Closure of a Set

Definition

Closure of a set $\mathbb{E}$ is the set $\mathbb{\bar E}$ includes $\mathbb{E}$ and $\mathbb{E'}$ which contains all limit points of $\mathbb{E}$.

$\mathbb{\bar E} = \mathbb{E} \cup \mathbb{E'}$

$\mathbb{\bar E}$ is a closed set.

Proof: To prove $\mathbb{\bar E}$ is closed, we need to show that any limit point of $\mathbb{\bar E}$ is in $\mathbb{\bar E}$. Since limit points of $\mathbb{\bar E}$ are either in $\mathbb{E}$ or in $\mathbb{E'}$, they are also limit points of $\mathbb{E}$.

Consider p a limit point of $\mathbb{\bar E}$ , and $\mathbb{N(p)}$ a neighborhood of p.

$\mathbb{N}$ must contains a point of $q, q \in \mathbb{\bar E}$

Assume $q \in \mathbb{E}$, we found such $q$.

Assume $q \in \mathbb{E'}$, q has a smaller neigborhood $\mathbb{N(q)} \in \mathbb{N(p)}$.

Theorm

Neighborhood are open.

$\mathbb{E}$ is closed if and only if $\mathbb{E} = \mathbb{\bar E}$

The complement of an open set is closed: if $\mathbb{E}$ is open, then $\mathbb{E} ^ C$ is closed.