Definition:Initial Topology

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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 $I$.


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


Also known as

The initial topology is also known as:

the projective topology
the weak topology on $X$ with respect to $\left \langle {f_i} \right \rangle_{i \mathop \in I}$


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