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Direct sum topology

(Redirected from Disjoint union (topology))

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.

The dual construction is called topological product.

Definition

Given two topological spaces $(X 1,τ 1)$ and $(X 2,τ 2)$ we call

$X_1 + X_2 := X_1 \times \lbrace 1 \rbrace \cup X_2 \times \lbrace 2 \rbrace$

the disjunct set union of $X 1$ and $X 2$ .

The functions

$i_1: X_1 \to X_1 + X_2$

defined as

$i_1: x_1 \mapsto (x_1,1) \mbox{ , } x_1 \in X_1$

and

$i_2: X_2 \to X_1 + X_2$

defined as

$i_2: x_2 \mapsto (x_2,2) \mbox{ , } x_2 \in X_2$

are called canonical injections.

The direct sum of two topological spaces is defined as

(X 1,τ 1) + (X 2,τ 2): = (X 1 + X 2,τ 1 + 2)

with the direct sum topology $τ 1 + 2$ defined as

$O \in \tau_{1+2} \Leftrightarrow i_1^{-1}[O] \in \tau_1 \mbox{ and } i_2^{-1}[O] \in \tau_2 .$

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

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