> [!theorem] > > Let $(\seqi{X}, \bracsn{f^i_j: i, j \in I, i \le j})$ be a [[Directed Set|directed]] system of [[Topological Space|topological spaces]], then there exists a topological space $X$ and a family $\bracsn{f^i_X: i \in I}$ of maps such that: > 1. $f^i_X: X_i \to X$ is a continuous map. > 2. $f^j_X \circ f^i_j = f^i_X$ for all $i, j \in I$ with $i \le j$. > 3. For any other family $(Y, \bracsn{f^i_Y: i \in I})$ satisfying $(1)$ and $(2)$, there exists a unique continuous map $f^X_Y: X \to Y$ such that $f^X_Y \circ f^i_X = f^i_Y$ for all $i \in I$. > > In other words, direct limits exist in the category of topological spaces. > > *Proof*. Let $X = \varinjlim X_i$ with $\bracsn{f^i_X: i \in I}$ be the set theoretic direct limit of $X_i$, then it's sufficient to place a topology on $X$ such that each $f^i_X$ and $f^X_Y$ is continuous. > > For each $U \subset X$, define $U$ to be [[Open Set|open]] if $(f^i_X)^{-1}(U)$ is open in $X_i$ for all $i \in I$, then the family of all such sets forms a topology on $X$ which makes each $f^i_X$ continuous. > > Now let $(Y, \bracsn{f_Y^i: i \in I})$ be another such family, and $f^X_Y: X \to Y$ be the map induced by the set theoretic direct limit, then $f^X_Y \circ f^i_X$ is a continuous map for each $i \in I$. For any open set $U \subset Y$, $(f^i_X)^{-1}[(f_Y^X)^{-1}(U)]$ is open in $X_i$ for all $i \in I$. Therefore $(f_Y^X)^{-1}(U)$ is open in $X$, and $f^X_Y$ is continuous.