I am reading Rudin's book on real analysis and am stuck on a few definitions.
First, here is the definition of a limit/interior point (not word to word from Rudin) but these definitions are worded from me (an undergrad student) so please correct me if they are not rigorous. The context here is basic topology and these are metric sets with the distance function as the metric.
A point $p$ of a set $E$ is a limit point if every neighborhood of $p$ contains a point $q \neq p$ such that $ q \in E$
Also, an interior point is defined as
A point $p$ of a set $E$ is an interior point if there is a neighborhood $N_r\{p\}$ that is contained in $E$ (ie, is a subset of E).
I understand interior points. Ofcourse given a point $p$ you can have any radius $r$ that makes this neighborhood fit into the set. Thats how I see it, thats how I picture it.
I can't understand limit points. It seems trivial to me that lets say you have a point $p$. Then one of its neighborhood is exactly the set in which it is contained, right? ie, you can pick a radius big enough that the neighborhood fits in the set.
Ofcourse I know this is false. Our professor gave us an example of a subset being the integers. He said this subset has no limit points, but I can't see how.