n-connected space

Builds on Homotopy group Topological space Path-connected space

Definition

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

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

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 $a \in A$ .

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Tag	`HZ5V3M1F54`
URI	`m2://HZ5V3M1F54`
Kind	definition
Writing	W0 stub
MSC	55Q05
Defines	n-connected space
Uses	Homotopy group, Topological space, Path-connected space
Changed	2 November 2026

n-connected space

Definition

Remarks

Reading path

Further reading

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