Baby Rudin Definition 2.18

Question

This are few Definitions stated in Principles of Mathematical Analysis by Walter Rudin:

Now, Consider the interval (1,2).

This interval is closed because according to $(b)$ the every point of this set contains another point in its neighbourhood, thus a limit point, hence closed.

Open at the same time according to the $(e), (f)$ given in the definition.

Please clarify.

Answer 1

There is a difference between

Every limit point of $E$ is a point of $E$ . . . . .$(1)$

and

Every point of $E$ is a limit point of $E$ . . . . . .$(2)$

You have reasoned that $(2)$ is true and concluded that $E$ is closed. But this is incorrect, because you need to check whether $(1)$ is true, since $E$ is defined to be closed if $(1)$ holds.

And indeed, $1$ is a limit point of $E = (1,2)$ but $1 \not\in E$, so $E$ is not closed.

Answer 2

$1$ and $2$ are limit points of $E= (1,2)$ but are not points of $E$. So $E$ is not closed.