Category:Rings of Continuous Mappings

This category contains results about Rings of Continuous Mappings.
Definitions specific to this category can be found in Definitions/Rings of Continuous Mappings.

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

Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring.

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

The ring of continuous mappings from $S$ to $R$, denoted $\map C {S, R}$, is the set of all continuous mappings in $R^S$ with (pointwise) ring operations $+$ and $*$ restricted to $\map C {S, R}$.

The (pointwise) ring operations on the ring of continuous mappings from $S$ to $R$ 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:

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