Exercise 10, § 6

Extend Definition 1, Proposition 1 and Theorem 3 to functions defined on a topological space $X$ which take their values in an arbitrary linearly ordered set $Y$. If $f$ and $g$ are two mappings of $X$ into $Y$, both lower semi-continuous at a point $a$, then $\inf (f, g)$ and $\sup (f, g)$ are lower semi-continuous at $a$. So is $f + g$ if $Y$ is a linearly ordered commutative group.

If $Y$ is compact in the topology $\mathfrak{T}_0(Y)$ (§ 2, Exercise 6), extend Theorem 4 and Propositions 3 and 4 to functions with values in $Y$.