Unity 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 unity $1_R$.

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

Then:

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

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

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

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

From Constant Mapping is Continuous:

$1_{R^S} \in \map C {S, R}$

From Subring Containing Ring Unity has Unity:

$1_{R^S}$ is the unity of $\map C {S, R}$

$\blacksquare$

Also see

• Ring of Continuous Mappings is Subring of All Mappings

• Zero of Ring of Continuous Mappings

• Additive Inverse in Ring of Continuous Mappings

• Commutativity of Ring of Continuous Mappings