---
tag: F2F143KQ2M
kind: concept
name: Homotopic maps
msc: [55Q05, 55P10]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy
---

## Summary

Two continuous maps $f, g \colon X \to Y$ are homotopic, written $f \simeq g$, when there is a [[homotopy|9YA6S6HD9V]] from $f$ to $g$.
Being homotopic is an equivalence relation on the set of continuous maps from $X$ to $Y$.
Its equivalence classes are the homotopy classes, and the set of them is written $[X, Y]$.
Algebraic topology studies spaces through such sets of homotopy classes.

## Explanations

### L1 human

Two maps are homotopic when one can be slid continuously into the other inside $Y$.
Homotopy forgets the fine details of a map and keeps only its shape up to deformation.
For example, every two maps into $\R^n$ are homotopic, by moving each point along a straight segment.
Two loops in the circle that wind around a different number of times are not homotopic.