The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
2answers
23 views

Isomorphism between quotient groups with normal subgroups

I'm looking at a problem in my textbook and it says: Let $ψ : G_1 → G_2$ be a surjective group homomorphism. Let $H_1$ be a normal subgroup of $G_1$ and suppose that $ψ(H_1) = H_2$. Prove or disprove ...
2
votes
2answers
32 views

The name for the quotient property.

We call a surjective $f:X\rightarrow Y$ a quotient mapping if it satisfies, for every $U\subset Y$ (continuity, continuous) $U$ is open $\Rightarrow$ $f^{-1}(U)$ is open and (???) ...
3
votes
1answer
42 views

Different identifications of the same sides of a polygon make the same quotient space

Let P$\subset\mathbb{R}^{2}$ be a polygon with sides $l_{1},...,l_{n}$ parametrized by the curves $\alpha_{1}(t),...,\alpha_{n}(t)$. Let $\beta_{1}(t),...,\beta_{n}(t)$ be another parametrization of ...
-1
votes
2answers
30 views

Find quotient of vector spaces [closed]

How can I find the solution of this quotient $$\frac{Span\{a,b,c,d\}}{Span\{a+b,c+d\}}$$?
1
vote
0answers
20 views

Proving that a quotient space is formed from a vector space with W-affine subspaces.

I have been given a 2-part question which first states given a vector space (V,K) and W$\subseteq$V is a subspace. that a W-affine subspace S$\subseteq$V is one in which s,s' $\in$ S, s-s' $\in$ W and ...
2
votes
0answers
17 views

Smoothly compatible for $\mathbb{R}^{2}/\sim$ ,where $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$

This is homework so no answers please. Consider the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$ ,where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$. Then ...
0
votes
0answers
27 views

Question about the third and fourth isomorphism theorems for groups

I am trying to work with both of the third and fourth isomorphism theorems for groups. I am considering the following situation: I wanted to take a subgroup in the quotient and see how it corresponds ...
0
votes
1answer
50 views

Is this the correct conclusions about a quotient topology on $\mathbb R$?

Can someone tell me if my conclusions about this space is correct? I have an equivalence relation for $x,y \in \mathbb R$ given by $$x\sim y \Longleftrightarrow (x = y) \lor (x,y \in (-a,a])$$ for ...
0
votes
1answer
22 views

Quotient Topology, why is this set “saturated”?

It says $[2,3]$ is saturated with respect to $q$, but not open in $Y$. BUt it doesn't make sense to me because $q(A) = q([0,1) \cup[2,3]) = [0,1) \cup [2-1,3-1] = [0,1) \cup [1,2] = [0,2] = ...
2
votes
2answers
41 views

Please explain the Quotient Rule

I am currently working on an equation but I'm having a hard time understanding how to get the answer. the answer is ${(x^2-4)(x^2+4)(2x+8)-(x^2+8x-4)(4x^3)\over (x^2-4)^2(x^2+4)^2}$ The equation is ...
1
vote
1answer
44 views

How Can I prove the three statements are equivalent?

Let $X$ be a compact Hausdorff space and $f:X \rightarrow Y$ be a quotient map. Show that the following are equivalent: (a)$Y$ is an Hausdorff space, (b)$f$ is closed map, (c)The set ...
0
votes
1answer
20 views

Is subspace of a vector space a quotient 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
vote
1answer
20 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
votes
1answer
31 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
votes
2answers
51 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
votes
0answers
76 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
votes
1answer
53 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
votes
1answer
52 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
votes
1answer
28 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
votes
2answers
43 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: ...
1
vote
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
votes
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
vote
0answers
45 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
54 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
votes
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
vote
1answer
42 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
vote
1answer
57 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
votes
0answers
17 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
votes
2answers
85 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
votes
1answer
105 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
votes
1answer
120 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
votes
1answer
77 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
104 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 ...
0
votes
0answers
32 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
votes
0answers
129 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
votes
1answer
78 views

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

My idea: how to make a connection between this with isomorphism.
4
votes
1answer
84 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
vote
1answer
29 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
76 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
votes
0answers
29 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
votes
2answers
87 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 ...
1
vote
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 ...
4
votes
1answer
38 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 ...
0
votes
0answers
66 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
vote
1answer
30 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
votes
1answer
57 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 ...
1
vote
0answers
34 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
51 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 \, ...
4
votes
2answers
108 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 ...