Limit point clarification

Question

Proposition 3.1.6: A is closed iff A contains its limit points

and the def of a limit point. https://en.wikipedia.org/wiki/Limit_point

I usually think closed sets have limit points because of 3.1.6. So l recently thought of open sets, and found they do too.

If both open and closed sets have limit points why is 3.1.6 useful?

Where is the defining difference in using it

Ok . I thought more. An open set could have more outside the set

Answer 1

Because while, indeed, usually open sets have limit poits, they do not contain all limit points.

Answer 2

There is a difference between "having" limit points and "containing" them. An open interval $(0,1)$ has $0$ and $1$ as limit points, but they do not belong to the interval. For closed sets, their limit points belong to them.

Answer 3

Proposition 3.1.6 is a characterization of closed sets. The original statement in the book (bold font mine):

Let $A$ be a subset of a topological space $(X,\tau)$. Then $A$ is closed in $(X,\tau)$ if and only if $A$ contains all of its limit points.

Your book comment on the previous page of the proposition that

"The next proposition provides a useful way of testing whether a set is closed or not."

and right after Proposition 3.1.6, it gives an example on how it is useful: