Are the closed intervals of $\mathbb{R}$ precisely the compact connected sets? Equip $\mathbb{R}$ with the topology generated by open intervals $(a, b)$. A subset of $\mathbb{R}$ is compact iff it's closed and bounded. Is every closed bounded connected subset of $\mathbb{R}$ a closed interval $[a, b]$ (and conversely)? Is every open bounded connected subset of $\mathbb{R}$ an open interval $(a, b)$ (and conversely)? Is this somehow related to the fact that removing one point from $\mathbb{R}$ splits it into 2 disconnected pieces (how is this property called anyway)?

Clearly a closed and bounded interval is compact and connected.

Conversely, if a set is connected, then it is an interval, meaning it is a set $I$ with the following property: for all $x,y \in I$, if $x < z