# A subset of a metric space is dense if...

Let $$(X,d)$$ be a metric space, $$E$$ a subset of $$X$$ and $$p$$ a point of $$X$$.

• A neighborhood of $$p$$ is a subset of $$X$$ consisting of all points $$q$$ of $$X$$ with $$d(p,q), some $$r>0$$.

• $$p$$ is a limit point of $$E$$ if every neighborhood of $$p$$ contains a point $$q\neq p$$ of $$X$$ which is a point of $$E$$

Definition 1 $$E$$ is said to be dense in $$X$$ if every point of $$X$$ is a limit point of $$E$$.

Definition 2 $$E$$ is said to be dense in $$X$$ if every point of $$X$$ is a limit point of $$E$$, or a point of $$E$$ (or both)

Is "..or a point of $$E$$, (or both)" really necessary in definition 2? In other words, are the two definitions defining the same class of subsets of $$X$$? I was thinking: If a point of $$X$$ is a point of $$E$$, then it is a limit point of $$E$$, unless $$E$$ has some isolated points, in that case I would have that $$E$$ is not dense into itself, according to definition 1. Is that the only circumstance where I need to specify "..or a point of $$E$$", or there are many cases of spaces $$X$$ and subspaces $$E$$ where the two definitions are actually not equivalent?

Consider any set $$X \neq \emptyset$$ with the discrete metric. Then $$X$$ is dense in itself but no point of $$X$$ is a limit point of $$X$$, so we must require ".. or a point of $$E$$" in the definition.
In fact, instead of asking for limit point as you defined, you can just define an adherent point of $$E$$ as a point $$x \in X$$ such that for every neighbood of $$x$$ contains a point of $$E$$. Then it is easy to see that a set $$E$$ is dense in $$X$$ iff every point of $$X$$ is an adherent point of $$E$$, so you don't need to distinguish cases. Note also that the set of adherent points of $$E$$ is exactly the closure of $$E$$, so basically the definition then becomes $$X= \overline{E}$$.