Im trying to understand the proof of theorem $2.5$ part $2)$ of Daniel Lackers paper : A Strong form of Propagation of Chaos.
In essence :
$\textbf{Assume the following :}$ we have a metric space $X$ and a lower-semi-continuous non-negative function $H:X\to \mathbb{R}$ ($\{x : H(x)\leq c\}$ is compact). Also $H(x)=0 ~\iff~ x=y$.
$\textbf{The question is :}$ why for any neighborhood $U$ of $y$ we have,
$$\inf_{x\notin U}H(x)>0. $$
Im sure we need to use Any lower semicontinuous function $f: X \to \mathbb{R}$ on a compact set $K \subseteq X$ attains a min on $K$. ?
The set $K=\{x:H(x)\leq 1\}$ is compact and thus so is $K\setminus U$. Thus as long as $K\setminus U$ is nonempty, $H$ achieves a minimum on $K\setminus U$, which is then also the minimum (and hence infimum) of $H$ on $X\setminus U$. Since $y\not\in K\setminus U$, this minimum must be positive. If $K\setminus U$ is empty, then $H(x)>1$ for all $x\not\in U$ and so the infimum of $H$ on $X\setminus U$ must be at least $1$.