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  "body": "\n## Definition\n\nLet $X$ be a [[locally compact|0FE2SB8WVJ]] [[metric space|0VKD1TP7F3]] and let $\\varphi \\colon \\R \\times X \\to X$ be a continuous [[flow|BBZP1R872N]].\nFor a set $N \\subseteq X$, let $\\operatorname{Inv}(N) = \\set{x \\in X : \\varphi_t(x) \\in N \\text{ for all } t \\in \\R}$, the largest [[invariant set|WWZTW11664]] in $N$.\nA compact set $N$ is an *isolating neighbourhood* if $\\operatorname{Inv}(N)$ lies in the interior of $N$.\nAn invariant set $S$ is *isolated* if $S = \\operatorname{Inv}(N)$ for some isolating neighbourhood $N$.\nAn *index pair* for an isolated invariant set $S$ is a pair of compact sets $L \\subseteq N \\subseteq X$ such that:\n\n1. The closure of $N \\setminus L$ is an isolating neighbourhood with $\\operatorname{Inv}(\\overline{N \\setminus L}) = S$.\n2. $L$ is positively invariant in $N$: if $x \\in L$, $t \\ge 0$ and $\\varphi_s(x) \\in N$ for all $s \\in [0, t]$, then $\\varphi_t(x) \\in L$.\n3. $L$ is an exit set for $N$: if $x \\in N$ and $\\varphi_t(x) \\notin N$ for some $t \u003e 0$, then there is $s \\in [0, t]$ with $\\varphi_s(x) \\in L$ and $\\varphi_r(x) \\in N$ for all $r \\in [0, s]$.\n\nEvery isolated invariant set has an index pair.\nThe homotopy type of the [[pointed space|T9DG8DDE75]] $(N/L, [L])$ does not depend on the choice of index pair.\nThis [[homotopy type|SK0ZJGWXDP]] is the *Conley index* $h(S)$ of $S$.\n\n## Remarks\n\nHere $N/L$ is the [[quotient space|0BR3FY5Q8M]] that collapses $L$ to one point $[L]$; when $L = \\emptyset$, one sets $N/L = N \\sqcup \\set{\\ast}$ with base point $\\ast$.\nExample: let $0$ be a hyperbolic rest point in $\\R^n$ with an unstable subspace of dimension $k$.\nIn adapted coordinates take $N = D^k \\times D^{n-k}$, a product of closed balls, and the exit set $L = \\partial D^k \\times D^{n-k}$.\nThen $N/L$ has the homotopy type of $D^k / \\partial D^k$, the pointed sphere $S^k$; for an attracting point, $k = 0$ and the index is $S^0$.\nNon-example: for the trivial flow $\\varphi_t(x) = x$ on $\\R$, the invariant set $S = \\set{0}$ is not isolated, since every compact $N$ satisfies $\\operatorname{Inv}(N) = N$, which is not contained in the interior of $N$.\nWażewski principle: the empty set has index the pointed one-point space, so if $h(S)$ is not that homotopy type, then $S \\neq \\emptyset$.\n\n## Explanations\n\n### L1 human\n\nThe Conley index is the shape left after collapsing the exit of a box around an isolated invariant set.\n",
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