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 :
[tex]D(x,r)\subset S\text{ therefore }x\in S[/tex]
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:
