# Definition:Initial Topology/Definition 2

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## 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 $I$.

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 see

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

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.3$: Sub-bases and weak topologies