Difference between the support of a discrete random variable and the atoms of its probability distribution I'm confused about the difference between the support of a discrete random variable and the atoms of its probability distribution. 
Suppose I have a discrete random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ with support $\mathcal{A}\subseteq \mathbb{R}$. I am told that the distribution of $X$ is $H$ with $c$ atoms, $a_1,\ldots,a_c$. 
Question: Is $\{a_1,\ldots,a_c\}=\mathcal{A}$?
My thought is the following: they are different because, while $\mathcal{A}$ could contain also a point $b$ at which $P(X=b)=0$, $P(X=a_i)>0$ for $i=1,\ldots,c$. 
Is this correct? 
 A: Below I describe a discrete probability distribution whose support is bigger than its set of atoms.
A point $x$ is a member of the "support" of the probability distribution of $X$ precisely if for every open neighborhood $G$ of $x$, we have $\Pr(X\in G)>0$.
Now suppose
$$
X = \begin{cases}
1/2 & \text{with probability }1/2, \\[6pt]
1/3 \text{ or } 2/3 & \text{with equal probabilities totalling }1/4, \\[6pt]
1/4 \text{ or } 3/4 & \text{with equal probabilities totalling }1/8, \\
& (\text{We skipped $2/4$ since it's not in lowest terms.}) \\[6pt]
1/5,\  2/5,\  3/5,\text{ or }4/5 & \text{with equal probabilities totalling }1/16, \\[6pt]
1/6\text{ or }5/6 & \text{with equal probabilities totalling }1/32, \\[6pt]
1/7,\  2/7,\  3/7,\  4/7,\  5/7,\text{ or }6/7 & \text{with equal probabilities totalling }1/64, \\[6pt]
\text{and so on.}
\end{cases}
$$
Then the probability distribution of $X$ is discrete since the probabilities of the atoms add up to $1$, i.e. 100 percent of the probability is in point masses.  The set of atoms is just the set of all rational numbers between $0$ and $1$.
But the support is the whole set $[0,1]$, which contains $0$ and $1$ (which are not atoms) and every irrational number between $0$ and $1$.  The reason for that is for that every open interval about each such number, the probability that $X$ falls in that interval is positive.
