# intervals as unions and intersections

#### Meelas

Hello,

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.