Points of a dense set are not limit points I'm reading Rudin's Principles of Mathematical Analysis. Here is how the book defines the dense set:

$E$ is dense in $X$ if every point of $X$ is a limit point of E, or a point of $E$ (or both).

To fully understand the definition, here is my question: 
Is there a dense set $E$, at least one point in $E$ is not a limit point? 
Furthermore, is it possible, that all points of a dense set are not limit points of $E$ ?

Take any nonempty set $X$, and define $d(x,x)=0$, for all $x\in X$, and $d(x,y)=1$, for all $x,y\in X$ with $x\ne y$.

It's easily verified that $d$ is a metric on $X$ (it's called the discrete metric), and clearly, all points of $X$ are isolated. 

Now just let $E=X$.

Then no point of $E$ is a limit point, and by definition, $E$ is dense in $X$.
