Exercise 2.3.5 (Semiopen real intervals are homeomorphic)

Proof. It suffices to demonstrate that the intervals of the $[a,b),[a,+\infty )$ and $(-\infty ,b],(a,b]$ are homeomorphic to the semiopen unit interval $[0,1)$. To do so, we extend the function used in the proof of the previous exercise to include the endpoint, i.e.,

and

It is easy to verify that both functions remain continuous and bijective after the inclusion of the endpoint $a$. We can use the same functions for the cases $(-\infty ,b]$ and $(a,b]$ by simply reversing the direction of their mapping. □