ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2972 on D2424: Set of real intervals
The four classes of real intervals are each closed under dilation
Formulation 1
Let $a, b, \lambda \in \mathbb{R}$ each be a D993: Real number such that
(i) \begin{equation} a \leq b \end{equation}
(ii) \begin{equation} \lambda \in (0, \infty) \end{equation}
Then
(1) \begin{equation} \lambda (a, b) = (\lambda a, \lambda b) \end{equation}
(2) \begin{equation} \lambda [a, b) = [\lambda a, \lambda b) \end{equation}
(3) \begin{equation} \lambda (a, b] = (\lambda a, \lambda b] \end{equation}
(4) \begin{equation} \lambda [a, b] = [\lambda a, \lambda b] \end{equation}