In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Euclid, detail from The School of Athens by Raphael. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction. In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...

Definition

Let {X_i : i ∈ I} be a family of topological spaces indexed by I. Let In mathematics, it is a common practice to index or label a collection of objects by some set I called an index set. ...

be the disjoint union of the underlying sets. For each i in I, let In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

be the canonical injection. The disjoint union topology on X is defined as the finest topology on X for which the canonical injections are continuous (i.e. the final topology for the family of functions {φ_i}). In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In topology and related areas of mathematics, the final topology on a set is the strongest topology to make a family of functions into continuous. ...

Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in X if and only if its preimage is open in X_i for each i ∈ I. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...

Properties

The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: 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: In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f : X → Y is continuous iff f_i = f o φ_i is continuous for all i in I. In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... The category Top has topological spaces as objects and continuous maps as morphisms. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

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 embeddings so that each X_i may be canonically thought of as a subspace of X. In topology, an open map is a function between two topological spaces which maps open sets to open sets. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

Examples

If each X_i is homeomorphic to a fixed space A, then the disjoint union X will be homeomorphic to A × I where I is given the discrete topology. This word should not be confused with homomorphism. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

Preservation of topological properties

• every disjoint union of discrete spaces is discrete
• Separation
  • every disjoint union of T₀ spaces is T₀
  • every disjoint union of T₁ spaces is T₁
  • every disjoint union of Hausdorff spaces is Hausdorff
• Connectedness
  • the disjoint union of two or more topological spaces is disconnected

In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... The title given to this article is incorrect due to technical limitations. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

See also

• product topology, the dual construction
• subspace topology and its dual quotient topology