We consider the following collection of sets:
$$\mathcal{I}_0:=\left\{(a,b]\; |\; -\infty\le a\le b<\infty \right\}\cup\{(a,\infty)\;|\; a\in\mathbb{R}\}.$$
We denote with $\mathcal{I}$ the collection formed by the finite unions of elements of $\mathcal{I}_0$.
Question. Intervals of the type $[a,b)$ belong to $\mathcal{I}$?
In my opinion not, but I can't show it formally, could someone help me?
You're right, if you had $$ \begin{align*} \bigcup_{ k = 1}^{ n} ( a_{ k} , b _{ k} ] = [c, d) \end{align*} $$ then the left set's infimum is not contained in the set (it's $ \min_{ 1\leqslant k\leqslant n} a_{ k} $) whereas the infimum of $ [c, d)$ is contained in the set (note that the adding sets $(a, \infty)$ doesn't change the argument).