If there is no open interval containing $p$, is $p$ a limit point? First we begin with my understanding of the definition of a limit point. 
“$p$ is said to be a limit point of a point set $M$ if every open interval containing $p$ contains a point of $M$ distinct from $p$.“ With this definition, suppose there is no open interval containing $p’$, is $p’$ a limit point? Under this definition there’s an implication that an open interval exists, the definition can be rewritten as “If there exists an open interval containing $p$, it must contain a point of $M$ distinct from $p$.” With this restatement, since there does not exist an open interval containing $p’$ then one could argue that every open interval (if it existed) contains a point of $M$ distinct from $p’.$ With this same logic I think you could argue the opposite way as well however I’m wondering if my argument works. 
 A: Let $M$ be a subset in a space $X$ and consider a point $p \in X$.  You want to ask what happens in analyzing if $p$ is a limit point of $M$ if there happens to be no open sets about $p$.  Well, that can't happen, since $X$ is open and $p \in X$.  So, as we discussed upthread in comments, your question is pointless (pun intended).
You could ponder:  what if $X$ is the only open set about $p$?  In other words, no proper open set surrounds $p$.  In that case, $p$ usually is a limit point of $M$.  This will be the case as long as $M$ contains something different than $p$. 
An example of this occurring is to take $X=\{1, 2, 3\}$ and give it the topology 
$$
\varnothing, \{1\}, \{1,2\}, X.
$$ 
Take $p=3$ and note that $X$ is the only open set about $p$.  With $M=\{1,2\}$ we see that $p=3$ is a limit point of $M$:  the open set $X$ contains the point $1 \in M$, and $1$ is not $p$.  Since this is the only open candidate, we're done.  The same argument flies with $p=3$ being a limit point of $M'=\{1,3\}$.
But, with $M''=\{3\}$ we have trouble.  Here the unique open set about $p$, namely $X$, contains no point of $M''$ different from $p=3$.  Thus $p$ is not a limit point of $M''$.
Exercise:  with the same set $X$ with topology above, work out all limit points of $\{2\}$ and then $\{2,3\}$.
A: The definition "If every open interval containing $p$ contains a point of $M$ distinct from $p$ then $p$  is said to be a limit point of a point set $M$ " is a conditional statement.
For a conditional statement the inverse is not true in general. However its contrapositive is always true.
Take $p=1$ in the set $[0,1]$ for which your definition doesn't fit into the antecedant part even.
