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I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to date ... see below ... )

Theorem 1.3.7 reads as follows:

In the above proof by Singh we read the following:

" ... ... So \(\displaystyle U\) is contained in the complement of \(\displaystyle A \cup A'\), and hence \(\displaystyle A \cup A'\) is closed. It follows that \(\displaystyle \overline{A} \subseteq A \cup A'\) ... ... "

My question is as follows:

Why does \(\displaystyle U\) being contained in the complement of \(\displaystyle A \cup A'\) imply that \(\displaystyle A \cup A'\) is closed ... and further why then does it follow that \(\displaystyle \overline{A} \subseteq A \cup A'\) ... ...

Help will be appreciated ...

Peter

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It is important that any proof of the above remarks only rely on the definitions and results Singh has established to date ... namely Definition 1.3.3, Proposition 1.3.4, Theorem 1.3.5 and Definition 1.3.6 ... which read as follows ... :

Hope that helps ...

Peter