---
tag: 1A943BWBQT
kind: concept
name: n-connected space
msc: [55Q05]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopical_connectivity
same_as:
  wikidata: Q6358325
  wikipedia: Homotopical connectivity
---

## Summary

A space is $n$-connected if it is nonempty, path-connected, and its [[homotopy groups|P976WYEFMB]] vanish in degrees $1$ to $n$.
It generalises path-connected ($0$-connected) and simply connected ($1$-connected) spaces.
Connectivity sets the range in which the Hurewicz, Freudenthal and Blakers–Massey theorems apply.

## Explanations

### L1 human

A space is $n$-connected if every sphere of dimension at most $n$ inside it can be shrunk to a point.
Being $0$-connected means any two points can be joined by a path.
Being $1$-connected means also that every loop can be contracted.
The $3$-sphere is $2$-connected: every loop and every small balloon on it can be shrunk.