Ring of Continuous Real-Valued Functions is Ring
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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:
- $\struct {\map C {S, \R}, +, *}$ is a ring
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 Ring of Continuous Mappings is Subring of All Mappings:
- $\struct {\map C {S, \R}, +, *}$ is a subring
By definition of subring:
- $\struct {\map C {S, \R}, +, *}$ is a ring
$\blacksquare$
Also see
Sources
1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.3$