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1
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1answer
15 views

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 ...
2
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1answer
36 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: ...
1
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1answer
19 views

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!!!!!
1
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1answer
23 views

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, ...
4
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3answers
65 views

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 $\mathbb R^{n+1} \setminus \{0\}$ ...
0
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1answer
48 views

The monomials not inside $in_<(I)$ form a K-basis inside the Quotient ring

Given the quotient ring $T/I$, where $T=K[x_1,...,x_n]$ is a polynomial ring and $I$ is an ideal. I need to show that for any monomial $x^u:=x_1^{u_1}*...*x_n^{u_n}$, if the monomial is not inside ...
0
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1answer
48 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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2answers
53 views

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 ...
1
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1answer
35 views

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 $$ ...
1
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1answer
25 views

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 ...
0
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0answers
28 views

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: ...
3
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1answer
52 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 ...
3
votes
1answer
35 views

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 ...
2
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1answer
41 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 ...
7
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2answers
341 views

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 ...
4
votes
2answers
109 views

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 × ${$1$}$)$, ...
1
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1answer
54 views

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 $ΣX$ ...
0
votes
1answer
114 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 $X/A$ ...
3
votes
3answers
128 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 ...
0
votes
0answers
26 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 ...
1
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0answers
74 views

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 $\mathscr{B}\subset\mathscr{A}$ ...
0
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3answers
35 views

“The limit of a sequence is insensitive to finite changes in the sequence” - help me understand this sentence!

The context is p59 of "A primer on Hilbert Space Theory, by Alabiso and Weiss, where the example is given of a quotient $c_{00}/c_0$ which is "the quotient of the space $c_0$, the space of all ...
0
votes
1answer
35 views

How do I prove that this map is a homeomorphism?

Let $X$ be a topological space. Let $\{X_i\}$ be a family of mutually disjoint open subsets of $X$ such that $\bigcup X_i = X$. Let $a_i$ be a point of $X_i$ for each $i$. Consider a quotient map ...
0
votes
1answer
28 views

To Show that $S^n/(v\sim -v)$ is homeomorphic to $\mathbf RP^n$.

Let $S^n$ be the unit sphere in $\mathbf R^{n+1}$ and $\mathbf R P^n$ be the real projective space(see the definition of $\mathbf R P^n$ I am using in the References). Define a relation $\sim$ on ...
3
votes
0answers
71 views

When is a quotient by closed equivalence relation Hausdorff

Let us say for an arbitrary topological space $X$ that it has property $\dagger$ if for any closed equivalence relation $\sim$ on $X$ (closed as a subset of $X^2$), the quotient space $X/{\sim}$ is ...
1
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1answer
51 views

Quotient space topology

Let $X$ be a topological space and $A \subseteq X$, $q:X \rightarrow X/A$ the quotient map. Let $Y$ be another topological space and $f:Y \rightarrow X/A$ continuous. Is there a $g:Y\rightarrow X$ ...
2
votes
1answer
100 views

Prove that $\mathbb R/\mathbb Z$ is sequential and not first countable

Let's say I have a a space X, the quotient space of R (the reals) obtained by identifying all points of Z (the integers). How do I prove that X is sequential but not first countable? (sequential ...
3
votes
1answer
58 views

Can I assign a gravity field to an infinite grid of point masses?

So, in the classic arcade game Asteroids, you move in a game field where the top and bottom edges are identified and the left and right edges likewise, topologically a torus. I'm interested in how ...
1
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2answers
53 views

Some Difficult Questions on Quotient Spaces in Linear Algebra

I've stumbled across some problems in the quotient spaces and I solved many of them but I cannot figure out the following. These are not homework and I appreciate your help. a) Let $P$ be the space ...
0
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1answer
57 views

Quotient Space induced by $x \sim y \iff \overline{\{ x \}} = \overline{\{ y \}}$ is $T_0$

Ok, I've been struggling with the following problem; before I get into it I want to put a disclaimer that I only want a hint, not a solution. Let $X$ be a topological space, consider $\sim$ defined ...
1
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3answers
45 views

What is the Quotient on the Coproduct in Adjunction Spaces

Can someone please provide a detailed explanation of the equivalence relation used to construct adjunction spaces from the topological coproduct? In particular, most sources talk about "identifying an ...
2
votes
0answers
24 views

Quotient group of intersections

My question is motivated by a recent discussion of the Frattini subgroup with my professor, and in particular, the special structure of $G/\Phi(G)$. Given a group $G$ with normal subgroups $H_1, H_2, ...
0
votes
1answer
41 views

Are the following quotient spaces finite dimensional?

If we take $\mathbb{F}[x]$ to be the set of all polynomials over the field $\mathbb{F}$, $E$ to be the subset of all such even polynomials, $N$ to be the set of these polynomials that have degree less ...
4
votes
1answer
50 views

Universality with respect to quotients

Is there an infinite cardinal $\kappa$ for which the following statement (S) true? (S) : There is a topology $\tau_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq ...
1
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2answers
45 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
38 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
49 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 ...
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votes
2answers
37 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
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0answers
25 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
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0answers
18 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
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0answers
34 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
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1answer
52 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
28 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
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2answers
47 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
51 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
28 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
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1answer
30 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
49 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
55 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
77 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 ...