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1answer
15 views

Is subspace a Qtuotient space?

Reasoning: Given a vector space V and a subspace W of V. "x≡y (mod W) if x-y∈W" defines equivalence relation. equivalence relation partitions V into equivalence classes. equivalence classes is ...
1
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1answer
15 views

Qtuotient space and affine space

Sorry for many questions in this part. But I am still confused about the following: From textbook "Optimization by vector space"(Luenberger): Problem: I read the def. of quotient space many ...
0
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1answer
18 views

dimension of quotient space

I am confused about the following: In Wiki: => dim(vector space) - dim(subspace) = dim(quotient space) In S. Boyd's textbook of cvx (p.22) => dim(subspace) = dim(affine set) Problem: ...
2
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2answers
49 views

“Reverse” quotients.

Given a set $A$ and a equivalence relation $\sim$ on $A$, one can construct the quotient $A/{\sim}$. But what about the opposite? More precisely, given a set $X$, is there a way to know whether exists ...
2
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0answers
75 views

The cohomology of $S^3/D^*_k$

I have tried to compute the de Rham cohomology and the homology over the integers of the space $S^3/D^*_k$, where $D^*_k$ is the binary dihedral group of order $4k$ and I would like to know if ...
3
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1answer
49 views

Closed surjection that does not preserve regularity

Def Map $p\colon X\rightarrow Y$ is perfect if it is a closed surjection and $p^{-1}\left(\left\{y\right\}\right)$ is compact for each $y\in Y$ It is well known that perfect maps preserve regularity, ...
0
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1answer
49 views

To show closed subset of $R^2/{\sim}$ contains origin

I am working on the old preliminary exam from my university. I found trouble to solve the following problem. Could you please help me on it. Let $X = \mathbb R^2/{\sim}$ be quotient space where $x\sim ...
0
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1answer
27 views

If a topological space $S$ is second-countable, must necessarily every quotient space of $S$ be second-countable?

Let $S$ be a second countable topological space. Let $S^*$ be a quotient space of $S$ with quotient map $\pi$. If $\pi$ is open, it's easy to show that it transfers a basis of $S$ into a basis of ...
2
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2answers
40 views

For which values of $a\in\mathbb ℤ/3\mathbb ℤ$ is the quotient $\mathbb ℤ/3\mathbb ℤ[x]/(x^3+x^2+ax+1)$ a field?

I'm trying to solve the following problem: Determine for which values of $a\in\mathbb{Z}/3\mathbb{Z}$ the quotient $Q_a=(\mathbb{Z}/3\mathbb{Z})[x]/(x^3+x^2+ax+1)$ is a field. I see two options: ...
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2answers
29 views

Polynomial quotient rings

I have a quotient ring $R=\Bbb Z[t]/(1-t)^3$. It is asked to show that $\overline{2t^3-2}=\overline{6t^2-6t}$ and $\overline{1-4t^3}=\overline{4t-6t^2-t^4}$. I feel like I am missing some important ...
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 ...
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0answers
43 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 ...
-1
votes
1answer
51 views

An equivalence relation $\rho$ 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 $\mathbb R^2/ \rho.$
0
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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
38 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
55 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 $S^{2}\times S^{1}$, ...
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2answers
83 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
104 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
118 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?
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1answer
62 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 ...
1
vote
1answer
89 views

What is meant by gluing two metric spaces together?

"Gluing" constructions are common in topology: by gluing two disks along their boundaries we get a sphere; by gluing a cylindical "handle" to a sphere we get a torus, and so forth. If the original ...
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0answers
31 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 ...
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0answers
116 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
69 views

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

My idea: how to make a connection between this with isomorphism.
3
votes
1answer
74 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 ...
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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
vote
1answer
68 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
23 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 ...
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2answers
78 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
23 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 ...
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1answer
26 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
31 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
votes
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
votes
1answer
50 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
86 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
votes
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
68 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
39 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
59 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.
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1answer
47 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
44 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) + ... + ...
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1answer
173 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
102 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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vote
2answers
59 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 ...
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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 ...