I have recently started a graduate course on Probability Theory and have been introduced to Borel $\sigma$ - Algebra. But I am having a hard time understanding it 'fully'. I understood the definition but I can't seem to figure out which sets can generate Borel $\sigma$ - Algebra and which can't.
To make things easier for me, I have assumed that for all purposes Borel $\sigma$ - Algebra contains almost everything in R.
For example,check this Question on Borel Sigma Field
According to me all options are correct but I don't have proper reasoning behind selecting all the 4 options. I would really appearicate any help and hints in this regard.
Only the sets 1. and 3. generate the Borel $\sigma$-algebra, set 2. doesn't generate any intervals, and sets generated by 4. will all stick around $0$.