Refer to Principles of Mathematical Analysis by Walter Rudin (3rd edition), Page 32, Definitions 2.18 (a), (b), (d), (e) and (f).
Let $N_r(p)$ denote neighbourhood of $p \in X$ of radius $r>0$
Let $ I(p) : (\exists r > 0) N_r(p) \subseteq X $ denote an interior point $p$
Let $ L(p) : (\forall r > 0) ( \exists q \in X ) q \in N_r(p), q \ne p $ denote a limit point $p$
According to 2.18 (d) $ (\forall x \in X) L(x) $ is false, therefore $X$ does not have any limit points. Equivalently let $L = \phi$ be the set of limit points and as $L \subset X$, the interpretation says $X$ is Closed. (The same is reflected in the example provided in 2.21 (c) below from this book).
However, according to 2.18 (f) $ (\forall x \in X) I(x) $ is true, therefore $X$ should be an Open Set
My question is Whether the interpretation in (5) is Correct? If not why?
[However, example 2.21 (c) below from this book says, "A nonempty finite set" is not an Open Set, which is not clear.]
On the sidelines, one general issue I have observed is, not using propositional (or first order) logic to unambiguously state ideas in this text book, in this particular section. There are verbal or common language textual statements which may be causing issues in my interpretation.
The inference in point (5) applies to all finite metric spaces. Going by the definition 2.15 of this book where a set is called metric space (instead of $(X,d)$).
