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.

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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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$X/{\sim}$ 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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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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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
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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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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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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…
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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…
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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…
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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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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…
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About direct sum of abelian groups and quotient

I'm trying to understand properly the relations between quotient and direct sum. The first thing I wanted to know, and couldn't find online, is whether my guess is true or not: Assume $G_\alpha$ are abelian groups, and $H_\alpha \leq G_\alpha$ a…
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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 Hausdorff. Is there a more "intrinsic"…
tomasz
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Why is addition defined, and not implied, on quotient spaces?

Small question. In chapter 3, section E, page 96 of "Linear Algebra Done Right", addition in quotient vector spaces is defined this way: I understand why scalar multiplication has to be defined, because multiplying a subset of a vector space with a…
ludo1337
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Can the fundamental group detect all ways to not have a section?

A common homework problem in topology classes is to find a quotient map $p : X \to Y$ which does not admit a (continuous) section $s : Y \to X$. The standard example of such a phenomenon is the map $[0,1] \to S^1$ which identifies the endpoints (or…
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