Question

Let X be a metric space and t: X to X be a map that preserves distances: d(t(x), t(y)) = d(x, y). Give an example in whicht is not bijective.

Could let t: x to x+1,x non-negative, but how does this mean t is not surjective?

Any help will be much appreciated!

Answer 1

Note That : Every distance preserving map are injective but not necessarily bijective. That means Distance preserving map need not to be surjective, in this case it will not be bijective.

Yes, Your taken example is correct. But you have to consider X is set non negatives.

So, I am modifying your example by takin X=N (set of natural numbers).

Define a map,

   such that,

t(n) =n+1   

Fisrt, we prove that it is distance preserving map.

Cosider, Let

It follows definition of distance preserving.

Therefore, map t , is distance preserving.

Now, we prove it is not surjective.

For, this it is enough to show that there exist atleast one element in N (co domain) which has no pre-image in N ( domain)

Take, (co-domain)

Then   will be preimage of 1, as  

But,   ( domain)

Thus,   (co-domain) has no pre-image in N (domain)

Hence,  It is not surjective, so not bijective.