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0
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18 views

Quotient splits in direct product

I have a quotient, say $$Pert(\mathcal{A} \oplus \mathcal{B}) / \ker(\phi) \cong Im(\phi) = Pert(\mathcal{A}) \times Pert(\mathcal{B}),$$ and I know that $$Pert(\mathcal{A} \oplus \mathcal{B}) \cong ...
2
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1answer
28 views

the homeomorphisms betwen two spaces looks like broom

Let $Y=${$(x,x/n)\in \mathbb{R} \times \mathbb{R}: x\in [0,1],n \in \mathbb{N}$} and $X=\cup_{n\in \mathbb{N}}{[0,1]\times (n)}$ and $(0,n) R (0,m),\forall n,m \in \mathbb{N}$. Then does $X/R $ is ...
1
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0answers
37 views

Prove: If H and G/H are totally disconnected then G is also totally disconnected

Let $G$ be a topological group and $H$ a subgroup of $G$. If $H$ and $G/H$ are totally disconnected, then $G$ is also totally disconnected. With 'totally disconnected' we mean the every connected ...
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1answer
39 views

An equivalence reletion $\rho$ on $\mathbb R^2$

Define an equivalence relation $\rho$ on $\mathbb R^2$ by $(x_1,y_1)\rho(x_2,y_2)$ iff $x_1^2+y_1^2=x_2^2+y_2^2$ Then find the corresponding quotient space $\mathbb R^2/ \rho.$
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1answer
31 views

$\mu$ is an equivalence relation on $Y$ and $X/\rho= Y/\mu.$

Let $f$: $X\to Y$ be a homeomorphism and $\rho$ be an equivalence relation on $X$. For $x,y $ in $X$ let $f(x)\mu f(y)$ iff $x\rho y$. Then Show $\mu$ is an equivalence relation on $Y$ and ...
1
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1answer
28 views

Quotient of a locally compact space

I am looking for an example of a quotient of a locally compact space that isn't locally compact. Is there a not too complicated example ?
1
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1answer
52 views

quotient of an amenable group

I have a question about amenable groups. The notion of amenability I am using is: The action of $G$ on $k$ ($k$ locally compact in a topological vector space) is amenable if there exists a point $x ...
2
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0answers
16 views

Quotient by a finite group of fixed-point-free isometries

I'm reading one paper in Riemannian manifolds which very briefly mentions that quotients of $S^{2}\times R^{1}$ by a finite group of fixed point free isometries include $S^{2}\times S^{1}$, ...
9
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2answers
80 views

A relation between product and quotient topology.

I was studying a topic about Algebraic Topology and a question popped into my mind: Suppose that we have two topological spaces $X$ and $Y$. Let $\sim_X$ and $\sim_Y$ equivalence relations in X and ...
7
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1answer
103 views

The topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$.

I'm looking for an example of a topological space $X$ together with an equivalence relation $\sim$ where the product topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$ as a final ...
2
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1answer
114 views

Is the quotient space of a contractible space contractible?

Let $X$ and $Y$ be topological spaces, where $X$ is contractible. Is the quotient space $X/Y$ contractible?
0
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1answer
57 views

Does the continuity at $0$ of the addition map in a vector space imply its continuity?

I have a question about the proof of Theorem 1.41 in Rudin, Functional Analysis, 2/e. The theorem states Let $N$ be a closed subspace of a topological vector space (t.v.s.) $X$. Let $\tau$ be the ...
0
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0answers
30 views

A peculiar fact about 3-dimensional complex projective space

I'm working on a result for my master's thesis, that right now involves translating a proof I don't quite follow, to something that is a bit more in line with what I already know. We define ...
0
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0answers
101 views

Integrating quotients with polynomials in numerator and denominator that are raised to fractional powers

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
0
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1answer
68 views

Find the quotient field of $\mathbb{Z}[\sqrt {2}]$

My idea: how to make a connection between this with isomorphism.
3
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1answer
60 views

The First Homology Group Obtained by Attaching a Möbius Strip to a Torus in a Certain Way.

Let $ M $ and $ \mathbb{T}^{2} $ denote the Möbius strip and the torus respectively. Suppose that we attach $ M $ to $ \mathbb{T}^{2} $ by wrapping the boundary circle $ C $ of $ M $ around the first ...
1
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1answer
27 views

Find a subspace homeomorphic to the quotient.

I'm giving the equivalence relation $V\sim W$ when $\exists\lambda>0:\lambda V=W$. I have to prove that $\sim$ is an equivalence relation and find a subspace of $\mathbb{R}^2$ homeomorphic to the ...
1
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1answer
62 views

Are these quotient spaces homeomorphic to a cylinder and to the Möbius Strip?

