homeomorphism..???

Question

Prove that (0,1] is homeomorphic to (1/2,1] by defining the homeomorphism.

Answer 1

Consider the function:

f: (0, 1] → (1/2, 1], f(x) = (x+1)/2

This function is bijective, with inverse f⁻¹(x) = 2x-1. Both the function and its inverse are continuous. Therefore, it is a homeomorphism between (0, 1] and (1/2, 1]. So we are done.