# Mysteriously simple containment questions

1. May 20, 2013

### trap101

Prove that the set i) S$\subseteq$ S(Closure)
ii) (Sint)int = Sint

Ok these supposedly simple containment questions irk me every time, how simplistic do I have to unravel the darn definitions:

Attempts: i) Let x $\in$ S(Closure)
==> x$\in$S or x$\in$$\partial$S (Boundary of S)
==> if x$\in$S the there exists a B(r,x) $\subseteq$ S
likewise if x$\in$$\partial$S then there exists the
B(r,x)$\cap$S ≠∅ and B(r,x)$\cap$Sc≠∅

==> S$\subseteq$ S(Closure) I really don't see what else can be done

ii) isn't this just direct from the definition of the Sint? I mean it's the interior of an interior.

2. May 20, 2013

### verty

i) First try to prove a simpler statement that is analogous to what you are asked to prove. For example, how would you prove that S⊆ S?

PS. This is all the help I can give here.