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I need help in order to fully understand Singh's proof of Theorem 1.3.10 ...

Theorem 1.3.10 (plus the definition of boundary) reads as follows:

In the above proof by Singh we read the following:

" ... ... Conversely, if \(\displaystyle x \in \overline{A} - A\), then \(\displaystyle x \in \overline{A} \cap (X - A) \subseteq \partial A\) and the reverse inclusion follows. ... ... "

My questions are as follows:

**Question 1**Why is it true that \(\displaystyle \overline{A} \cap (X - A) \subseteq \partial A\) ... ?

I suspect that this is because \(\displaystyle (X - A) \subseteq \overline{ (X - A) }\) ... ... is that correct ... ?

**Question 2**How does \(\displaystyle x \in \overline{A} \cap (X - A) \subseteq \partial A\) lead to the reverse inclusion being true ... ... ?

(I am assuming that the reverse inclusion is \(\displaystyle \overline{A} \subseteq A \cup \partial A\) ... )

Help will be much appreciated ... ...

Peter