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Could anybody give me a hand to prove the following question that I have just seen on the book? I really appreciate your help!

Let $X$, $d(x, y)$ be a metric space and $A ⊂ X$, $B ⊂ X$. Prove the following formulas:

(a) $∂(A ∪ B) ⊂ ∂A ∪ ∂B$.

(b) $int(A ∪ B) ⊃ int(A) ∪ int(B)$.

(c) Show by examples that in general there is no equality in (a) and (b).

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Surely you can do some of this on your own? Finding examples, perhaps? –  Gerry Myerson Nov 18 '12 at 0:21

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up vote 1 down vote accepted

$\newcommand{\bdry}{\operatorname{bdry}}\newcommand{\int}{\operatorname{int}}$HINTS:

(a) Assume that $x\notin\bdry A\cup\bdry B$ and show that $x\notin\bdry(A\cup B)$. Use the fact that if $x\notin\bdry A$, then either $x\in\int A$ or $x\in\int(X\setminus A)$. (Why?)

(b) This is completely straightforward; it really shouldn’t have caused any trouble at all, since the most obvious approach works. Suppose that $x\in\int A\cup\int B$. Then $x\in\int A$ or $x\in\int B$. Without loss of generality suppose that $x\in\int A$. Then there an $r>0$ such that $B(x,r)\subseteq A$. Can you finish it from there?

(c) Consider the sets $A=[0,1]$ and $B=(1,2]$ in $\Bbb R$ for both (a) and (b).

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