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In topology and functional analysis, a quotient space is (intuitively speaking) the result of
identifying or "gluing together" certain points of some other space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct
new spaces from given ones. The operation can be thought of (very informally indeed) as the act of "dividing" the input space by
the equivalence relation, hence both the name "quotient", and the notation, which are both reminiscent of division.
Quotient of a topological space by an equivalence relation
Formally, suppose X is a topological space and ~ is
an equivalence relation on X. We define a topology on the quotient
set X/~ (the set consisting of all equivalence
classes of ~) as follows: a set of equivalence classes in X/~ is open
if and only if their union is open in X.
Examples
Consider the set X = R of all real numbers with
the ordinary topology, and write x ~ y iff x-y is an
integer. Then the quotient space X/~ (also written as
R/Z) is homeomorphic to the unit circle S1.
As another example, consider the unit square X = [0,1]×[0,1] and the equivalence relation ~ generated by the
requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then
X/~ is homeomorphic to the unit sphere S2.
Properties
Let p : X → X/~ be the projection map which sends each element of X to its
equivalence class. The map p is continuous; in fact, the topology on
X/~ is the finest (the one with the most open sets) which makes p continuous. The map p is in general
not open.
If Y is some other topological space, then a function f : X/~ → Y is continuous
if and only if fop is continuous.
If g : X → Y is a continuous map with the property that a~b implies
g(a)=g(b), then there exists a unique continuous map h : X/~ →
Y such that g = hop.
The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on
X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This
criterion is constantly being used when studying quotient spaces.
Compatibility with other topological notions
- Separation
- Connectedness
- Compactness
- If a space is compact, then so are all its quotient spaces.
- A quotient space of a locally compact space need not be locally
compact.
- please add more results like this
Quotient of a vector space by a subspace
If X is a vector space, then the quotient space can sometimes
also be seen as a vector space.
Specifically, let X be a vector space over K,
where either K = R or K = C, and let M be a subspace of X. We define an equivalence relation ~ on X by
stating that x ~ y if x − y ∈ M. Let [x] denote the
equivalence class containing x. We define scalar
multiplication and addition on the equivalent classes by α [x] = [αx] and
[x] + [y] = [x+y]. This definition makes sense, because the equivalence classes on
the right-hand side do not depend on the elements chosen to represent the equivalence classes on the left-hand side. This turns
the quotient space X/M into a vector space.
Quotient of a Banach space by a subspace
If X is a Banach space and M is a closed subspace of
X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector
space structure by the construction of the previous section. We define a norm on X/M by
-
The quotient space X/M is complete with respect to
the norm, so it is a Banach space.
Examples
Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1]. Denote the subspace of
all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some
function g is determined by its value at 0, and the quotient space C[0,1] / M is isomorphic
to R.
If X is a Hilbert space, then the quotient space
X/M is isomorphic to the orthogonal complement of M.
See also: Quotient group
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