In
general topology
In mathematics, general 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 differential topology, geometri ...
and related areas of
mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s is a space formed by equipping the
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of the underlying sets with a
natural topology
In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that ...
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
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the sou ...
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, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
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
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as th ...
on ''X'' for which all the canonical injections
are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
(i.e.: it is the
final topology
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
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'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
in ''X''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
its
preimage 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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
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 fr ...
: 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
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
:
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 coproduc ...
in the
category of topological spaces. 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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
''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
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, a ...
. 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 giv ...
s so that each ''X''
''i'' may be canonically thought of as a
subspace of ''X''.
Examples
If each ''X''
''i'' is
homeomorphic 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 top ...
.
Preservation of topological properties
* Every disjoint union of
discrete spaces 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 ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
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-s ...
, the dual construction
*
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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
{{DEFAULTSORT:Disjoint Union (Topology)
General topology