Connected set

Definition

Let $X$ be a topological space and let $S \subseteq X$. The set $S$ is connected when $S$, with the subspace topology, is a connected space.

Remarks

Equivalently, there are no open sets $U, V$ of $X$ with $S \subseteq U \cup V$, $S \cap U \cap V = \varnothing$, $S \cap U \neq \varnothing$ and $S \cap V \neq \varnothing$. The connected subsets of the real line are the intervals. The closure of a connected set is connected. A union of connected sets with a common point is connected.

Explanations

L1 human

A subset is connected when it is in one piece, as a space of its own. For example, the set $\mathbb{Q}$ of rationals is not connected in $\mathbb{R}$: the sets $(-\infty, \sqrt{2})$ and $(\sqrt{2}, \infty)$ split it.