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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.

Quotient space is a space where the underlying space is set of equivalence classes of some equivalence relation. This concept appears in various contexts. For example, quotient spaces can be defined for topological spaces, vector spaces and normed spaces.

As this concept appears in various areas, include also a tag specifying subject matter, such as , , , etc.

2015 questions
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When is a quotient map open?

Quotient map from $X$ to $Y$ is continuous and surjective with a property : $f^{-1}(U)$ is open in $X$ iff $U$ is open in $Y$. But when it is open map? What condition need?
Oh hyung seok
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61
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2 answers

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each $q_n$ is an equivalence class induced by $Y$, I…
Metta World Peace
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$X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$

$X$ is a Hausdorff space and $\sim$ is an equivalence relation. If the quotient map is open, then $X/{\sim}$ is a Hausdorff space if and only if $\sim$ is a closed subset of the product space $X \times X$. Necessity is obvious, but I don't know how…
yaoxiao
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47
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On the norm of a quotient of a Banach space.

Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert y\right\Vert_E\mid y\in x+F\}. $$ Unfortunately…
Jyrki Lahtonen
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43
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3 answers

How does the quotient $\mathbb{R}/\mathbb{Z}$ become the circle $S^1$?

I came to know that the circle $S^1$ is actually the quotient space $\mathbb{R}/\mathbb{Z}$. But I don't understand how. To my knowledge elements of the quotient space $X/Y$ are of the form $xY$, i.e. the cosets. Right? But how…
Sharabh
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38
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5 answers

Why is $n \bmod 0$ undefined? (by some authors)

I tried to find out what $n$ mod $0$ is, for some $n\in \mathbb{Z}$. Apparently it is an undefined operation - why?
Newb
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When is the product of two quotient maps a quotient map?

It is not true in general that the product of two quotient maps is a quotient maps (I don't know any examples though). Are any weaker statements true? For example, if $X, Y, Z$ are spaces and $f : X \to Y$ is a quotient map, is it true that $ f…
DBr
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30
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3 answers

Topological "Freshman's Dream"

When one learns about quotient and product spaces in topology for the first time, it is perhaps natural to expect that they would behave like mutual inverses: Topological Freshman's Dream (TFD). For a space $X$ and subspace $\emptyset \neq…
YiFan Tey
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29
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Example of quotient mapping that is not open

I have the following definition: Let ($X$,$\mathcal{T}$) and ($X'$, $\mathcal{T'}$) be topological spaces. A surjection $q: X \longrightarrow X'$ is a quotient mapping if $$U'\in \mathcal{T'} \Longleftrightarrow q^{-1}\left( U'\right) \in…
Luc M
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28
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Quotient Space of Hausdorff space

Is it true that quotient space of a Hausdorff space is necessarily Hausdorff? In the book Algebraic Curves and Riemann Surfaces, by Miranda, the author writes: $\mathbb{P}^2$ can be viewed as the quotient space of $\mathbb{C}^3-\{0\}$ by the…
user8186
22
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2 answers

Is $\mathbb{R}/\mathord{\sim}$ a Hausdorff space if $\{(x,y)\!:x\sim y\}$ is a closed subset of $\mathbb{R}\times\mathbb{R}$?

Let $\sim$ be an equivalence relation on a topological space $X$ such that $\{(x,y)\!:x\sim y\}$ is a closed subset of the product space $X\times X$. It is known that if $X$ is a compact Hausdorff space then the quotient space $X/\mathord{\sim}$ is…
Peter Elias
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22
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1 answer

universal property in quotient topology

The following is a theorem in topology: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/\sim$ be the canonical projection. If $g : X → Z$ is a continuous map such that $a \sim b$ implies $g(a) = g(b)$ for…
user9464
21
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0 answers

What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are continuous. $d_1$ and $d_2$ are uniformly…
Keshav Srinivasan
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20
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1 answer

Generalize exterior algebra: vectors are nilcube instead of nilsquare

The exterior product on a ($d$-dimensional) vector space $V$ is defined to be associative and bilinear, and to make any vector square to $0$, and is otherwise unrestricted. Formally, the exterior algebra $\Lambda V$ is a quotient of the tensor…
mr_e_man
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20
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The $n$-disk $D^n$ quotiented by its boundary $S^{n-1}$ gives $S^n$

Define $D^n = \{ x \in \mathbb{R}^n : |x| \leq 1 \}$. By identifying all the points of $S^{n-1}$ we get a topological space which is intuitively homeomorphic to $S^n$. If $n = 2$, this can be visualised by pushing the centre of the disc $D^2$ down…
DBr
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