# 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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### Quotient Topology, equivalence relation. I need help to prove if X is homeomorphic to X/~

Let $X = [0,3] \subset\mathbb{R}$ and consider the following equivalence relation: $$x\sim y \Leftrightarrow x=y \vee x,y \in [1,2]$$ and call $Y = X/\sim$. (1). Establish if $Y$ is ...
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### $T_0$-Identification of a topological space

In Willard's General Topology, section 13.2.c, for any topological space $X$ is defined a quotient space $X/\sim$ such that $x \sim y$ iff $cl(\{x\})=cl(\{y\})$ where $cl(.)$ is the topological ...
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### Proving a map in a commutative diagram is continuous.

Suppose $G$ and $H$ are topological groups, with $H \subset G$. I have the following commutative diagram with $f$ being continuous and $p$ being an open surjection (canonical projection). Does this ...
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### Question about prime ideals and Quotient rings

I`m struggling with the following solution I wanted to use: Let S := $\mathbb{C}[x,y,z] /(z^2-xy) \cong \mathbb{C}[x,y,\sqrt{xy}]$ Then $S/(x-y) \cong \mathbb{C}[x,x,\sqrt{x^2}] = \mathbb{C}[x]$, ...
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### A problem about invariant and quotient space

Assume V is finite dimensional,that $T:V \rightarrow V$ is a linear operator, and W is a T-invariant subspace. Define $\bar{T}: V/W \rightarrow V/W$ by $\bar{T}(v+W)=T(v)+W$ for any $v+W \in V/W$. Let ...
Consider the Möbius strip defined by the following equivalence relation on the subspace $[0,1]\times]-a,a[$ of $\mathbb{R}^2$: $$(x,y)\sim (x',y')\implies (x,y)=(x',y')\vee|x'-x|=1\:\text{and}\:y'... 1answer 23 views ### Quotient Topology of Multiplication Map We have f(x,y)=xy where f:\mathbb{R}_{[-)} \times \mathbb{R}_{[-)} \to \mathbb{R}. What is the quotient topology? I know that the topological basis in \mathbb{R}_{[-)} \times \mathbb{R}_{[-)} ... 2answers 40 views ### Understanding the 2-torus as a definition of equivalence classes Note: My questions are marked in boldface. In Devaney's book "An Introduction to Chaotic Dynamical Systems", Westview Press (2003), on p. 190 he defines the 2-torus by To describe the torus, let ... 0answers 26 views ### Dimension of quotient space and basis. I'm quickly reviewing the concept of quotient space. Suppose V is a vector space of dimension m and W is a vector subspace of V. Then the quotient vector space is defined as V/W, which is a ... 0answers 33 views ### Relative GIT quotient. I´m studying Mumford's geometric invariant theory and I came to the following question. Let us suppose that k is an algebraically closed field of characteristic 0, and G is a reductive linear ... 0answers 27 views ### The maximal normal subgroup in U(n) and SU(n) Let U(n) be a the unitary group the quotient$$\frac{U(n)}{U(n-1)} \simeq S^{2n-1}$$and I believe that$$\frac{SU(n)}{SU(n-1)} \simeq S^{2n-1}.$$Obviously the U(n-1) is not a normal subgroup ... 0answers 28 views ### The magic properties of the quotient space from U(2^{l-1})/{\rm Spin}(2 l) Inspired by a previous post, Embed a Spin group to a special unitary group I am wondering what are the magic properties of the quotient space from$$U(2^{l-1})/{\rm Spin}(2 l)$$that makes such an ... 1answer 40 views ### Quotient space (vector space) and linear maps my professor give an exercise about quotient spaces envolving vector spaces and I have some questions about that. Consider f:V \to V a linear map and W \subset V a subspace. Take \phi: V/W \to ... 1answer 99 views ### Quotient manifold of a finite group action Let R be a (crystallographic) root system on an Euclidian space (E,⟨−,−⟩) and$$W:=gen\{σ_r∣r∈R\}=gen\{σ_r∣r∈R^+\}$$its associated reflection group. Taking$$M=E-\cup_{r\in R^+} H_r,$$where H_r ... 1answer 56 views ### Describe the elements of the quotient space and find a basis for it Describe the elements of the quotient space of V/S and also find a basis for it. V = P_{10}, the space of all real polynomials of degree at most 10,$$S=\operatorname{span}(\{{x,x^3,x^5,x^7,x^9}\}...
Let $X$ be a compact Hausdorff space with a faithful continuous $S^1$-action. Further, assume that the orbit space is $[0,1]\times[0,1]$, and denote the quotient map by $\pi:X\to[0,1]\times[0,1]$. Now,...