Definition:Initial Topology

Definition
Let $X$ be a set.

Let $I$ be an indexing set.

Let $\left \langle {\left({Y_i, \tau_i}\right)} \right \rangle_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$.

Let $\left \langle {f_i: X \to Y_i} \right \rangle_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$.

Also known as
The initial topology is also known as:
 * the projective topology
 * the weak topology on $X$ with respect to $\left \langle {f_i} \right \rangle_{i \mathop \in I}$

If only a single topological space $\left({Y, \tau_Y}\right)$ and a single mapping $f: X \to Y$ are under consideration, the initial topology on $X$ with respect to $f$ is additionally known as:
 * the pullback topology on $X$ under $f$
 * the topology on $X$ induced by $f$
 * the inverse image of $\tau_Y$ under $f$

and is often denoted by $f^* \left({\tau_Y}\right)$ or $f^{-1} \left({\tau_Y}\right)$.

Also see

 * Equivalence of Definitions of Initial Topology
 * Initial Topology with respect to Mapping equals Set of Preimages
 * Domain Topology Contains Initial Topology iff Mappings are Continuous
 * Definition:Final Topology