Definition:Ring of Continuous Real-Valued Functions

Definition

Let $\struct {S, \tau }$ be a topological space.

Let $\R$ denote the real number line.

The ring of continuous mappings from $S$ to $\R$, denoted $\map C {S, \R}$, is called the ring of continuous real-valued functions from $S$.

The (pointwise) ring operations on the ring of continuous real-valued functions from $S$ are defined as:

$\forall f, g \in \map C {S, \R} : f + g : S \to \R$ is defined by:

$\forall s \in S : \map {\paren{f + g}} s = \map f x + \map g s$

$\forall f, g \in \map C {S, \R} : f g : S \to \R$ is defined by:

$\forall s \in S : \map {\paren{f g}} s = \map f x \map g s$

Also see

• Ring of Continuous Real-Valued Functions is Ring

• Zero of Ring of Continuous Real-Valued Functions

• Additive Inverse in Ring of Continuous Real-Valued Functions

• Unity of Ring of Continuous Real-Valued Functions

• Ring of Continuous Real-Valued Functions is Commutative

Sources

1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.3$