Question

Prove the isolated and the interior points of a set must belong to the set, but give examples to show the limit points and boundary points of a set may or may not belong to the set.

Answer 1

By definition, an isolated point x of a set S is a point of S. 
Also, an interior point of a set S can be defined by the mean of the 
existence of a positive number r>0 such that :
$D(x,r)\subset S\text{ therefore }x\in S$
wherein D(x,r) is the disc of center x and radius r. 

Example of a set whose boundary is not a subset of S: