# Definition:Initial Topology

## Definition

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 $I$.

### Definition 1

Let:

- $\SS = \set {f_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$

where $f_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $f_i$.

The topology $\tau$ on $X$ generated by $\SS$ is called the **initial topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$**.

### Definition 2

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

Then $\tau$ is known as the **initial topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$**.

## Also known as

The **initial topology** is also known as:

- the
**projective topology** - the
**weak topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$**

If only a single topological space $\struct {Y, \tau_Y}$ 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 $\map {f^*} {\tau_Y}$ or $\map {f^{-1} } {\tau_Y}$.

## Also see

- Equivalence of Definitions of Initial Topology
- Initial Topology with respect to Mapping equals Set of Preimages
- Domain Topology Contains Initial Topology iff Mappings are Continuous
- Definition:Final Topology

- Results about
**the initial topology**can be found**here**.