Suppose \((X_\alpha)_{\alpha\in A}\) is a family of topological spaces . The disjoint union space \(\coprod_{\alpha\in A} X_\alpha\) is the set of tuples \((x,\alpha)\) where \(\alpha\in A\) and \(x\in X_\alpha\). For each \(\alpha\in A\), there is a canonical injection \(\iota_\alpha\colon X_\alpha\to \coprod_{\alpha\in A} X_\alpha\) given by \(\iota_\alpha(x)=(x,\alpha)\). We usually identify \(X_\alpha\) with \(X_\alpha^*=\iota_\alpha(X_\alpha)\), where \(\iota_\alpha\) denotes the canonical injection .
The disjoint union topology on the disjoint union space consists of subsets that are intersected with \(X_\alpha\) are open.
Examples
Remarks
- The disjoint union space topology satisfies the characteristic property of disjoint union spaces