I have the following set:

\(\displaystyle \mathcal{J}= \left\{ (c,c+1), c \in \mathbb{R} \right\}\)

I would like to show that the sets:

\(\displaystyle (0, \frac{1}{2})\) , \(\displaystyle (0, \frac{1}{2}]\) , \(\displaystyle \left\{\frac{1}{2} \right\}\) , \(\displaystyle [0,1]\)

belong to the \(\displaystyle \sigma\)-algebra \(\displaystyle \sigma(\mathcal{J})\).

The elements of \(\displaystyle \sigma(\mathcal{J})\) are obtained by countable unions/intersections/differences of the elements of \(\displaystyle \mathcal{J}\).

I am having trouble constructing these intersections and unions. Assistance would be appreciated.