# 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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### Unique Haar measure on quotient is pushforward

Let $G$ be an abelian locally compact Hausdorff group with discrete subgroup $H$. Let $\mu$ be a Haar measure on $G$ and $\lambda$ the usual counting measure on $H$. Then we obtain a unique Haar ...
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### fundamental group of quotient (singular) tori

What we know about fundamental group of quotient singular manifold? I'm particularly interesting in the case of quotient given by $X:=T/G$ where $T$ is a complex torus and $G$ is finite. Since $\pi_1$ ...
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### Projective special linear group and projective space as homogeneous space

The real projective special linear group $PSL(n+1,\mathbb{R})$ acts transitively and effectively on the real projective space $\mathbb{R}P^n$. As a homogeneous space, it can be considered as a ...
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### Quotient of affine space under negation

Suppose we are working over a field $k$. Consider the affine space $k^n$ and the $\mathbb{Z}/(2)$-action on it given by $(x_1,x_2,\dots,x_n) \mapsto (-x_1,-x_2,\dots,-x_n)$. I would like to compute ...
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### What are these quotient spaces of sphere products homeomorphic to?

For $A,B \subset \mathbb{R}^n$, consider the equivalence relation $(a,b) \sim(-a,-b)$ on $A \times B$. I'm trying to see what the quotient spaces of $S^2 \times S^1$ and $S^1 \times S^1$ by $\sim$ ...
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### Question on the connectedness of the orthogonal group

I want to show that the quotient $O_2^- = O_2/SO_2$ is connected. My idea was as follows: It's easy to show that $SO_2$ is connected. $S0_2$ is a topological group (normal subgroup of a topological ...
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### Can germs be defined as a quotient of vector spaces?

Summary: Let $M$ be a smooth manifold and $p\in M$. I know of two notions of germs of functions at $p$, the more restrictive of which can be written as a quotient vector space. I am curious whether ...
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### Is this space with the trivial metric, a nontrivial quotient of a subspace of the Sorgenfrey topology?

I guess this is a question about how accurate my intuition is about the Sorgenfrey line being a richer topology than the standard metric space on $\Bbb Q$, with reference to a specific application. ...
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### Prove that $(S^2/\{\pm 1\})/S^1 \cong S^2$

I'm searching an homeomorphism between $S^2$ and the space $A=\dfrac{S^2/\{\pm 1\}}{S^1}$, where the numerator is the quotient space given by the group action by multiplication of $\{\pm 1\}$ on $S^2$,...
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### Partitioning the 2-sphere into compact homeomorphic subsets

This question is related to a previous one Covering the plane with compact sets, II (as yet unanswered). Can the 2-sphere be partitioned into an infinite family of disjoint compact subsets, all of ...
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### What is the shape I get after moding out the sphere $S^2$ by $A$?

I am trying to answer the following question: $(a)$ Compute the homology groups $H_n(X, A)$ when $X$ is $S^2$ or $S^1 \times S^1$ and $A$ is a finite set of points in $X.$ And I am trying to use the ...
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### Criterion for proving that a quotient space is finite dimensional

In the book of Introductory Real Analysis of Kolmogorov and Fomin (page 122), there is some theorem about the dimension of quotient spaces. Let $L$ be vector space and $L'$ be a vector subspace. It ...
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### The Axiom of Choice and a definition of addition in a quotient space of a vector space

I am thinking about the Axiom of Choice and I am trying to understand the Axiom with some but a little progress. Some time ago I could not understand why the obvious "proof" of the Axiom of ...