Equivalence of Definitions of Initial Topology

Theorem

The following definitions of the concept of Initial Topology are equivalent:

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 $I$.

Definition 1

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 $\left \langle {f_i}\right \rangle_{i \mathop \in I}$.

Definition 2

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 $\left \langle {f_i} \right \rangle_{i \mathop \in I}$.

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

$\Box$

Definition 1 provides the coarsest such topology

Suppose that the mappings are continuous from $\left({X, \vartheta}\right)$.

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

Then for some $i \in I$ and some $V \in \tau_i$,

$U = f^{-1} \left({V}\right)$

Then since the mappings are continuous from $\left({X, \vartheta}\right)$:

$U \in \upsilon$

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

$\blacksquare$