Definition:Lattice of Real-Valued Functions

Definition

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

Let $\R$ denote the real number line.

Let $\R^S$ be the set of mappings from $S$ to $\R$.

The lattice of real-valued mappings from $S$, denoted $\struct{\R^S, \vee, \wedge, \le}$, is the set of all mappings $\R^S$ with (pointwise) lattice operations $\vee$ and $\wedge$ defined by:

$\forall f, g \in \R^S : f \vee g : S \to \R$ is defined by:

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

$\forall f, g \in \R^S : f \wedge g : S \to \R$ is defined by:

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

and ordering defined by:

$\forall f, g \in \R^S : f \le g$ if and only if:

$\forall x \in S : \map f x \le \map g x$

Also see

• Lattice of Real-Valued Functions forms Distributive Lattice

Sources

1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.2$