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Math Help - metric space

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    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 x\inint( (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\inint(A) \wedge x\inint(B)

    For the converse use again (1) identity
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