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1) Let S1 = {x E R^n | f(x)>0 or =0}

Let S2 = {x E R^n | f(x)=0}

Both sets S1 and S2 are "closed"

>>>>>I understand why S1 is closed, but I don't get why S2 is closed, can anyone explain?<<<<<

2) "A set X is a "metric space" if it admits a function d: X x X -> R such that:

(i) d(x,y)>0 or =0

(ii) d(x,y) = 0 iff x=y

(iii) d(x,y) = d(y,x)

(iv) d(x,z) < or = d(x,y) + d(x,z)"

>>>>>Now, I just don't get the "d: X x X -> R" part at all. How can you multiply two sets together? I don't get the notation, can someone please explain?<<<<<

3) "The mapping f: A->B is "one-to-one" if f(x)=f(y) implies x=y, and f is said to map A "onto" B if f(A)=B. A mapping f: A->B is said to be "invertible" if there is another mapping g: B->A such that g(f(x)) = x for all x E A and f(g(y)) = y for all y E B.

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The equation g(f(x)) = x can be valid for all x E A only if f is one-to-one, and the equation f(g(y)) = y can be valid for all y E B only if f maps A onto B. Conversely, if these two conditions are satisfied, then f is invertible." <<<<<-These are quoted from my textbook, but I don't understand how they come up with the conclusions in these last two sentences. Could someone kindly explain the reasonings?>>>>>

Thank you!