---
tag: HZ5V3M1F54
kind: definition
name: n-connected space
msc: [55Q05]
defines: 1A943BWBQT
uses: [P976WYEFMB, 0CQHFPGJ45, 0WB6DH499T]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopical_connectivity
---

## Definition

Let $n \ge 0$ be an integer, and let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *$n$-connected* if both of the following hold:

1. $X$ is nonempty and [[path-connected|0WB6DH499T]];
2. $\pi_k(X, x) = 0$ for every $x \in X$ and every $k$ with $1 \le k \le n$, where $\pi_k$ is the [[homotopy group|P976WYEFMB]].

## Remarks

Equivalently, $X$ is nonempty and for $0 \le k \le n$ every continuous map $S^k \to X$ extends to a continuous map $D^{k+1} \to X$.
So $0$-connected means path-connected, and $1$-connected means simply connected.
By convention, $(-1)$-connected means nonempty.

Example: the sphere $S^m$ is $(m-1)$-connected, since $\pi_k(S^m) = 0$ for $k < m$.

Non-example: $S^m$ is not $m$-connected, because $\pi_m(S^m) \cong \Z$, so clause 2 fails for $k = m$.

A pair $(X, A)$ is called $n$-connected if every path component of $X$ meets $A$ and $\pi_k(X, A, a) = 0$ for $1 \le k \le n$ and all $a \in A$.