove that there's a morphism between $R/(gh) \rightarrow R/(g) \times R/(h)$ and I proved that the canonical projection is in fact well defined there. Then it asked to ...
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How to verify the inverse of a polynomial in mod polynomial?

This is in $F_2$. This might sound silly but I know that the inverse of $(x^3+x)$ in mod $(x^4+x+1)$ is $(x^3 + x^2)$ but I am not sure how to verify that. It should be that when I multiply $(x^3+x)$ ...
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43 views

Is quotient space a subspace of the underlying vector space

If $W$ is a subspace of the finite dimensional vector space $V$, is the quotient space $V/W$ a subspace of $V$ ? How do I prove or disprove this statement ?
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Understanding the Quotient norm, constructed example

I just read the definition of the quotient norm, but it had no examples, so I constructed my own. Is this correct? Let $(X=\Bbb R^3,\|\cdot\|)$ with Euclidean norm and $Y\subset X$, $\quad ...
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define orientation of quotient space

How can one expand the definition of orientation of vector spaces to corresponding quotient space? For example, if I am given two vector spaces V and W, and I know the definition of orientation for ...