Connected Sets

Separated Sets

Two subsets $A,B$ of Metric Space $X$ are said to be separated if both $A\cap \overline{B}$ and $\overline{A}\cap B$ are empty.

Connected Sets

A set $E\subset X$ is said to be connected if $E$ is not a union of two nonempty separated sets.

Proposition (Connectedness in $\mathbb{R}$)

Let $C\subset (\mathbb{R},d)$. Then $C$ is connected if and only if $\forall x,y\in C$ and $z\in (x,y)$, we have that $z\in C$, such that $C$ is an interval.

Proof:

$(⟹)$ Suppose $C$ is connected. We want to show that $z\in C$. We prove by contradiction. Suppose $\exists x,y\in C$ and $x<z<y$ but $z\notin C$. Then

$$C=(C\cap (-\infty ,z))\cup (C\cap (z,\infty ))$$

where LHS is $A$ and RHS is $B$. But then $C$ is not connected and thus a contradiction.

$(⟸)$ TODO

Remark

$A,B$ separated $⟹$ $A,B$ are disjoint. The opposite need not happen.