In the mathematical field of topology a direct sum, direct disjoint sum or coproduct is an important universal construction for topological spaces. The canonical topology on the newly constructed space is called direct sum topology. Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

The dual construction is called topological product. 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...

Definition

Given two topological spaces (X₁,τ₁) and (X₂,τ₂) we call

the disjunct set union of X₁ and X₂.

The functions

defined as

and

defined as

are called canonical injections.

The direct sum of two topological spaces is defined as

(X₁,τ₁) + (X₂,τ₂): = (X₁ + X₂,τ_{1 + 2})

with the direct sum topology τ_{1 + 2} defined as

The direct sum topology is the finest topology such that the canonical injections are continuous.

Preservation of topological properties

• the direct sum of two topological spaces is disconnected

See also

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