We say that a set of real numbers is closed if and only if it contains its limit points. Can I say that since a set of real numbers is open, it does not contain its limit points.
Can I say that if a set is open then it does not contain its limit points
general-topologyreal-analysis
Best Answer
No.
To begin with OPEN does not mean not CLOSED and CLOSED does not mean not OPEN. Consider $\emptyset$ is both closed and open. So is $\mathbb R$. And consider $[0,1)$ is neither open nor closed.
Second; closed means contains all its limit points if it has any. But if it also means that if a set doesn't have any limit points then it is also closed. This is because there are no limit points so that the set isn't missing any limit points; it has them all. So for example: $\{0,1,5\}$ is closed because it has no limit points so no limit point is outside the set.
And an open set may have limit points that are inside it. Consider $(0,1)$ then point $\frac 12$ is a limit point because every neighborhood of $\frac 12$ has a point (other then $\frac 12$) that is in $(0,1)$. So $A$ has some of its limit points (but not all-- it is missing $0$ and $1$.
But an ope set doesn't need to be missing any of it's limit points. $\mathbb R$ is open and it contains all its limit points. In this case $\mathbb R$ is both open and closed.