Is Every Point in a Subset of ℝ Either a Limit Point or an Isolated Point?

[Bipolarity]

Would it be correct to say that out of the following two statements, exactly one is always true and one is always false?

1) x is a limit point of S, where S is a subset of ℝ
2) x is an isolated point of S, where S is a subset of ℝ

In other words, every point is either a limit point of a set or an isolated point of that set.

Also, for a point to be a limit point/isolated point of a set, does it have to be in the set?

Thanks!

BiP

 

[jbunniii]

Every point in $S$ is either an isolated point of $S$ or a limit point of $S$. The two characterizations are mutually exclusive: a point in $S$ is an isolated point if and only if it is not a limit point of $S$.

$S$ need not contain all of its limit points. $S$ is closed if and only if it does contain them all.

Isolated points of $S$ are always contained in $S$.