> [!definition] > > Let $\bracs{(X_i, \topo_i)}_{i \in I}$ be a family of [[Topological Space|topological spaces]], $Y$ be a set, and $\bracs{f_i: X_i \to Y}$ be a family of mappings, then there exists a unique topology $\topo_F$ on $Y$ such that: > 1. $f_i: (X_i, \topo_i) \to (Y, \topo_F)$ is [[Continuity|continuous]]. > 2. For every topological space $Z$, a mapping $g: (Y, \topo_F) \to Z$ is continuous if and only if $g \circ f_i: X_i \to Z$ is continuous. > 3. For any other topology $\topo$ on $Y$ such that $f_i: X_i \to (Y, \topo)$ is continuous, $\topo_F \subset \topo'$. > > The topology $\topo_F$ is known as the **final topology** on $Y$. > > *Proof*. $(3)$ follows from $(2)$, and uniqueness follows from $(3)$. For $(1)$, let > $ > \topo_Y = \bracs{U \subset Y: f_i^{-1}(U) \in \topo_i \forall i \in I} > $ > then since preimages commute with unions and complements, $\topo_Y$ is a topology on $Y$ satisfying $(1)$. > > For $(2)$, $g \circ f_i: X_i \to Z$ is continuous if and only if $f_i^{-1}(g^{-1}(U)) \in \topo_i$ for each $U$ [[Open Set|open]] in $Z$. Thus $g$ is continuous if and only if $g \circ f_i$ is continuous for all $i \in I$. > [!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.