If A and B are subsets of a metric space (X,d), show that int(A∩B) = int(A)∩int(B). Find an example where int(A∪B)≠int(A)∪int(B)
I've done the second part of the question, but could anyone help me to do the first part of the question?
LET $\displaystyle x\in$int($\displaystyle (A\cap B)$, then there exists a δ>0 and such that: $\displaystyle B(x,\delta)\subset A\cap B$
BUT from set theory we have that for any S,T,R SETS :
$\displaystyle S\subset (T\cap R) = (S\subset T)\cap(S\subset R)$
AND according to that:
$\displaystyle B(x,\delta)\subset(A\cap B) = (B(x,\delta)\subset A)\cap(B(x,\delta)\subset B)$.................................................. ..........................................2
hence $\displaystyle x\in$int(A)$\displaystyle \wedge$$\displaystyle x\in$int(B)
For the converse use again (1) identity