Definition:Initial Topology

Definition
Let $X$ be a set.

Let $I$ be an indexing set.

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

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

Let:
 * $\mathcal S = \left\{{f_i^{-1} \left({U}\right): i \in I, \, U \in \tau_i}\right\} \subseteq \mathcal P \left({X}\right)$

where $f_i^{-1} \left({U}\right)$ denotes the preimage of $U$ under $f_i$.

The topology $\tau$ on $X$ generated by $\mathcal S$ is called the initial topology on $X$ with respect to $\left \langle {f_i}\right \rangle_{i \in I}$.

Equivalently, $\tau$ is the coarsest topology on $X$ such that $f_i$ is continuous for all $i \in I$.

Also known as
The initial topology is also known as the projective topology.

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 $f$
 * topology on $X$ induced by $f$
 * inverse image of $\tau_Y$ under $f$

and can also be denoted by $f^* \left({\tau_Y}\right)$ or $f^{-1} \left({\tau_Y}\right)$.

Also see

 * Final Topology
 * Initial Topology with respect to Mapping