Q: Show that every point of $\mathbb{Q}$ is a limit point of $\mathbb{Q}$ How can one show that every point of the set of natural numbers, $\mathbb{Q}$,   is a limit point of $\mathbb{Q}$.

A: The fancy term for "every point of $\mathbb Q$ is a limit point" is that $\mathbb Q$ is dense in itself.
A topological space is always a "neighborhood of all of its points", so that is not a useful way of saying it.
On the other hand, $\mathbb Q$ is not a neighborhood in $\mathbb R$ of anything. Being a neighborhood is always something that is relative to a particular topological space.
If you have a subset of a topological space that happens to be a neighborhood of each of its points, that means neither more nor less than the set is open. ($\mathbb Q$ is definitely not open in $\mathbb R$).