Consider for $[0,1]\times [0,2]$ the function $f:\{0\}\times [0,1]\to \{1\}\times [0,1]$ given by $f(0,x)=(1,x)$. Prove that the quotient space given by this $f$ is homeomorphic to the cylinder and ...
0
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0answers
20 views

Automorphism group of torus fixing origin

I've got a short question: Suppose that you have some lattice $\Lambda$, say $\Lambda=\mathbb{Z}+\mathbb{Z}i$, and let $T$ be the torus $\mathbb{C}/\Lambda$, coming with the quotient map ...
2
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2answers
73 views

Proving that $X/R$ is Hausdorff $\implies$ $R$ closed.

As the title says, I'm trying to prove that if $X/R$ is a Hausdorff space then $R\subset X\times X$ is closed. I have several questions about this: $(1)$ What exactly is $R$? I thought of $R$ as an ...
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1answer
20 views

Openess of sets given by equivalence relations in the quotient topology.

I'm trying to prove this: Let $R$ be an equivalence relation in $X$. Show that $A$ is open in $X/R .\iff \bigcup_{[x]\in A}[x]$ is open in $X$ One of the first things that come to my mind is ...
3
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1answer
33 views

Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
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0answers
65 views

$\mathbb{R}^n$ $\backslash$ $\mathbb{R}^k$ what does this mean?

$\mathbb{R}^n$ $\backslash$ $\mathbb{R}^k$ I saw this in my topology assignment. The question was about quotient spaces and homeomorphisms. I have never seen this expression before so it doesn't ...
1
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1answer
23 views

Need help on Quotient Spaces

Definition: Suppose $X, \tau$ is a topological space and $R$ is an equivalence relation on $X$. Let $X/R$ denote the set of $R$-equivalence classes. Define the function $f$ from $X$ to $X/R$ by $f(x) ...
2
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1answer
55 views

Relation Between Free Quotients and Modules

Here is my question: Let $M$ and $M'$ be $R$-modules, where $R$ is a commutative ring, and $N \subseteq M$ and $N' \subseteq M'$ submodules. Suppose that $N \cong N'$ and $M/N \cong M'/N'$. Determine ...
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0answers
30 views

How to show that the dimension of a quotient space in the field of polynomials is not finite?

I have to show that if I have a quotient of the form $\mathbb{K}[x_1,x_2,\dots,x_n]/\langle f_1,f_2,\dots,f_s\rangle$, $\operatorname{char}(\mathbb{K})\not=0$, and on which the class of $[x_i^l]$ is ...
0
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2answers
39 views

Realizations of Circle

I really don't know how to prove without brute force that: $\mathbb{R}/{\sim}\cong[0,1]/{\sim}$ I know already that: $[0,1]/{\sim}\cong\mathbb{S}^1$ (simply use closed map lemma and uniqueness of ...
0
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1answer
48 views

Proof check for $(X/M)^{*} \cong M^{\perp}$

I would like to know if the proof I have is correct. Statement: Let $M$ be a closed subspace if a Banach space $X$. Let $\pi: X \rightarrow X/M$ be the quotient map. Put $Y= X/M$ for each $\varphi \, ...
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2answers
79 views

Quotient Topology vs Subspace Topology.

Let $X$ and $Y$ be topological space and let $\pi:X\to Y$ be a quotient map (it is surjective map and $Y$ has the quotient topology induced by $\pi$). A subset $U\subset X$ is said to be saturated if ...
0
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0answers
17 views

How to… quotient set on a fractal continuous curve.

I'm really not good at math so I can't really formulate my problem in a closed form :) There is a curve $C$ in $R^2$. There are some rulers of length ${L1,L2,L3,L4,L5,....}$ I need to find a way to ...
2
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1answer
67 views

What does quotienting by a congruence mean?

I have come across quotient algebras in my different mathematics courses. I know of quotienting with normal groups, quotienting with ideals etc. While studying Boolean Algebra I encounter quotienting ...
0
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1answer
36 views

Is there example for isomorphic closed subspaces of a Banach space with non isomorphic quotient?

$Y_1$ and $Y_2$ are closed subspaces of a Banach space X and $Y_1 \simeq Y_2$. I can't find a way to show $X/Y_1 \simeq X/Y_2$ and it made be think that it's not true. Is there a counter example?
0
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1answer
48 views

Quotient of projective variety by finite group

Suppose I have a projective variety $X$ (for which I have explicit equations) and an involution $\iota$ on it (again, explicit). I'd like to write down explicit equations for $X/\langle \iota ...
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2answers
72 views

Prove that dimension is finite

I want to prove that $\dim V/(X \cap Y)$ in finite, if $V$ be a vector space and $X$, $Y$ two sub spaces of $V$ such that $\dim V/Y$ and $\dim V/X$ are finite.
0
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1answer
41 views

Showing a projection map on restricted to a subset is not an open map

I'm working on a problem from Munkres about open and closed maps. Here's the problem: "Let $\pi_1 : \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ be projection onto the first coordinate. Let ...
2
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0answers
40 views

When does the quotient metric reduce to the infimum of the distances of only two points?

