Definition:Limit of Topological Spaces

Definition
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 $\mathbf {Top}$.

Notation
$\lim D$ also stands for the categorical limit of $D$, which is only unique up to unique isomorphism.

By Limit of Topological Spaces is Limit this notation makes sense for the limit as defined in this article.

Also see

 * Limit of Topological Spaces is Limit
 * Top is Complete