Definition:Initial Topology/Definition 2

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 $I$.

Let $\tau$ be the coarsest topology on $X$ such that each $f_i: X \to Y_i$ is $\left({\tau, \tau_i}\right)$-continuous.

Then $\tau$ is known as the initial topology on $X$ with respect to $\left \langle {f_i} \right \rangle_{i \mathop \in I}$.

Equivalence of Definitions
This definition is shown to be equivalent to Initial Topology: Definition 1 in Equivalence of Definitions of Initial Topology.

Also known as
As a consequence of this definition, the initial topology is also known as the weak topology on $X$ with respect to $\left \langle {f_i} \right \rangle_{i \mathop \in I}$

Also see

 * Equivalence of Definitions of Initial Topology


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