Remark

To show that all bounded open intervals are homeomorphic to each other, it would be enough to remark that, if $a \neq b$ and $a' \neq b'$, there is a homeomorphism of $\mathbf{R}$ onto itself, of the form $x \to ax + \beta$ (and only one) which maps $a$ to $a'$ and $b$ to $b'$, and hence maps the open (resp. half-open, closed) interval with end-points $a, b$ onto the open (resp. half-open, closed) interval with end-points $a', b'$; the reader may easily verify this by calculating $\alpha$ and $\beta$.