Topological field
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps
where R × R carries the product topology. That means R is an additive topological group and a multiplicative topological semigroup.
The group of units R× of R is a topological group when endowed with the topology coming from the embedding of R× into the product R × R as (x,x−1). However, if the unit group is endowed with the subspace topology as a subspace of R, it may not be a topological group, because inversion on R× need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on R× is continuous in the subspace topology of R then these two topologies on R× are the same.
If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring which is a topological group (for +) in which multiplication is continuous, too.
Topological rings occur in mathematical analysis, for example as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space; all Banach algebras 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 numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples.
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