TOPOLOGT 541 Let M be a set with a real valued function D M

TOPOLOGT 541

Let M be a set with a real valued function D : M × M satisfying the following:

(1) D(a, a) = 0;
(2) D(a, b) = 0 for a = b;
(3) D(a,b)+D(b,c)D(a,c)foralla,b,andc.

Prove that (M,D) is a metric space. (Note: It is not assumed that D(a, b) 0 and D(a, b) = D(b, a). You need to prove them.)

Given that D(a,b)+D(b,c) >=D(a,c)

D(a,b) -D(b,a) = D(a,b) +D(-b,a) >=D(a,a)

Hence D(a,b) >= D(b,a) since D(a,a,) =0

Similarly it can be proved that

D(b,a) >= D(a,b)

Hence it follows that D(a,b) = D(b,a)

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Hence D(x,y) + D(y,x) >= D(x,x) (Since d(x,y) = d(y,x) it follows that 2d(x,y))

Or 2d(x,y)>=0

It follows that d(x,y) >=0 for all x and y.

Hence I condition is prove that D(a,b) >=0 for all a and b.

So M,D is a metric space.