Definition:Limit Topology

Definition
Let $\II$ be a small category.

Let $D : \II \to \mathbf {Top}$ be a diagram in the category of topological spaces $\mathbf {Top}$.

Let $\lim D$ be a limit of sets of $D$.

Let $\pi_i : \lim D \to \map D i$ for all $i \mathop \in \II$ be the corresponding projection.

The limit topology on $\lim D$ is the initial topology with respect to the family of all projections $\family {\pi_i}_{i \mathop \in \II}$.