# Questions tagged [quotient-spaces]

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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### Show that $( X \sqcup_f Y)/Y$ is homeomorphic to $X/A$.

Let $X$ be a topological space and let $A \subset X$ be a subset. We define the topological space $X/A$ to be the quotient space $X/\mathcal{R}$ where $\mathcal{R}$ is the equivalence relation defined ...
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### Quotient space not the same as the original space

I am trying to understand why quotient space is not the same as the original space. Let $V$ be a vector space and $W$ be its vector subspace. If I define $a+W:=\{x=a+w; w\in W\}$ then the quotient ...
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### Factor Rings over Finite Fields

Given a polynomial ring over a field $F[x]$, I can factor, for example, the ideal generated by an irreducible polynomial $ax^2 + bx + c$: $F[x]/\left<ax^2 + bx + c\right>$, and guarantee that ...
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### Is there an explicit function from the quotient space of an annulus to a torus?

The annulus is ${1\leq x^2+y^2\leq4}$ I would like to show that the quotient space of the annulus given by the equivalence relation $(x,y)\sim(x,y)$ and $(x,y)\sim(2x,2y)$ if $x^2+y^2=1$ is ...
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### Relation between quotient map and quotient space

Given a surjective function $p$ that takes a space $X$ to it’s decomposition $X*$ (decomposition here means a partition into equivalence classes), is it always the case that the function $p$ is a ...
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### Saturated subsets of quotient map.

From an example in Munkres Topology:$$\\$$ Consider the projection map $\pi_{1}: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ onto the first coordinate; it is continuous and surjective. It is also ...
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### Proper actions in complex geometry

[skip to next blockquote for the actual question] Complex geometry books often treat quotients by 'properly discontinuous' actions, such as the action of a lattice $L \subseteq \mathbb{C}$ on the ...
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### Using the quotient manifold theorem to show real projective spaces are smooth manifolds

The goal of this post is, as stated in the title, Use the quotient manifold theorem to show real projective spaces are smooth manifolds. I know this is an overkill, but I am just curious about ...
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### quotient module $\mathbb{Z}^3/(2,0,3)\mathbb{Z}$

Is $\mathbb{Z}^n/(m,a_2,...,a_2)\mathbb{Z}\cong\mathbb{Z}/m\mathbb{Z}\oplus\mathbb{Z}^{n-1}$ for $0<m<a_2,...,a_n\in\mathbb{Z}$ (as $\mathbb{Z}$-modules)? If not, how can you compute something ...
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### The Rank of an Endomorphism over a Quotient space, which is generated by an invariant subspace

I have problems answering the second part of the following question: Let $U$ be a $K$-vectorspace with finite dimension, $W \subset U$ a subset of $U$ and $\varphi :U\to U$ an endomorphism. 1.) Show:...
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### Is $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ free?

I'm trying to solve a question which asks me to determine whether the quotient $\mathbb{Z}$-module $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ is free. I'm then supposed to find some ...
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### Quotient Space Metric with Nice Equivalence Classes

Quotient Space Metric: The quotient metric for arbitrary quotient spaces is defined as If $M$ is a metric space with metric $d$, and $\sim$ is an equivalence relation on $M$, then we can endow ...
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### Is $(S^1)^3/{\sim}$ a manifold? $\sim$ is equivalent relation on all triple permutations $(a,b,c)\sim (a,c,b)\sim (c,a,b)\sim …$ on $(S^1)^3$

Let $\sim_1$ be an equivalent relation on $(S^1)^2$, $(a,b)\sim(b,a)$, then $M_2:=(S^1)^2/{\sim_1}$ is homeomorphic to Mobius band and is a $2$-manifold,. What can we say about $M_3=(S^1)^3/{\sim_2}$ ...
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### Quotient space $[0,1]/C$ is homeomorphic to $[0,1]/\left\{0,1,\frac12,\frac13,…\right\}$, $C$ denotes Cantor set. [closed]

How to prove quotient space $[0,1]/C$, where $C$ denotes Cantor set, is homeomorphic to $[0,1]/\left\{0,1,\frac12,\frac13,...\right\}$?
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### Quotient topology clarification, what happens if we glue together a point on a boundary with a point in the middle

When I first learned about the quotient topology $X/\sim$ on a topological space $X$ the quotient space was defined to be $X$ with all the points identified by $\sim$ glued together. So if $X = [0,2]$ ...
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### Quotient of Second Countable Space

I'm looking for an example for a second countable topological space $T$ such that there exist a quotient structure $T/\sim$ which is not second countable. Does there exist an example where $T$ is a ...
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### Prove that quotient space is Hausdorff

Let $A$ be a closed subset of the interval $[0,1]$. Show that the quotient space $$W=(-2,2)\big/A$$ is Hausdorff. I guess we need to explicitly find the disjoint open sets for each two $x\neq y$ from ...
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### Map from circle to real projective plane

can you help me with these proofs? Let $f: S^1 \to \mathbb{R}P^2$ be a map from circle to real projective plane and $\pi: S^2 \to S^2/\sim$ be the quotient map where $\sim$ identifies antipodal ...
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### Intuition on norm of quotient space

Theorem. Let $(X,\| \cdot \|$) be a normed space. Then $$p(x+U) = \inf_{z \in U} \|z-x \|$$ defines a semi-norm on $X/U$ with $p(x+U) \leq \|x \|$. a) If $U$ is closed, then $p$ is a norm. b) If $U$...
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### Tensor Product of Quotient.

Let $K$ be a field and $L/K$ a field extension. Suppose $A$ is a $K$-algebra and $I$ is an ideal. I want to show that $$(A/I\otimes_K L) \to (A\otimes_K L)/(I\otimes_K L)$$ So i define a map f: A\...
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### Volume of a 3D torus as a form on a quotient space

I'm trying to calculate the volume form $dx\wedge dy\wedge dz$ on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$: $\int_{\mathbb{T}^3}dx\wedge dy\wedge dz$ I've been told that it's simply the volume of ...
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### Show that the space $X$ is not a surface

I would like to show that the following space is not a surface: $X$ is made as an identification space of the unit square $Q=\{(x,y)\mid 0\leq x,y\leq1\}$ with the identifications: $(0,y)\sim(1,y)$ ...
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### Unique complex structure on the modular curve $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$

Is the complex structure on the modular curve coming from the quotient $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$ unique? (Here $\mathbb{H}$ is the upper half plane in $\mathbb{C}$) According to ...
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### Computing Coordinate Rings of Varieites

I am a complex analyst who was been screwed over by fate and now has to work with elliptic curves for my doctoral dissertation. This entails learning about (non-category-theoretic) algebraic geometry. ...
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### When is a space homeomorphic to a quotient space?

Is the following theorem true? It seems straightforward but I haven't seen it published anywhere, not even as a corollary, so I'm concerned I've missed something. Discussions that introduce quotient ...