Top is Complete

Theorem

The category of topological spaces is complete.

Proof

Let $\II$ be a small category.

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

Let $\family {\lim D, \family {\pi_i}_{i \mathop \in \II}}$ be the limit of topological spaces of $D$.

By Limit of Topological Spaces is Limit, $\family {\lim D, \family {\pi_i}_{i \mathop \in \II}}$ is a categorical limit of $D$.

$\blacksquare$