Activity 15.4.
Let \(a\) and \(b\) be real numbers with \(a \lt b\text{.}\) To show that \((\R, d_E)\) is homeomorphic to \((a,b)\text{,}\) we need a continuous bijection from \(\R\) to \((a,b)\) whose inverse is also continuous.
(a)
First we demonstrate that \((0,1)\) and \(\R\) are homeomorphic using the Euclidean metric topology. Let \(f : (0,1) \to \R\) be defined by
\begin{equation*}
f(x) = \tan\left(\pi\left(x-\frac{1}{2}\right)\right)\text{.}
\end{equation*}
(i)
Explain why \(f\) maps \((0,1)\) to \(\R\text{.}\)
(ii)
Explain why \(f\) is an injection.
(iii)
Explain why \(f\) is a surjection.
(iv)
Explain why \(f\) and \(f^{-1}\) are continuous.
Hint.
Use a result from calculus.
(b)
The result of (a) is that \(\R\) and \((0,1)\) are homeomorphic spaces. To complete the argument that \(\R\) is homeomorphic to \((a,b)\text{,}\) define a function \(g: (0,1) \to (a,b)\) and explain why your \(g\) is a homeomorphism.
