---
tag: 1NWC3710WW
kind: concept
name: Initial topology
msc: [54A10, 54B30]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Initial_topology
same_as:
  wikidata: Q13416298
  wikipedia: Initial topology
---

## Summary

Take a set and a family of maps from it into topological spaces.
The initial topology on the set is the coarsest topology that makes all these maps continuous.
The [[subspace topology|0WS9P12FQR]] and the [[product topology|0GBG8B49AA]] are both initial topologies.
It is characterised by a universal property: a map into the set is continuous exactly when its composites with all the given maps are continuous.
The dual notion is the [[final topology|6ERJRRZK32]].

## Explanations

### L1 human

Suppose you have a set and some maps from it into spaces you already understand.
You want the fewest open sets that still make every map continuous.
So you take the preimages of open sets under the maps, and only add what you must to get a topology.
For example, the projections $\R^2 \to \R$ onto the two coordinates give the usual topology of the plane.
Open rectangles are intersections of two such preimages.