Separated Sets

Any two sets, say $\mathbb{A}$ and $\mathbb{B}$, are separated if both $\mathbb{A} \cap \bar{\mathbb{B}}$ and $\bar{\mathbb{A}} \cap \mathbb{B}$ are empty

Connected Sets

A set $\mathbb{E}$ is connected if $\mathbb{E}$ is not the union of two separated sets in $\mathbb{E}$.

A set $\mathbb{E}$ is not the union of two disjoint and nonempty open sets in $\mathbb{E}$.

$[a, b]$ is connected.

proof: suppose not, $\exists$ two separated sets, say $\mathbb{A}$ and $\mathbb{B}$.

Let s = sup $\mathbb{A}$, then if $s \in \bar{\mathbb{A}}$, so $s \notin \mathbb{B}$; if $s \in \mathbb{A}$, so $s \notin \bar{\mathbb{B}}$.

Thus, $\exists (s-\epsilon, s+\epsilon)$ containing no points of $\mathbb{B}$.

$\rightarrow \forall$in A, which is contradictory to $s=sup A$.