---
tag: GRZKVSMJXF
kind: definition
name: Initial topology
msc: [54A10, 54B30]
defines: 1NWC3710WW
uses: [0CQHFPGJ45, 0VGFHTM75Y, 09S8DV34Y0, 01AHG4G9RT, 072J68M6HX]
sources:
  - url: https://en.wikipedia.org/wiki/Initial_topology
---

## Definition

Let $X$ be a set, let $I$ be a set, and for each $i \in I$ let $Y_i$ be a [[topological space|0CQHFPGJ45]] and $f_i \colon X \to Y_i$ a map.
The *initial topology* on $X$ for the family $(f_i)_{i \in I}$ is the [[coarsest|072J68M6HX]] topology on $X$ for which every $f_i$ is [[continuous|0VGFHTM75Y]].
Equivalently, it is the topology generated by the [[subbasis|09S8DV34Y0]]
$$\set{f_i^{-1}(V) : i \in I, \ V \text{ open in } Y_i}.$$

## Remarks

The [[preimages|01AHG4G9RT]] $f_i^{-1}(V)$ make each $f_i$ continuous, and any topology that makes each $f_i$ continuous must contain them; this gives the equivalence of the two forms.
Universal property: for a topological space $Z$, a map $g \colon Z \to X$ is continuous for the initial topology if and only if $f_i \circ g$ is continuous for every $i$.

Example: for a subset $X$ of a space $Y$ and the inclusion $\iota \colon X \to Y$, the initial topology is the [[subspace topology|0WS9P12FQR]].
Its open sets are the sets $\iota^{-1}(V) = X \cap V$ with $V$ open in $Y$.
Example: for a product $X = \prod_i Y_i$ and the projections, the initial topology is the [[product topology|0GBG8B49AA]].

Non-example: the discrete topology on $\R$ makes the identity map to $\R$ with its usual topology continuous, but it is not the initial topology for this map.
It is strictly finer than needed: the initial topology here is the usual topology, in which $\set{0}$ is not open.

The initial topology is also called the weak topology or the induced topology for the family.