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If I said that $x \in \overline{A}$ (closure of A), does this mean that $\forall G_{x}$ (means an open set containing x) $G_{x}\cap A \neq \varnothing$?

Is not any interior point satisfies this also ?

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    $\begingroup$ Yes, that is correct. $\endgroup$ – drhab May 1 '17 at 12:40
  • $\begingroup$ But is not any interior point satisfies this also ? $\endgroup$ – Intuition May 1 '17 at 16:09
  • $\begingroup$ Yes but interior points also satisfy stronger conditions the boundary points fail to satisfy. $\endgroup$ – avs May 1 '17 at 16:12
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    $\begingroup$ Read Kolmogorov and Fomin's "introduction to real analysis". $\endgroup$ – avs May 1 '17 at 16:13
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    $\begingroup$ Yes, they do, the set of interior points of $A$ is a subset of $A$ which on its turn is a subset of $\overline{A}$. $\endgroup$ – drhab May 1 '17 at 16:15
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This definition of $\overline{A}$ is quite correct, it's the set of adherence points of $A$ i.e. $$\overline{A} = \{x \in X: \forall O \subseteq X \text{ open } : x \in O \rightarrow O \cap A \neq \emptyset \}$$

If $x \in A'$, so $x$ is an accumulation point of $A$, i.e $$A'= \{x \in X : \forall O \subseteq X \text{ open }: x \in O : O \cap (A \setminus \{x\}) = (O \setminus \{x\} ) \cap A \neq \emptyset\}$$

It's clear that $A' \subseteq \overline{A}$: if $O \setminus \{x\}$ intersects $A$, then also the larger set $O$ will intersect $A$.

$A$ is closed iff $A = \overline{A}$ iff $A' \subseteq A$. Both are often used as the definition of a closed set. Also, if $x \in \overline{A}$ and $x \notin A$ then $x \in A'$, also clear from the definitions.

It's also clear that $\operatorname{int}(A) \subseteq A \subseteq \overline{A}$ I don't see any contradiction there.

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  • $\begingroup$ about your first definition of $\bar{A}$: what is $X$ and what is $O$? Is $O$ an open ball centered at $x$? $\endgroup$ – johnny09 Apr 20 at 17:49
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    $\begingroup$ @johnny09 $X$ is the whole space we’re working in. $O$ is any open set containing $x$; it could be a an open ball centered om $x$. $\endgroup$ – Henno Brandsma Apr 20 at 18:28

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