Connected space

Definition

A topological space $X$ is connected when there are no open sets $U, V \subseteq X$ with $U \neq \varnothing$, $V \neq \varnothing$, $U \cap V = \varnothing$ and $U \cup V = X$.

Remarks

Equivalently, the only subsets of $X$ that are both open and closed are $\varnothing$ and $X$. Equivalently, each continuous map from $X$ to the discrete space $\{0, 1\}$ is constant. The interval $[0, 1]$ is connected. The set $[0, 1] \cup [2, 3]$ is not connected. Under this definition the empty space is connected. Some authors require a connected space to be nonempty.

Explanations

L1 human

A connected space cannot be split into two separate open parts. Connectedness is the topological form of the idea that a space is in one piece.