nLab
topological ring

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic 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 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)