> [!definition] > > Let $I$ be a [[Directed Set|directed set]], and $(X_i, \bracsn{f^i_j})$ be a directed family of [[Topological Space|topological spaces]], then the [[Direct Limit|direct limit]] $\varinjlim A_i$ exists. The topology on the limit is known as the **final topology**. > > *Proof*. Assume without loss of generality that $\seqi{X}$ is disjoint. Let $\ol X = \bigsqcup_{i \in I}X_i$ be equipped with the disjoint union topology. Similar to the set theoretic limit, if $x_i \in A_i$ and $x_j \in A_j$, define $x_i \sim x_j$ if there exists $k \ge i, j$ such that $f^i_k(x_i) = f^j_k(x_j)$. Define $X = \ol X / \sim$, and for each $i \in I$, let $f^i_X$ be the following composition > $ > \begin{CD}\ > X_i @>{\iota}>> \ol X @>{\text{can}}>> X > \end{CD} > $ > then $(X, \bracsn{f^i_X})$ is the desired limit.