Let \(X\) be a topological space and let \((X_\alpha)_{\alpha}\) be a family of subspaces with \(X = \Union_\alpha X_\alpha\). The weak topology determined by \((X_\alpha)_\alpha\) is the topology on \(X\) in which a subset \(A\subseteq X\) is closed if and only if \(A\intersect X_\alpha\) is closed in \(X_\alpha\) for every \(\alpha\).
The weak topology is the finest topology on the set \(X\) for which every inclusion \(X_\alpha\injects X\) is continuous. A CW complex \(X\) carries the weak topology determined by its skeleta \(X^n\), equivalently by its closed cells.
[@Hat02].