This entry is about the concept in topology. For quotient vector spaces in linear algebras see there.
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A quotient space is a quotient object in some category of spaces, such as Top (of topological spaces), or Loc (of locales), etc.
Often the construction is used for the quotient $X/A$ by a subspace $A \subset X$ (example 2 below).
Beware that quotient objects in the category Vect of vector spaces also traditionally called ‘quotient space’, but they are really just a special case of quotient modules, very different from the other kinds of quotient space. However in topological vector spaces both concepts come together.
(quotient topological space)
Let $(X,\tau_X)$ be a topological space and let
be an equivalence relation on its underlying set. Then the quotient topological space has
and
a subset $O \subset X/\sim$ is declared to be an open subset precisely if its preimage $\pi^{-1}(O)$ under the canonical projection map
is open in $X$.
To see that this indeed does define a topology on $X/\sim$ it is sufficient to observe that taking pre-images commutes with taking unions and with taking intersections.
Often one considers this with input datum not the equivalence relation, but any surjection
of sets. Of course this identifies $Y = X/\sim$ with $(x_1 \sim x_2) \Leftrightarrow (\pi(x_1) = \pi(x_2))$. Hence the quotient topology on the codomain set of a function out of any topological space has as open subsets those whose pre-images are open.
Equivalently this is the final topology or strong topology induced on $Y$ by the function $X \to Y$, see at Top – Universal constructions.
For this construction the function $X \to Y$ need not even 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 also at topological concrete category.
A quotient space in $Loc$ is given by a regular subobject in Frm.
(More details needed.)
from the closed interval with its Euclidean metric topology to the unit circle equipped with the subspace topology of the Euclidean plane
descends to a homeomorphism on the quotient space $[0,2 \pi]/(0 \sim 2 \pi)$ by the equivalence relation which identifies the two endpoints of the open interval.
By the universal property of the quotient it follows that $[0,2\pi]/(0 \sim 2\pi) \to S^1$ is a continuous function. Moreover, it is a bijection on the underlying sets by the $2\pi$-periodicity of sine and coside. Hence it is sufficient to see that it is an open map (by this prop.).
Since the open subsets of $[0,2\pi]$ are unions of
the open intervals $(a,b)$ with $0 \lt a \lt b \lt 2\pi$,
the half-open intervals $[0,b)$ and $(a,2\pi]$ with $0 \lt a,b \lt 2\pi$
and since the projection map $\pi \colon [0,2\pi] \to [0,2\pi]/(0 \sim 2\pi)$ is injective on $(0, 2\pi)$, the open subsets of $[0,2\pi]/(0 \sim 2\pi)$ are unions of
the open intervals $(a,b)$ with $0 \lt a \lt b \lt 2\pi$,
the glued half-open intervals $(b,2\pi]/(0\sim 2\pi) \cup [0,a)/(0 \sim 2\pi)$ for $0 \lt a,b \lt 2\pi$.
By the $2\pi$-periodicity of $(cos(-),sin(-))$, the image of the latter under $(cos(-),sin(-))$ is the same as the image of $(b, 2\pi + a)$. Since the function $(cos(-),sin(-)) \colon \mathbb{R} \to S^1$ is clearly an open map, it follows that the images of these open subsets in $S^1$ are open.
(quotient by a subspace)
Let $X$ be a topological space and $A \subset X$ a non-empty subset. Consider the equivalence relation on $X$ which identifies all points in $A$ with each other. The resulting quotient space (def. 1) is often simply denoted $X/A$.
Notice that $X/A$ is canonically a pointed topological space, with base point the equivalence class $A/A \subset X/A$ of $A$.
If $A = \emptyset$ is the empty space, then one defines
to be the disjoint union space of $X$ with the point space. This is no longer a quotient space, but both constructions are unified by the pushout $i \colon A \to X$ along the map $A \to \ast$, equivalently the cokernel of the inclusion:
This kind of quotient space plays a central role in the discussion of long exact sequences in cohomology, see at generalized (Eilenberg-Steenrod) cohomology.
Consider the real numbers $\mathbb{R}$ equipped with their Euclidean metric topology. Consider on $\mathbb{R}$ the equivalence relation which identifies all real numbers that differ by a rational number:
Then the quotient space $\mathbb{R}/\sim_{\mathbb{Q}}$ is a codiscrete topological space.
We need to check that the only open subsets of $X/\sim_{\mathbb{Q}}$ are the empty set and the entire set $X/\sim_{\mathbb{Q}}$.
So let $U \subset \mathbb{R}/\sim$ be a non-empty subset.
Write $\pi \colon \mathbb{R} \to \mathbb{R}/\sim_{\mathbb{Q}}$ for the quotient projection. By definition $U$ is open precisely if its pre-image $\pi^{-1}(U) \subset \mathbb{R}$ is open. By the Euclidean topology, this is the case precissely if $\pi^{-1}(U)$ is a union of open intervals. Since by assumption $\pi^{-1}(U)$ is non-empty, it contains at least one open interval $(a,b) \subset \mathbb{R}$, with $a \lt b$. By the density of the rational numbers, there exists a rational number $q \in \mathbb{Q} \subset \mathbb{R}$ with
By definition of $\sim_{\mathbb{Q}}$ we have for all $n \in \mathbb{Z}$ that all elements in $(a + n q, b + n q) \subset \mathbb{R}$ are $\sim_{\mathbb{Q}}$-equivalent to elements in $(a,b)$, hence that also $(a+q,b+q) \subset \pi^{-1}(U)$. But the union of these open intervals is all of $\mathbb{R}$
and so $\pi^{-1}(U) = \mathbb{R}$.
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 $S^1$.
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.)
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.
whose underlying function $f \colon X \longrightarrow Y$ is surjective exhibits $\tau_Y$ as the corresponding quotient topology precisely if $f$ sends open and $f$-saturated subsets in $X$ to open subsets of $Y$. By this lemma this is the case precisely if it sends closed and $f$-saturated subsets to closed subsets.
examples of universal constructions of topological spaces:
$\phantom{AAAA}$limits | $\phantom{AAAA}$colimits |
---|---|
$\,$ point space$\,$ | $\,$ empty space $\,$ |
$\,$ product topological space $\,$ | $\,$ disjoint union topological space $\,$ |
$\,$ topological subspace $\,$ | $\,$ quotient topological space $\,$ |
$\,$ fiber space $\,$ | $\,$ space attachment $\,$ |
$\,$ mapping cocylinder, mapping cocone $\,$ | $\,$ mapping cylinder, mapping cone, mapping telescope $\,$ |
$\,$ cell complex, CW-complex $\,$ |