Definition:Initial Topology

Definition
Let $X$ be a set.

Let $I$ be an indexing set.

Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$.

Let $\family {f_i: X \to Y_i}_{i \mathop \in I}$ 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 $\family {f_i}_{i \mathop \in I}$

If only a single topological space $\struct {Y, \tau_Y}$ 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 $\map {f^*} {\tau_Y}$ or $\map {f^{-1} } {\tau_Y}$.

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