Disjoint unions

A disjoint union, or coproduct, is a space $Q$ such that given two spaces $X,Y$, a continuous map from $Q$ has the same data as a map from each of $X$ and $Y$. Compare this to a product of topological spaces which, in the standard product topology, where a continuous map some $X\to Y\times Z$ is the same data as a continuous map to each of $Y,Z$.

Definition: Disjoint union

Given a collection of sets $\{X_{\alpha }{\}}_{\alpha \in J}$, their disjoint union, or coproduct, is

$$\coprod _{\alpha \in J}{X}_{\alpha }=\{(x,\alpha )|\alpha \in J,x\in X_{\alpha }\}.$$

For a single set $X$, while the usual union of $X$ with itself is simply the original set $X\cup X=X$, while $X\bigsqcup X$ involves two distinct copies of $X$.

Further, there exist natural injective inclusion maps $i_{\alpha }:X_{\alpha }\to {\coprod }_{\alpha }{X}_{\alpha }$ given by $x\mapsto (x,\alpha )$.

Note that if $Y$ is discrete and $X$ is any space, then $X\times Y\cong {\coprod }_{y\in Y}X$ is a homeomorphism.

The universal property of the disjoint union motivates quotient spaces: we would like each inclusion $i_{\alpha }:X_{\alpha }\to \coprod X_{\alpha }$ to be continuous, so we should define a set $U\subseteq \coprod X_{\alpha }$ if and only if$i_{\alpha }^{-1}(U)\subseteq X_{\alpha }$ is open for all $\alpha$.