Definition:Initial Topology

Equivalence of Definitions
These definitions are shown to be equivalent in Equivalence of Definitions of Initial Topology.

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

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

 * Definition:Final Topology
 * Initial Topology with respect to Mapping