$\overline{X}=X\cup\partial X$
I'm stuck in this problem. That is the definition of boundary I should use:
$\partial X=\overline{X}\cap \overline{X^c}$
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your definition of a boundary is the empty set. – infinity Feb 03 '20 at 14:55
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$\partial X = \bar{X} \cap \bar{(X^c)}$ the "()" makes the diferency, Sorry. – Matheus José Feb 03 '20 at 15:01
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Small latex tip: In order to denote the closure of the complement of $X$, use \overline{X^c} instead of \bar{X^c}. The former is clearer. – bliipbluup Feb 03 '20 at 15:02
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Thank you. I edited. – Matheus José Feb 03 '20 at 15:03
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Have you tried substituting in the definition of a boundary and then used the distributive property of sets? – Feb 03 '20 at 15:49
1 Answers
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The inclusion $"\supseteq"$ follows from $X \subseteq \overline{X}$ and $\partial X= \overline{X} \cap \overline{X^c} \subseteq \overline{X}$.
For the reverse inclusion we take $x\in \overline{X}$. If it is in $X$, then we are happy. On the other hand, if $x\in \overline{X} \setminus X$, then we need to show that $x\in \partial X$. We already have $x\in \overline{X}$, so we are left to show that $x\in \overline{X^c}$. This follows from $x\in X^c \subseteq \overline{X^c}$.
Severin Schraven
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