nLab
topological ring

Context

Topology

Higher algebra

Contents

Definition

A topological ring is a ring internal to Top, a ring object in Top:

a topological space RR equipped with the structure of a ring on its underlying set, such that addition and multiplication are continuous functions. Of course this makes RR a uniform space.

A topological field is a topological ring KK whose underlying ring is in fact a field and such that reciprocation () 1:K{0}K{0}(-)^{-1}: K \setminus \{0\} \to K \setminus \{0\} is continuous. This latter condition is the same as demanding that the subspace topology on K{0}K \setminus \{0\} induced by the embedding K{0}KK \setminus \{0\} \hookrightarrow K coincide with the subspace topology induced by the embedding K{0}K×K:x(x,x 1)K \setminus \{0\} \to K \times K: x \mapsto (x, x^{-1}). More at topological field.

Remark

In a topological ring, the closure of {0}\{0\} is an ideal. It follows that for a topological field FF, either 00 is a closed point (so that FF is T 1T_1 and therefore completely regular Hausdorff, by standard arguments in the theory of uniform spaces), or is a codiscrete space.

A topological algebra over a topological ring RR is a topological ring SS together with a topological ring map RSR \to S that makes SS an RR-algebra at the underlying set level (a topological associative algebra).

Examples

Revised on August 19, 2015 15:01:00 by Todd Trimble (67.81.95.215)