Lattice of Continuous Functions is Sublattice of All Real-Valued Functions

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Theorem

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

Let $\R$ denote the real number line.

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

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

Then:

$\struct{\map C {S, R}, \vee, \wedge, \le}$ is a sublattice of $\struct {\R^S, \vee, \wedge, \le}$

Proof

To show that $\struct{\map C {S, R}, \vee, \wedge, \le}$ is a sublattice of $\struct {\R^S, \vee, \wedge, \le}$ it is sufficient to show that $\map C {S, R}$ is closed under $\vee$ and $\wedge$.

From Maximum Rule for Continuous Real-Valued Functions:

$\map C {S, R}$ is closed under $\vee$

From Minimum Rule for Continuous Real-Valued Functions:

$\map C {S, R}$ is closed under $\wedge$

$\blacksquare$