Definition of connectedness? Connectedness is defined as: "A metric space $E$ is connected if the only subsets of $E$ which are both open and closed are $E$ and $\varnothing$. A subset $S$ of a metric space is a connected subset if the subspace $S$ is connected."
Can someone provide me with a more trivial/simple definition of connectedness?

A space $X$ is disconnected if there is a continuous function $f \colon X \to \{0,1\}$ that is not constant.
For example: $\mathbb R$ is connected because every map $f \colon \mathbb R \to \{0,1\}$ is constant (intermediate value theorem). The space $[0,1] \cup [2,3]$ is disconnected, because the function can be defined piecewise.
