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
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
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 by a subspace (example 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 be a topological space and let
be an equivalence relation on its underlying set. Then the quotient topological space has
and
a subset is declared to be an open subset precisely if its preimage under the canonical projection map
is open in .
To see that this indeed does define a topology on 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 with . 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 by the function , see at Top – Universal constructions.
For this construction the function 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 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 by the equivalence relation which identifies the two endpoints of the open interval.
By the universal property of the quotient it follows that is a continuous function. Moreover, it is a bijection on the underlying sets by the -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 are unions of
the open intervals with ,
the half-open intervals and with
and since the projection map is injective on , the open subsets of are unions of
the open intervals with ,
the glued half-open intervals for .
By the -periodicity of , the image of the latter under is the same as the image of . Since the function is clearly an open map, it follows that the images of these open subsets in are open.
(quotient by a subspace)
Let be a topological space and a non-empty subset. Consider the equivalence relation on which identifies all points in with each other. The resulting quotient space (def. ) is often simply denoted .
Notice that is canonically a pointed topological space, with base point the equivalence class of .
If is the empty space, then one defines
to be the disjoint union space of with the point space. This is no longer a quotient space, but both constructions are unified by the pushout along the map , 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 equipped with their Euclidean metric topology. Consider on the equivalence relation which identifies all real numbers that differ by a rational number:
Then the quotient space is a codiscrete topological space.
We need to check that the only open subsets of are the empty set and the entire set .
So let be a non-empty subset.
Write for the quotient projection. By definition is open precisely if its pre-image is open. By the Euclidean topology, this is the case precissely if is a union of open intervals. Since by assumption is non-empty, it contains at least one open interval , with . By the density of the rational numbers, there exists a rational number with
By definition of we have for all that all elements in are -equivalent to elements in , hence that also . But the union of these open intervals is all of
and so .
Recall that a map is open if is open in whenever is open in . It is not the case that a quotient map is necessarily open. Indeed, the identification map , where the endpoints of are identified with , takes the open point of the domain to a non-open point in .
Nor is it the case that a quotient map is necessarily a closed map; the classic example is the projection map , which projects the closed locus onto a non-closed subset of . (This is a quotient map, by the next remark.)
It is easy to prove that a continuous open surjection is a quotient map. For instance, projection maps are quotient maps, provided that is inhabited. Likewise, a continuous closed surjection is a quotient map: is open is closed is closed is open. For example, a continuous surjection from a compact space to a Hausdorff space is a quotient map.
whose underlying function is surjective exhibits as the corresponding quotient topology precisely if sends open and -saturated subsets in to open subsets of . By this lemma this is the case precisely if it sends closed and -saturated subsets to closed subsets.
Sullivan model of finite G-quotient?
examples of universal constructions of topological spaces:
Last revised on October 17, 2019 at 08:38:59. See the history of this page for a list of all contributions to it.