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  "body": "\n## Definition\n\nLet $p = (p_1, p_2)$ and $q = (q_1, q_2)$ be distinct points of $\\Z^2$.\n\n1. $p$ and $q$ are *$4$-adjacent* if $\\abs{p_1 - q_1} + \\abs{p_2 - q_2} = 1$.\n2. $p$ and $q$ are *$8$-adjacent* if $\\max(\\abs{p_1 - q_1}, \\abs{p_2 - q_2}) = 1$.\n3. For $k \\in \\set{4, 8}$, a *$k$-path* in a set $S \\subseteq \\Z^2$ is a finite sequence of points of $S$ in which consecutive points are $k$-adjacent.\n4. $S$ is *$k$-connected* if any two points of $S$ are joined by a $k$-path in $S$; the maximal $k$-connected subsets of $S$ are its *$k$-components*.\n\n*Digital topology* is the study of such adjacency, connectedness and separation notions on $\\Z^2$ and their analogues on $\\Z^n$.\nFor each $k$, the $k$-adjacency makes $S$ a [[graph|00C0BR7D7H]] whose [[connected components|01VVP10MWJ]] are the $k$-components of $S$.\n\n## Remarks\n\nExample: $S = \\set{(0, 0), (1, 1)}$ is $8$-connected, since the two points are $8$-adjacent.\nNon-example: the same $S$ is not $4$-connected, because $\\abs{0 - 1} + \\abs{0 - 1} = 2$ and $S$ has no other points.\nLet $S = \\set{(1, 0), (0, 1), (-1, 0), (0, -1)}$.\nWith $8$-adjacency for $\\Z^2 \\setminus S$, the point $(0, 0)$ is $8$-adjacent to $(1, 1) \\in \\Z^2 \\setminus S$, so the complement is $8$-connected and the \"curve\" $S$ encloses nothing.\nWith $4$-adjacency for the complement, $(0, 0)$ is a $4$-component by itself, so $S$ separates it from the outside.\nFor this reason one pairs $8$-adjacency for a set with $4$-adjacency for its complement, or the reverse.\nThe Khalimsky topology on $\\Z$ is a genuine topology: the sets $\\set{2n + 1}$ and $\\set{2n - 1, 2n, 2n + 1}$ for $n \\in \\Z$ form a base, and $\\Z$ is connected in it.\n",
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