In the literature the support, $S$, of a random variable $X$ is defined as the smallest closed subset of real line $\mathbb{R}$ with probability $1$. Looking to prove that $S$ is where the graph of $X$'s cdf, $F$, is not “flat”. More formally, with $O_x$ an open interval containing $x$ with probability $P(O_x)$, show that: $S=\{x: \forall O_x\quad P(O_x)\neq 0\}$.
To clarify, prove that $S=\{x: \forall O_x\quad P(O_x)\neq 0\}$ is closed with probability 1 and is the smallest such closed certain set.