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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer, and let $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *$n$-connected* if both of the following hold:\n\n1. $X$ is nonempty and [[path-connected|0WB6DH499T]];\n2. $\\pi_k(X, x) = 0$ for every $x \\in X$ and every $k$ with $1 \\le k \\le n$, where $\\pi_k$ is the [[homotopy group|P976WYEFMB]].\n\n## Remarks\n\nEquivalently, $X$ is nonempty and for $0 \\le k \\le n$ every continuous map $S^k \\to X$ extends to a continuous map $D^{k+1} \\to X$.\nSo $0$-connected means path-connected, and $1$-connected means simply connected.\nBy convention, $(-1)$-connected means nonempty.\n\nExample: the sphere $S^m$ is $(m-1)$-connected, since $\\pi_k(S^m) = 0$ for $k \u003c m$.\n\nNon-example: $S^m$ is not $m$-connected, because $\\pi_m(S^m) \\cong \\Z$, so clause 2 fails for $k = m$.\n\nA 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 $a \\in A$.\n",
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