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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.

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

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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 $0$.

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    $\begingroup$ More presicely 2) generates the sigma algebra of countable sets and their complements. $\endgroup$ Commented Sep 12, 2020 at 7:33

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