Support vs range of a random variable

Is there any difference between the two? I have not met any formal definition of the support of a random variable. I know that for the function $f$ the support is a closure of the set $\{y:\;y=f(x)\ne0\}$.

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The support of the probability distribution of a random variable $X$ is the set of all points whose every open neighborhood $N$ has the property that $\Pr(X\in N)>0$.

It is more accurate to speak of the support of the distribution than that of the support of the random variable.

The complement of the support is the union of all open sets $G$ such that $\Pr(X\in G)=0$. Since the complement is a union of open sets, the complement is open. Therefore the support is closed.