Unity of Ring of Continuous Real-Valued Functions

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Theorem

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

Let $\R$ denote the real number line.

Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.

Then:

the unity of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $1_{\R^S} : S \to \R$ defined by:

$\forall s \in S : \map {1_{\R^S}} s = 1$.

Proof

By definition of ring of continuous real-valued functions:

$\struct {\map C {S, \R}, +, *}$ is the ring of continuous mappings from $S$ to $\R$.

From Unity of Ring of Continuous Mappings:

the unity of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $1_{\R^S} : S \to \R$ defined by:

$\forall s \in S : \map {1_{\R^S}} s = 1$.

$\blacksquare$

Also see

• Ring of Continuous Real-Valued Functions is Ring

• Zero 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, $\S 1.3$