| Topological Ring |
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| CATEGORIES ABOUT TOPOLOGICAL RING | |
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R where ''R'' × ''R'' carries the Product Topology . Examples Topological rings occur in Mathematical Analysis , for examples as rings of continuous real-valued Function s on some topological space (where the topology is given by pointwise convergence), or as rings of continuous Linear Operator s on some Normed Vector Space ; all Banach Algebra s are topological rings. The Rational , Real , Complex and ''p''-adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, Split-complex Number s and Dual Numbers form alternative topological rings. See Hypercomplex Numbers for other low dimensional examples. In ''U'' of ''R'' is open Iff for every ''x'' in ''U'' there exists a natural number ''n'' such that ''x'' + ''I''''n'' ⊆ ''U''. This turns ''R'' into a topological ring. The ''I''-adic topology is Hausdorff if and only if the Intersection of all powers of ''I'' is the zero ideal (0). The ''p''-adic topology on the Integer s is an example of an ''I''-adic topology (with ''I'' = (''p'')). Completion Every topological ring is a Subring such that the given topology on ''R'' equals the Subspace Topology arising from ''S''. The ring ''S'' can be constructed as a set of equivalence classes of Cauchy Sequences in ''R''. The rings of Formal Power Series and the ''p''-adic Integers are most naturally defined as completions of certain topological rings carrying ''I''-adic topologies. TOPOLOGICAL FIELDS Some of the most important examples are also Field s ''F''. To have a topological field we should also specify that Inversion is continuous, when restricted to ''F''\{0}. |
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