nLab
topological ring

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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 \cdot are continuous functions.

Remarks:

  • The structure of a topological ring (R,+,)(R,+,\cdot) makes RR a uniform space.

  • It is automatic that negation in a topological unital ring is continuous, since it is the operation of multiplication with 1-1, and multiplication is continuous in each argument. Hence for (R,+,)(R,+,\cdot) a topological ring, then (R,+)(R,+) is a topological group.

  • 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

References

Lecture notes include

See also

Revised on May 30, 2017 05:42:30 by Urs Schreiber (131.220.184.222)