Upper semicontinuous real map on a topological space

Let $T$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) and let $f:T\to \mathbb{R}$. The map $f$ is upper semicontinuous if, for every $a\in \mathbb{R}$, the strict sublevel set

$\{x\in T:f(x)<a\}$

is open in $T$. Equivalently, every superlevel set $\{x\in T:f(x)\ge a\}$ is closed, because it is the complement of the strict sublevel set.

When $T=A\subseteq \mathbb{R}$ has the subspace topology, this agrees with the existing pointwise definition: the equivalence with openness of all strict sublevels is exactly $f$ is upper semicontinuous on $A$ if and only if $\{x\in A:f(x)<\alpha \}$ is relatively open in $A$ for every real $\alpha$, lower semicontinuous if and only if $\{x\in A:f(x)>\alpha \}$ is, and continuous if and only if it is both, claim

1. The empty-domain condition is vacuous, constant functions are upper semicontinuous, and the inequalities deliberately distinguish the open threshold $f<a$ from the closed threshold $f\ge a$.