The definition of a limit point I'm using is as follows:
$x$ is a limit point of $A \subset X$ if every neighbourhood of $x$ has a non-empty intersection with $A \setminus \{x\}$
The definition of a neighbourhood I'm using is a set which contains an open set containing the point in question.
Can you provide an intuition as to why neighbourhoods are used in this definition and not just open sets?
Best Answer
You can!
If you consider any open set containing $x$, then it will necessarily contain a neighbourhood containing $x$ (by definition of openness). If you were to define this limit point in terms of open sets, you might say:
$x$ is a limit point of $A$ if every open set $E$ containing $x$ has a non-empty intersection with $A$. And since any $E$ contains a neighbourhood of $x$, call it $V(x)$. We know by your definition of a neighbourhood that $V(x)$ contains $E'$ some open subset, so it would follow that every neighbourhood of $x$ would need to intersect $A$.