1. ## metric space

If A and B are subsets of a metric space (X,d), show that int(AB) = int(A)∩int(B). Find an example where int(AB)≠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?

2. Originally Posted by jin_nzzang
If A and B are subsets of a metric space (X,d), show that int(AB) = int(A)∩int(B). Find an example where int(AB)≠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