In
general topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
and related areas of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the disjoint union
(also called the direct sum, free union, free sum, topological sum, or coproduct) of a
family
Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
of
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s is a space formed by equipping the
disjoint union
In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
of the underlying sets with a
natural topology called the disjoint union topology. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.
The name ''coproduct'' originates from the fact that the disjoint union is the
categorical dual of the
product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
construction.
Definition
Let be a family of topological spaces indexed by ''I''. Let
:
be the
disjoint union
In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
of the underlying sets. For each ''i'' in ''I'', let
:
be the canonical injection (defined by
). The disjoint union topology on ''X'' is defined as the
finest topology on ''X'' for which all the canonical injections
are
continuous (i.e.: it is the
final topology
In general topology and related areas of mathematics, the final topology (or coinduced, weak, colimit, or inductive topology) on a Set (mathematics), set X, with respect to a family of functions from Topological space, topological spaces into X, is ...
on ''X'' induced by the canonical injections).
Explicitly, the disjoint union topology can be described as follows. A subset ''U'' of ''X'' is
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979
* ''Open'' (Go ...
in ''X''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
its
preimage
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y.
More generally, evaluating f at each ...
is open in ''X''
''i'' for each ''i'' ∈ ''I''. Yet another formulation is that a subset ''V'' of ''X'' is open relative to ''X''
iff
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both ...
its intersection with ''X
i'' is open relative to ''X
i'' for each ''i''.
Properties
The disjoint union space ''X'', together with the canonical injections, can be characterized by the following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: If ''Y'' is a topological space, and ''f
i'' : ''X
i'' → ''Y'' is a continuous map for each ''i'' ∈ ''I'', then there exists ''precisely one'' continuous map ''f'' : ''X'' → ''Y'' such that the following set of diagrams
commute:
This shows that the disjoint union is the
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...
in the
category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...
. It follows from the above universal property that a map ''f'' : ''X'' → ''Y'' is continuous
iff
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both ...
''f
i'' = ''f'' o φ
''i'' is continuous for all ''i'' in ''I''.
In addition to being continuous, the canonical injections φ
''i'' : ''X''
''i'' → ''X'' are
open and closed maps. It follows that the injections are
topological embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
s so that each ''X''
''i'' may be canonically thought of as a
subspace of ''X''.
Examples
If each ''X''
''i'' is
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to a fixed space ''A'', then the disjoint union ''X'' is homeomorphic to the
product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
''A'' × ''I'' where ''I'' has the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
.
Preservation of topological properties
* Every disjoint union of
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
s is discrete
*''Separation''
** Every disjoint union of
T0 spaces is T
0
** Every disjoint union of
T1 spaces is T
1
** Every disjoint union of
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
s is Hausdorff
*''Connectedness''
** The disjoint union of two or more nonempty topological spaces is
disconnected
See also
*
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
, the dual construction
*
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
and its dual
quotient topology
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
*
topological union, a generalization to the case where the pieces are not disjoint
References
{{DEFAULTSORT:Disjoint Union (Topology)
General topology