Disjoint union

Definition

For two spaces

Suppose $X$ and $Y$ are topological spaces. The disjoint union of $X$ and $Y$ , denoted $X ⊔ Y$ , is defined as follows:

As a set, it is a disjoint union of the spaces $X$ and $Y$ , i.e., it is a union of two subsets with trivial intersection, identified with $X$ and $Y$ respectively.
A subset $U$ of $X ⊔ Y$ is defined to be an open subset if its intersection with $X$ is open in the $X$ -copy and its intersection with $Y$ is open in the $Y$ -copy.

Note that disjoint union in particular means that, even if $X = Y$ , the space $X ⊔ Y$ contains two separate copies of the space.

The topology on the disjoint union is a special case of a coherent topology.

For an arbitrary number of spaces

Suppose $I$ is an indexing set, and $X_{i}, i \in I$ , are all topological spaces. The disjoint union $⨆_{i \in I} X_{i} < / m a t h . i s d e f i n e d a s f o l l o w s : * A s a s e t, i t i s t h e d i s j o i n t u n i o n o f t h e s p a c e s < m a t h > X_{i}$ . In other words, it is the union pairwise disjoint subspaces, each identified with the different $X_{i}$ s.

A subset $U$ of $⨆_{i \in I} X_{i}$ is defined to be open if the intersection of $U$ with the $X_{i}$ -copy is an open subset of $X_{i}$ .

As a coproduct

Disjoint union is the coproduct in the category of topological spaces.

Algebraic topology

Homology groups

Further information: homology of disjoint union is direct sum of homologies

Cohomology groups

Further information: cohomology of disjoint union is direct sum of cohomologies

Homotopy groups

When we take a disjoint union, the path component of a point in the disjoint union is the same as the path component inside whichever piece it originated from. Since homotopy groups, including the fundamental group, depend only on the homeomorphism type of the path component, the homotopy groups inside the disjoint union at a basepoint remain the same as the homotopy groups inside whichever piece the basepoint comes from.