Equivalence of Definitions of Initial Topology/Definition 2 Implies Definition 1

Theorem
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 $I$.

Proof
As Definition 2 implies uniqueness, we need only show that the topology defined by Definition 1 satisfies the requirements of Definition 2.

Mappings are continuous in definition 1
Let $i \in I$.

Let $U \in \tau_i$.

Then $\map {f_i^{-1}} {U}$ is an element of the natural subbase of the initial topology, and is therefore trivially in $\tau$.

Definition 1 provides the coarsest such topology
Suppose that the mappings are continuous from $\struct{X, \vartheta}$.

Let $U$ be a member of the subbase from Definition 1.

Then for some $i \in I$ and some $V \in \tau_i$,
 * $U = \map {f^{-1}} {V}$

Then since the mappings are continuous from $\struct{X, \vartheta}$:
 * $U \in \vartheta$

Since $\upsilon$ is a topology containing a subbase of $\tau$, $\tau$ is coarser than $\vartheta$.