Given a metric space $X$ and an equivalence relation $\sim$, the quotient (pseudo-)metric on $X/\sim$ is defined as follows: $d'([x],[y]) = \inf \left \{ d(p_1,q_1) + d(p_2,q_2) + ... + ...
1
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1answer
155 views

Quotient space of closed unit ball and the unit 2-sphere $S^2$

This is an example from Munkres's Topology (Example 4 in Section 22 titled "The Quotient Topology", 2nd edition). Example 4: Let $X$ be the closed unit ball $$\{ x \times y \mid x^2 + y^2 \le ...
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0answers
92 views

How many melodies are there?

Clearly if we assume only 12 chromatic notes to the scale, not all of which sound good next to each other, a melody of length $N$ chooses from less than $12^N$ potential melodies. Allowing melodies to ...
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2answers
56 views

Why does the quotient space V/W not equal the vectorspace V?

Let's say $V=\Re^3$ and $W$ is a plane through the origin. The way I understand the quotient space $V/W$ is that it's formed by taking every vector $\vec{v}^{\,} \in V$ and adding it to the subspace ...
2
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0answers
55 views

Show that a certain function $\tilde{f}:S^3\to\mathbb{R}$ induces a function $f:S^3/S^1\to\mathbb{R}$ (group actions)

I am a bit stuck on a homework assignment and I'm hoping someone can push me in the right direction. Consider the 3-dimensional sphere: $$S^3=\{(z_1,z_2)\in\mathbb{C}:|z_1|^2+|z_2|^2=1\}$$ and the ...
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1answer
47 views

Quotient space: T2 and pseudo-metric?

I've got a question on Quotient Space but I just can't visualise the space. This is the definition: X = $\mathbb{R}$ x $\mathbb{Z}$ (x,z) and (x',z') are said to be equivalent iff ...
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1answer
62 views

Why does the natural quotient metric works in this case?

There have been at least two discussions in this forum about why a (pseudo) metric is defined in the quotient of a metric space by: $$ d([x],[y]) = \inf \{ d(p_{1}, q_{1}) + \ldots, d(p_{n}, q_{n}) \} ...
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1answer
95 views

Basis of quotient space

Given the follwing vectors in $\mathbb{R}^4$ $v:= \begin {pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}\,$, $a:=\begin {pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix}\,$, $b:=\begin {pmatrix} 1 \\ 0 \\ 1 \\ 2 ...
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3answers
102 views

Is this identity for the Dihedral group correct?

Let $D_n$ represents the Dihedral group with $2n$ elements, and my question(based on some physics backgrounds) is: Does $Z_2$ a normal subgroup of $Q_8$? If it is, then is the indentity $D_2\cong ...
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0answers
37 views

Algebra “A/I”: what does it mean?

I have an Algebra $A$ and an Ideal $I \subset A$. What is the algebra $A/I$ ? I've seen it several times but I can't find the definition... Maybe is it the subspace of $A$ that is $A \ominus I$ ?
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2answers
47 views

Can someone explain the concept of Fundamental Domain to me?

Hi for a Geometries course we're dealing with fundamental and I don't quite understand the definition of a fundamental domain. The definition in my book is that a fundamental domain: is a region of ...
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0answers
43 views

Tensor power of a quotient space

Let $E$ and $F$ be vector spaces, $F$ a subspace of $E$. Is there any canonical isomorphism between $E/F \otimes E/F$ and a quotient of the form $E \otimes E/G$, where $G$ is a subspace of $E \otimes ...
2
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1answer
61 views

Equivalence Relation, Quotient Space, Compactness

Define an equivalence relation on $\mathbb{R}^2$ by $a \times b$ ~ $c \times d$ iff $a-c$ and $b-d$ are both integers and let $T=(\mathbb{R}^2)^*$ denote the corresponding quotient space. (a) Show ...
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0answers
45 views

Hausdorff and Quotient Spaces

Let $L$ be a subset of $\mathbb{R}^{2}$ and let $N = \mathbb{R}^{2}/L$ be the quotient space obtained by identifying all points in $L$ to a single point. I need to prove that $N$ is Hausdorff ...
6
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1answer
218 views

Images of composed homomorphisms

Let $R$ be a ring and $M,N,S$ be $R$-modules. Let $\varphi_1 : M\to N$ and $\varphi_2 : N\to S$ be homomorphisms. Then $\operatorname{im}(\varphi_2\circ\varphi_1) \subseteq ...