the function F:(-1,1)toR dfined by F(x)=x/(1-x^2) is homeomorphism in a topology.. please explain the answer for...

the function F:(-1,1)toR dfined by F(x)=x/(1-x^2) is homeomorphism in a topology..

please explain the answer for me

Expert Solution

Now, for surjectiveness:

Since f(1-h) >f(c) > f(-1 +h), where h tends to zero.

As we know, any bijective, open continuous map is homeorphism.

If we dont want to use this theorem, in that case:

Since F is continuous , bijective and monotone. Hence its inverse exists, say g is the inverse, henece g is also monotone.

Without loss of generality, let F is strictly increasing, so g is.

Thus g satisfy the epsilon delta definition for continuity.

As we know, any bijective, bicontinuous function is homeorphism.

Colby Messinger answered 2 years ago

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