# nLab quotient space

### Context

#### Topology

topology

algebraic topology

# Quotient spaces

## Idea

A quotient space is a quotient object in some category of spaces, such as Top (of topological spaces), or Loc (of locales), etc.

Quotient objects in the category $Vect$ of vector spaces also traditionally use the term ‘quotient space’, but they are really just a special case of quotient modules, very different from the other kinds of quotient space. Quotient TVSes, however, combine both aspects.

## Definitions

### In $Top$

Let $X$ be a topological space and $\sim$ an equivalence relation on (the underlying set of) $X$. (Since monomorphisms in Top are just injective continuous maps, to give an equivalence relation on the underlying set of a topological space is the same as to give a congruence on that space in $Top$.) Let $Y = X/{\sim}$ be the quotient set and $q\colon X\to Y$ the quotient map.

The quotient topology, or identification topology, induced on $Y$ from $X$ says that a subset $U\subseteq Y$ is open if and only if $q^{-1}(U)\subseteq X$ is open. With this topology $Y$ is a quotient space or identification space of $X$.

Obviously, up to homeomorphism, all that matters is the surjective function $X\to Y$. For the above definition, we don’t even need it to be surjective, and we could generalize to a sink instead of a single map; in such a case one generally says final topology or strong topology. See topological concrete category.

Remarks:

1. Recall that a map $q \colon X \to Y$ is open if $q(U)$ is open in $Y$ whenever $U$ is open in $X$. It is not the case that a quotient map $q \colon X \to Y$ is necessarily open. Indeed, the identification map $q \colon I \sqcup \{\ast\} \to S^1$, where the endpoints of $I$ are identified with $\ast$, takes the open point $\ast$ of the domain to a non-open point in $S^1$.

2. Nor is it the case that a quotient map is necessarily a closed map; the classic example is the projection map $\pi_1 \colon \mathbb{R}^2 \to \mathbb{R}$, which projects the closed locus $x y = 1$ onto a non-closed subset of $\mathbb{R}$. (This is a quotient map, by the next remark.)

3. It is easy to prove that a continuous open surjection $p \colon X \to Y$ is a quotient map. For instance, projection maps $\pi \colon X \times Y \to Y$ are quotient maps, provided that $X$ is inhabited. Likewise, a continuous closed surjection $p: X \to Y$ is a quotient map: $p^{-1}(U)$ is open $\Rightarrow$ $p^{-1}(\neg U)$ is closed $\Rightarrow$ $p(p^{-1}(\neg U)) = \neg U$ is closed $\Rightarrow$ $U$ is open. For example, a continuous surjection from a compact space to a Hausdorff space is a quotient map.

### In $Loc$

A quotient space in $Loc$ is given by a regular subobject in Frm.

(More details needed.)

Revised on May 25, 2014 20:06:40 by Todd Trimble (67.81.95.215)