# Tagged Questions

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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### What is the meaning of $\Bbb{Q}[x]/f(x)$?

I am very confused with the meaning of $\Bbb{Q}(x)/f(x)$. Does it mean the set of all polynomials modulo $f(x)$? If it does then how can we say that $\Bbb{R}[x]/(x^2+1)$ is isomorphic to set of ...
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### $17\mathbb{Z}[\sqrt {10}]$ is prime ideal in $\mathbb{Z}[\sqrt {10}]$

This seems tedious since I would have to show $(a+b\sqrt {10})(c+d\sqrt {10})=17k+17j\sqrt {10}$ implies that one of the factors belongs to $17\mathbb{Z}[\sqrt {10}]$. It's easier to show something ...
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### In the quotient topology $D^2/{S^1} \cong S^2$

Let $X = D^2 = \{(x, y) \in R^2\ : x^2 + y^2 \le 1\}$ be the closed unit disc (in the standard topology). Identify $S^1$ with the boundary of $D^2$. Now I have to prove that $$D^2/S^1\cong S^2$$ I ...
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### What is the quotient space $\mathbb{R}^2/F$ where $F =a[1,1]$?

What is the quotient space $\mathbb{R}^2/F$ where $F =a[1,1]$? How do you find a basis of it? What is a good way to think of it in mind?
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### Proving these equivalent conditions for an open map using boundary of a set

Let $X,Y$ be topological spaces. Prove the following statements are equivalent. $(1)$ $f\colon X\to Y$ is an open map. $(2)$ For all $x\in X$ and open set $U \ni x$ there exists open set $V$...
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### $R/\langle p^k\rangle$ is an associator (i.e. if $\langle a\rangle = \langle b\rangle,$ then $a$ and $b$ are associates) when $R$ is a PID.

As the title says, I want to show that when two principal ideals are equal in $R/\langle p^k\rangle,$ where $R$ is a principal ideal domain and $p\in R$ is a prime element, then their generators are ...
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### convex polyhedral cone and faces, one to one inclusion preserving map from faces of cone to faces of quotient cones.

I'm doing with a problem in 'lectures on toric variety' written by David Cox. In page 15, exercise 1.5, Suppose that $\sigma$ $\subset$ in $\mathbb R^n$ is convex polyhedral cone of dim d, and assume ...
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### Clarification of ideas concerning a quotient space.

Suppose I have a vector space $V$, and I identify $x\in V$ with $\lambda x\in V$, where $x\neq 0$ and $\lambda>0$, $\lambda\in\mathbb{R}$. I'm confused about two things: (1) Can I define a norm on ...
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### Not fully understanding polynomial quotient rings.

This is my (informal) understanding of a quotient ring. I understand that this is very flimsy, but I hope you can get the main idea. You have some ring $R$ and you want to quotient out an ideal $I$...
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### Restricting the quotient map of rings to a subring

When $q$ maps $R$ to $R/I$ and $p$ is the restriction of $q$ to a subring $A$ of $R$, why is the image of $p$ $(A+I)/I$? $q$ maps $r$ to $r+I$, so shouldn't $p$ map $a \in A$ to $a+I$, so image of $p$...
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### quotient groups and SLOCC

I have a math-physics question, which is based on an interest in SLOCC systems for black hole entanglement. The Cartan decomposition of a group $G$ such that $H = G/K$ is such that the derivation or ...
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### Why is this function well definded?

I have to take a look at the relation of $x \sim y: \Leftrightarrow \exists k \in \mathbb{Z} : x-y=5k$ in $\mathbb{Z}$. I have no idea how to show at $\mathbb{Z} / \sim$ that the addition is well ...
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### Projection of measure with bowen - walters metric.

Given $X$ a compact metric space, $f:X\to X$ be a homeomorphism and consider the quotient space $Y^{1,f}=(X\times [0,1])/\sim$, where $(x,1)\sim(f(x),0)$ for all $x\in X$. Let $d^{1,f}$ be the Bowen-...
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### A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n$

(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...
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### Quotient topology on unit sphere

Let $\sim$ be the equivalence relation $$a\sim b\iff a=b\text{ or }a=-b,$$ for $a,b$ on the unit sphere $S^2$. Let $Q$ be the quotient space. How do I show that the quotient map is a covering ...
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### Quotient topology, topological spaces

This is a practice final exam. My questions: 1) When the question defines X/~ as quotient topology. Does that mean I can write: $q: X \rightarrow X/\sim$ 2) Specifically, I am trying to prove that ...
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### Invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$

If $A$ is a ring with unit element $1 \ne 0$ let $A^{\times}=\{a \in A: a$ invertible$\}$. Find the invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$. ...
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### Construct a homemorphism $\phi : T^2/A \rightarrow X/B$

Construct a homemorphism $\phi : T^2/A \rightarrow X/B$ $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by $A=S^1 \times\{1\}$. $X=S^1 \times [-1, 1]$ and $B = S^1 \times\{-1, 1\}$. $T^2/A$ ...
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### Examples of a quotient map not closed and quotient space not Hausdorff

Is there any example of a closed relation $\sim$ on a Hausdorff space $X$ such that $X/\sim$ is not Hausdorff? Also, is there any example of a closed relation ~ on a Hausdorff space $X$ such that a ...
### Order in a quotient space of $\mathbb{R}^\mathbb{N}$ ($\mathbb{R}^\omega$)
Let $\mathcal{F}$ be a filter in $\mathbb{N}$ finer than Fréchet filter. In $\mathbb{R}^\mathbb{N}$ we define the equivalente relation : $(a_n) \equiv (b_n)$ iff $\{n | a_n = b_n\} \in \mathcal{F}$. ...