# 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.

1,884 questions
Filter by
Sorted by
Tagged with
1 vote
31 views

### How does the wedge product appear in the quotient approach to constructing an exterior algebra?

Let $V$ be a finite dimensional vector space and let $T(V)$ denote the tensor algebra $$T(V) = \bigoplus_{k=0}^\infty T^k(V)$$ where $$T^k(V) = \underbrace{V \otimes \cdots \otimes V}_k.$$ Let $I(V)$ ...
1 vote
58 views

### Closed curve in a torus diffeomorphic to a circle?

In Example 15.9 of Tu's "An Introduction to Manifolds: Second Edition", it is written Let G be the torus $R^2/Z^2$ and L a line through the origin in R2. The torus can also be represented ...
1 vote
61 views

### G-invariant Tietze Extension Theorem

The Tietze Extension Theorem states that if $V$ is a closed subset of a metric space $X$ and $f$ is a continuous function on $V$ then there exists $F$ continuous on $X$ such that $F|_{V} = f$. Suppose ...
24 views

### X/A with respect to the quotient topology is Hausdorff?

Consider $X = [0,1]$ and $A = (a,b)$ (where $0 < a < b < 1$). I want to prove $X/A$ is connected and compact. Here's my approach : If I take an open cover of $X/A$, the corresponding cover in ...
102 views

### Can every "noodly" space be decomposed into one or more manifolds?

The Premise: Let $X$ be a topological space. For any pair of points $x_0,x_1\in X$, let $\mathrm N_X(x_0,x_1)$ be the set of all subspaces $S$ of $X$ such that $x_0,x_1\in S$ and $S\cong I$ for some ...
45 views

### Set of positive definite Hermitian matrices as quotient

Let $$\mathcal{H}_n^{++} := \{ A \in \mathrm{Mat}_n(\mathbb{C}) \vert A = A^{\dagger} \text{ and } A > 0 \}$$ denote the space of positive definite Hermitian matrices. I have read in some ...
94 views

### Topological Quotient vs Group Quotient [closed]

I was wondering about if topological quotients and group quotients always agree for a topological group. By this, I mean if G is a topological group, G/~ is a topological quotient space, G/N is a ...
1k views

### When can we "switch" isomorphic things

I'm finishing a course on basic abstract algebra, which covers groups, rings, modules, and finite group representations. However, up to this point, I am not very sure about the concept of isomorphism. ...
13 views

### How to identify the number of degrees of freedom in a system of particles using the Principle of Relativity

So consider the following laws of motion in an inertial system : $$m_k \partial_{tt} x_k = - \partial_{x_k} V(x_1, .., x_k)$$ where $x_i \in \mathbb{R}^3$ is a particle of mass $m_i$. By principle of ...
48 views

83 views

### Definition of Universal enveloping algebra of a lie algebra

I am a physics student, and I am trying to understand the Casimir operator from a formal perspective; therefore, I come to learn what's Universal enveloping algebra. Two definitions of Universal ...
31 views

### When is a quotient a differentiable manifold?

The two most popular ways to generate a topological manifold from others are picking a subspace of a topological space and computing the quotient space of a topological space. In differential geometry,...
44 views

### Does the quotient induced topology coincide with the "corresponding" topology on the quotient space?

Let $X$ be a Banach space, and $Y$ a closed subspace, and let the quotient map be $q : X \to X/Y$. Now, if $X$ and $X/Y$ are given the norm topology (where the norm on $X/Y$ is $\|x+Y\| = d(x,Y)$), ...
1 vote
88 views

54 views

### What does it mean to 'attach a cone along a subspace'?

I am a little bit confused about the terminology used in Section 2.4 of this paper. We have a finite pointed CW complex $B$ and a subcomplex $A$. Then "let $C$ be obtained from $B$ by attaching a ...
37 views

### Upper bound the dimension of Quotient Space

Let $X$ be a Banach space and let $Y$ be the kernel of a bounded linear functional on $X$, say $M: X \rightarrow \mathbb{R}$. Is it possible to upper-bound the dimension of the quotient space $X/Y$. I ...
72 views

### Sheaf cohomology on quotient stacks

Suppose I have a scheme $X$ over $\mathbb C$, acted on by a finite group $G$. Let $\mathcal F$ be a $G$-equivariant coherent sheaf of $\mathcal O_X$-modules. Then I can form the stack quotient $[X/G]$,...
1 vote
75 views

1 vote
55 views

### Quotient of projective group scheme by a finite group action

Let $X$ be a projective group scheme, over some base $S$, and let $G$ be a finite group acting on $X$ by $S$-isomorphisms. I would like to understand if/when the quotient $X/G$ is representable by a ...
1 vote
50 views

1 vote
31 views

### Restriction of quotient spaces.

Let $V_{1,2}$ be two $\mathbb{R}$-vector spaces and $U_{1,2}\subset V_{1,2}$ two linear subspaces. If $f:V_{1}\to V_{2}$ is a linear map such that $f(U_{1})\to U_{2}$, it induces a well-defined map on ...
1 vote
104 views

### What are cells in CW complex?

I was studying Hatcher's Algebraic Topology regarding CW complex. I am having trouble in understanding definition of CW Complex. (1) Start with a discrete set $X^0$, the $0$-cells of $X$. (2) ...
### Characterization of torus $I/\mathbb{Z}^{d}$?
The quotient space $\mathbb{R}/\mathbb{Z}$ be defined by the set of all equivalence classes with respect to the equivalence relation $x \sim y \iff x-y \in \mathbb{Z}$. This quotient space is then ...
I was wondering if this methodology is sound. I've checked that for any element in $\mathbb{Z}[3i]$ if we take the quotient map with ideal $(3i)$, its kernel consists of the elements $\{ 9a+3bi \}$. ...