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Oster
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(1) (X,d) is a COMPACT metric space.
(2) f:X->X is a function such that
d(f(x),f(y))=d(x,y) for all x and y in (X,d)
Prove f is onto.
Things I know:
(2) => f is one-one.
(2) => f is uniformly continuous.
I tried to proceed by assuming the existence of y in X such that y has no pre-image.
That, and the fact that f is 1-1, implies that the sequence y(n)={f applied to y n times} is a sequence of distinct points. X is compact and hence y(n) has a convergent subsequence.
Also, X, f(X), f(f(X)),...and so on are all closed and nested (because f is continuous and X is compact?). Their intersection is non-empty because y(n) has a limit point which should be in the intersection? So, f restricted to the intersection is a continuous bijection.
Note: the case where X is finite can be solved by using the pigeonhole principle to show that Image(f) =/= X implies f is not one-one. And, loosely, compactness can be thought of as a generalization of finiteness...so...??
I really don't think I'm getting anywhere...
WHERE ARE YOU CONTRADICTION?
Please help. This is really bugging me.
(2) f:X->X is a function such that
d(f(x),f(y))=d(x,y) for all x and y in (X,d)
Prove f is onto.
Things I know:
(2) => f is one-one.
(2) => f is uniformly continuous.
I tried to proceed by assuming the existence of y in X such that y has no pre-image.
That, and the fact that f is 1-1, implies that the sequence y(n)={f applied to y n times} is a sequence of distinct points. X is compact and hence y(n) has a convergent subsequence.
Also, X, f(X), f(f(X)),...and so on are all closed and nested (because f is continuous and X is compact?). Their intersection is non-empty because y(n) has a limit point which should be in the intersection? So, f restricted to the intersection is a continuous bijection.
Note: the case where X is finite can be solved by using the pigeonhole principle to show that Image(f) =/= X implies f is not one-one. And, loosely, compactness can be thought of as a generalization of finiteness...so...??
I really don't think I'm getting anywhere...
WHERE ARE YOU CONTRADICTION?
Please help. This is really bugging me.
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