A Real Function with Closed Superlevel Sets on a Subset of a Metric Space

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, whose open sets form the topology $\mathcal{T}_{d}$ of Metric Open Sets Form a Topology, let $A\subseteq X$, and let $u:A\to\mathbb{R}$.

1. (Superlevel sets)¶ For $t\in\mathbb{R}$, the superlevel set of $u$ at height $t$ is the subset

of $X$.

2. (Closed superlevel sets)¶ The function $u$ has closed superlevel sets in $X$ if for every $t\in\mathbb{R}$ the set $\{u\ge t\}$ is closed in $(X,\mathcal{T}_{d})$.

This is a condition on the function $u$ together with the pair $A\subseteq X$: the superlevel sets are subsets of $A$, but they are required to be closed in the ambient space $X$, so the condition constrains the behaviour of $u$ near points of $X$ that do not lie in $A$.