If there is no open interval containing $p$, is $p$ a limit point?

Question

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.

Answer 1

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 $\{2,3\}$.

Answer 2

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.