Definition:Lattice of Bounded Continuous Real-Valued Functions

Definition

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

Let $\R$ denote the real number line.

Let $\struct {\map C {S, \R}, \vee, \wedge}$ be the lattice of continuous real-valued functions from $S$.

The lattice of bounded continuous real-valued functions from $S$, denoted $\map {C^*} {S, \R}$, is the set of all bounded continuous mappings in $\map C {S, \R}$ with (pointwise) lattice operations $\vee$ and $\wedge$ restricted to $\map {C^*} {S, R}$.

The (pointwise) lattice operations on the lattice of bounded continuous real-valued functions from $S$ are defined as:

$\forall f, g \in \map {C^*} {S, \R} : f \vee g : S \to \R$ is defined by:

$\forall s \in S : \map {\paren{f \vee g}} s = \max \set{\map f s, \map g s}$

$\forall f, g \in \map {C^*} {S, \R} : f \wedge g : S \to \R$ is defined by:

$\forall s \in S : \map {\paren{f \wedge g}} s = \min \set{\map f s, \map g s}$

Also see

• Lattice of Bounded Continuous Functions is Sublattice of Continuous Real-Valued Functions

Sources

1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.4$