Category:Lattices of Bounded Continuous Real-Valued Functions
This category contains results about Lattices of Bounded Continuous Real-Valued Functions.
Definitions specific to this category can be found in Definitions/Lattices of Bounded Continuous Real-Valued Functions.
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:
- $\forall s \in S : \map {\paren{f \wedge g}} s = \min \set{\map f s, \map g s}$
Pages in category "Lattices of Bounded Continuous Real-Valued Functions"
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