# Metric Space: Closure and Interior

$(X,d)$ be a metric space and $E \subset X$.

$\bar{E} = E \cup \{\text{limit points of$E$} \}$

$int(E) = \{\text{collection of all interior points of$E$}\}$, where a point $e \in E$ is an interior point if there exists a neighborhood of $e$ contained in $E$. $\bar{E}$ denotes closure of $E$ and $int(E)$ denotes the interior of $E$. Then prove that

a) $\bar{E} = \bigcap_{E \subset F} F$, $F \subset X$ and $F$ is closed.

b) $int(E) = \bigcup_{V \subset E} V$, $V$ is open.

In (b) I am trying to show that LHS is contained in RHS and vice versa. LHS ⊂ RHS can be shown by considering a neighborhood N around a point x ∈E. N is an open set and N⊂E. So it is a subset of the union on the RHS. Now coming to other implication. Consider a point x in the RHS, so it must be element of some V⊂E. Would this V be a neighborhood? I know V is an open set but how do I ensure it is a neighborhood of $x$. My definition for neighborhood is as given in Rudin's Principles of Mathematical Analysis which states that a neighborhood of a point $p$ denoted by $Nr(p)$ consists of all points $q$ such that $d(p,q)<r$.

Please provide me with directions for part (a).

• Nice definitions. What is your question? – drhab Feb 5 '15 at 16:25
• In (b) I am trying to show that LHS is contained in RHS and vice versa. LHS $\subset$ RHS can be shown by considering a neighborhood N around a point x $\in E$. N is an open set and $N \subset E$. So it is a subset of the union on the RHS. Now coming to other implication. Consider a point x in the RHS, so it must be element of some $V \subset E$. Would this $V$ be a neighborhood? I know $V$ is an open set but how do I ensure it is a neighborhood of $x$. – saurav90 Feb 5 '15 at 16:44
• Normally a neighborhood of $x$ is defined to be a set that contains an open set wich on its turn contains $x$ as element. Consequently any open set that contains $x$ as element is a neighborhood of $x$. Are you practicizing another definition of neighborhood? – drhab Feb 5 '15 at 17:39
• To avoid closure of your question it is useful to add information that you placed in your comment (a testimony that you have been working on this) in your question. – drhab Feb 5 '15 at 17:41
• I am using the definition mentioned in Rudin. A neighborhood of a point $p$ denoted by $N_r(p)$ consists of all points $q$ such that $d(p,q) < r$. Please provide me with directions for part (a). – saurav90 Feb 5 '15 at 17:43