\begin{definition}[Definition:Limit Topology]
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 $\family {\pi_i}_{i \mathop \in \II}$.
Category:Definitions/Topology
Category:Definitions/Topological Spaces
Category:Definitions/Category Theory
\end{definition}