# 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.

1,940 questions
Filter by
Sorted by
Tagged with
27 views

43 views

### Definition of homology group as quotient in chain complex

I am working through some theory about abelian categories and complexes from "An Introduction to Homological Algebra" by Rotman. I don't understand one of the sections which I will explain ...
• 695
546 views

### Why don't we simply say that the open sets in the quotient topology are projection of open sets of initial topology? [duplicate]

When we define the quotient topology, we say that an open set in it, are those sets which have pre image as an open set. But, why not just define it to be the image of open set of the initial set? ...
231 views

### How do we understand intuitively how the quotient topology changes as we make the relation bigger or smaller?

Provided a relation on the set of a topological space, we can turn that relation into an generated equivalence relation, and hence induce a quotient topology on the quotient set. The open sets of the ...
22 views

### Feedback on and assistance with this proof about a particular quotient space of $\mathbb{C}P^1$

The goal here is to define the particular equivalence relation I'm attempting to describe, and then provide an equation (in this case, (2)) that can be used to determine whether or not two given ...
• 887
53 views

### Embedding the space to be attached into the adjunction

(Note that I am NOT asking about the space onto which glueing is done.) I am skeptical of the following statement: Let $A\subseteq X$ and $f\colon A\to Y$ be continuous and injective. Then $X$ embeds ...
• 4,057
56 views

### Two dimensional cylinder

I am reading a paper referring a two dimensional cylinder, if i understand the definition is 2d cylinder= $\mathbb{R} \times \mathbb{R}/ \mathbb{Z}$ What's the intuition behind this? Does anyone have ...
30 views

### Axler Example 3.100 on quotient spaces

In Axler's linear algebra textbook, he gives an example of a quotient space as $\mathbb{R}^3/U$, where $U$ is a line in $\mathbb{R}^3$ containing the origin. The quotient, he claims, is the set of ...
• 1,243
25 views

### Inequality relating quotient norm and norm

In an comment under an answer to this question How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? it is claimed that we have the ...
• 15
61 views

117 views

### Is $\mathbb R/\mathbb N$ homeomorphic to $\mathbb R/\mathbb Z?$

Where $\mathbb R/\mathbb N$ is the quotient topological space, where $\mathbb N$ is collapsed onto a point. Same with $\mathbb R/\mathbb Z.$ I was thinking I could prove it wasn't homeomorphic using ...
50 views

### Is there a name for the arc $\mathbb{S}^1 / (x \sim360 - x)$

I was playing with some ideas in a vague way and I have encountered this structure that arises from taking the space of angles $\mathbb{S}^1$ and quotienting it by the relation $(x, 360-x)$ (here $360$...
• 17.5k
186 views
+200

### How does the quotient affect the complex valued metric?

Take a probability distribution without the normalization factor $f_t(x)=e^{{tL}}$ for suffcient statistic $L(x)=\frac{1}{\log x}.$ Take a map from the open unit interval $\mathcal M: I \to \Bbb C$ ...
• 752
77 views

### Coherent topology with subspaces

EXAMPLE 2. Let $J$ be a discrete space, and let $E = [0,1] \times J$. Then the quotient space $X$ obtained from $E$ by collapsing the set $\{0\} \times J$ to a point $p$ is a linear graph. ...
45 views

### When $S^d/X$ is a manifold $S^d-X$ is homology equivalent to a point?

Upon reviewing for an algebraic topology final, I have found the following question which has stumped me: Let $S^d$ be a sphere of dimension $d\geq 1$, and $X\subset S^d$ a proper subset of $S$ such ...
• 3,431
33 views

1 vote
42 views

### Topological Quotients: Understanding $X/\sim$ and $X/Y$ with Insights into the disk Structure.

I could use some assistance in clarifying a concept. In topology, when we have a space denoted as $X$, we can create a quotient space (a space of equivalence classes) denoted as $X/\sim$, where $\sim$ ...
• 123
1 vote
106 views

### Is the adjunction space of two Hausdorff spaces also Hausdorff?

I was reading the definition of CW-complex in terms of pushouts given by Lück's Algebraische Topologie: Homologie und Mannigfaltigkeiten (Chapter 3). It is stated (though not proven) that such a ...
50 views

• 1,344
1 vote
28 views

### Reference for quotient lattices and universal property?

Question: Are there any references explaining the definition (or definitions) of quotient objects in the category of lattices? In particular a characterization in terms of universal properties would ...
103 views

### Free Commutative Monoid Quotient by Relations?

Say I have a commutative monoid $M$ that is generated by three elements $A,B,C$, where I have that $A+C=2B$. I want to write this a free (does that even mean anything?) monoid $\mathbb N^3$ with ...
• 3,431
1 vote
55 views

• 7,789
87 views

### Does $Spec(k[G/[P,P]])$ have worse than quotient singularities?

Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a connected reductive linear algebraic group over $k$, and let $P$ be a parabolic subgroup of $G$. So we have the flag ...
• 13.6k
44 views

### Different topologies for the real projective base as a manifold

I am stuck with the following problem. I am given the $d$-dimensional real projective space $\mathbb{R}P^d$ as the set of equivalence classes of lines in $\mathbb{R}^{d+1}$, i.e. \...
• 101
### The norm on $\mathbb{R}^2$ is a quotient map
I think this is a simple example of a quotient map: $$f:\mathbb{R}^2 \rightarrow \mathbb{R}_{\geq 0} \text{ defined by } f\left(x\right) = \left|x\right|$$ where $\mathbb{R}^2$ and \$\mathbb{R}_{\geq ...