\begin{definition}[Definition:Limit of Topological Spaces]
Let $\II$ be a small category.
Let $D : \II \to \mathbf {Top}$ be a diagram in the category of topological spaces $\mathbf {Top}$.
The limit of topological spaces of $D$ is defined as the limit of sets of $D$ equipped with the limit topology.
The corresponding projections $\pi_j : \lim D \to \map D j$ are defined by:
:$\forall j \in \II: \map {\pi_j} {\family {a_i}_{i \mathop \in \II}} := a_j$
By Equivalence of Definitions of Initial Topology the projections are continuous and thus define morphisms in $\mathbf {Top}$.
\end{definition}