How to identify which sets genenrate Borel Sigma Algebra?

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.

• Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures. Commented Sep 12, 2020 at 6:45
• I understand, but please bear with me for the first time. In the mean time, I will learn Latex and get better at it. Commented Sep 12, 2020 at 6:48

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 $$0$$.