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  "body": "\n## Definition\n\nLet $\\F$ be a [[field|04C5ZBG1WT]] and let $K$ be a finite [[simplicial complex|V8GZ0W1ZZ5]] with a filtration by subcomplexes\n$$K_0 \\subseteq K_1 \\subseteq \\cdots \\subseteq K_m = K.$$\nFor $i \\le j$ and $p \\ge 0$, the inclusion $K_i \\subseteq K_j$ induces a linear map $f_p^{i,j} \\colon H_p(K_i; \\F) \\to H_p(K_j; \\F)$ of [[simplicial homology|C6J40QF82R]] groups, which are [[vector spaces|000AYPA9AJ]] over $\\F$.\nThe *$p$-th persistent homology group* from $i$ to $j$ is the image\n$$H_p^{i,j} = \\im f_p^{i,j} \\subseteq H_p(K_j; \\F),$$\nand its dimension $\\beta_p^{i,j}$ is the *persistent Betti number*.\nThe family of all $H_p(K_i; \\F)$ with all maps $f_p^{i,j}$ is the *$p$-th persistence module* of the filtration.\n\n## Remarks\n\nA class is *born* at $i$ if it lies in $H_p(K_i; \\F)$ but not in the image of $f_p^{i-1,i}$.\nIt *dies* at $j$ if it merges with an older class or becomes $0$ when passing from $K_{j-1}$ to $K_j$.\nThe persistence module splits, uniquely up to order, as a direct sum of interval modules.\nThe list of their intervals $[b, d)$ is the *barcode*.\nThe same data, as points $(b, d)$ in the plane, form the *persistence diagram*.\n\nExample: let $K_0$ consist of two vertices $a, b$ and let $K_1$ add the edge $ab$.\nThen $H_0(K_0; \\F) \\cong \\F^2$, $H_0(K_1; \\F) \\cong \\F$, and $f_0^{0,1}$ is onto, so $\\beta_0^{0,1} = 1$.\nThe barcode in degree $0$ is $\\set{[0, \\infty), [0, 1)}$: one component lives forever and one dies when the edge joins them.\n\nNon-example: a family of complexes $K_0, K_1$ where $K_1$ does not contain $K_0$ is not a filtration, and no maps $f_p^{0,1}$ are induced.\nZigzag persistence treats such cases with maps in both directions.\n\nFor a finite set of points in a metric space, the standard filtrations are the Vietoris–Rips and Čech complexes at increasing scale.\n",
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