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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] with a [[topology|0CNQXVBHCX]].\nGive $K \\times K$ the product topology and $K^\\times = K \\setminus \\set{0}$ the subspace topology.\nThe field $K$ is a *topological field* if the following maps are [[continuous|0P27W9RW0G]]:\n\n1. addition $K \\times K \\to K$, $(x, y) \\mapsto x + y$;\n2. negation $K \\to K$, $x \\mapsto -x$;\n3. multiplication $K \\times K \\to K$, $(x, y) \\mapsto xy$;\n4. inversion $K^\\times \\to K^\\times$, $x \\mapsto x^{-1}$.\n\n## Remarks\n\nClauses 1 to 3 say that $K$ is a topological ring.\nMany authors also require the topology to be Hausdorff and not discrete.\nExample: $\\R$ with the usual topology is a topological field.\nClauses 1 to 3 are the limit rules for sums, negatives and products, and clause 4 holds because $x \\mapsto 1/x$ is continuous on $\\R \\setminus \\set{0}$.\nMore generally, every field with an absolute value, with the topology of the metric $d(x, y) = \\abs{x - y}$, is a topological field.\nNon-example: $\\R$ with the lower limit topology, generated by the intervals $[a, b)$, is not a topological field.\nNegation fails clause 2: the preimage of the open set $[0, 1)$ under $x \\mapsto -x$ is $(-1, 0]$, which is not open.\nEvery field is a topological field for the discrete topology and for the indiscrete topology.\n",
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