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  "body": "\n## Definition\n\nA *surface* is a topological space $S$ such that:\n\n1. $S$ is [[Hausdorff|0S1FJ8VEX9]];\n2. $S$ is second countable, that is, its topology has a countable base;\n3. every point of $S$ has an open neighbourhood [[homeomorphic|0QT00PB5G2]] to an open subset of $\\R^2$.\n\nEquivalently, a surface is a [[topological manifold|QQQ9B1DK4P]] of dimension $2$.\nA *surface with boundary* is defined in the same way, with open subsets of the closed half-plane $\\set{(x, y) \\in \\R^2 : y \\ge 0}$ in clause 3.\nA *closed surface* is a compact connected surface without boundary.\n\n## Remarks\n\nExample: the unit sphere $S^2 \\subset \\R^3$ is a surface.\nIt is Hausdorff and second countable as a subspace of $\\R^3$.\nStereographic projections from the north pole and from the south pole are homeomorphisms of two open subsets of $S^2$ onto $\\R^2$, and these subsets cover $S^2$.\n\nNon-example: let $X$ be the union of the planes $z = 0$ and $y = 0$ in $\\R^3$.\nClauses 1 and 2 hold, but clause 3 fails at points of the $x$-axis.\nA small sphere around a point $P$ of this axis meets $X$ in two great circles crossing at two points, a graph $G$ with $2$ vertices and $4$ edges.\nSo $P$ has a neighbourhood that is a cone on $G$, and the local homology group $H_2(X, X \\setminus \\set{P})$ is $\\tilde H_1(G) \\cong \\Z^3$.\nAt a point of an open subset of $\\R^2$ this group is $\\tilde H_1(S^1) \\cong \\Z$, and local homology is preserved by homeomorphisms of neighbourhoods.\n\n## Explanations\n\n### L1 human\n\nThe three clauses say: points can be separated, the space is not too big, and near each point the space is a patch of the plane.\nThe last clause is the heart of the matter.\nThe first two rule out pathological spaces, such as the plane with two origins, which fails clause 1.\n",
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