metric space

**jin_nzzang** · April 22nd 2009, 03:27 AM

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?

**xalk** · April 22nd 2009, 07:22 AM

Originally Posted by jin_nzzang

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 $x\in$ int( $(A\cap B)$ , then there exists a δ>0 and such that: $B(x,\delta)\subset A\cap B$

BUT from set theory we have that for any S,T,R SETS :

$S\subset (T\cap R) = (S\subset T)\cap(S\subset R)$

AND according to that:

$B(x,\delta)\subset(A\cap B) = (B(x,\delta)\subset A)\cap(B(x,\delta)\subset B)$ .................................................. ..........................................2

hence $x\in$ int(A) $\wedge$ $x\in$ int(B)

For the converse use again (1) identity

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