Zero of Ring of Continuous Mappings
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Theorem
Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring with zero $0_R$.
Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.
Then:
- the zero of $\struct{\map C {S, R}, +, *}$ is the constant mapping $0_{R^S} : S \to R$ defined by:
- $\forall s \in S : \map {0_{R^S}} s = 0_R$
Proof
Let $\struct {R^S, +, *}$ be the ring of mappings from $S$ to $R$.
From Ring of Continuous Mappings is Subring of All Mappings:
- $\struct{\map C {S, R}, +, *}$ is a subring of $\struct {R^S, +, *}$
From Induced Structure Identity:
- the zero of $\struct {R^S, +, *}$ is the constant mapping $0_{R^S} : S \to R$ defined by:
- $\forall s \in S : \map {0_{R^S}} s = 0_R$
From Zero of Subring is Zero of Ring:
- $0_{R^S} \in \map C {S, R}$
$\blacksquare$