Definition 2

A subset $A$ of a topological space $X$ is said to be a connected set if the subspace $A$ of $X$ is connected.

For $A$ to be a connected subset of $X$ it is necessary and sufficient that, for each covering of $A$ by two open (or closed) subsets $B, C$ of $X$ such that $A \cap B$ and $A \cap C$ are non-empty, we have $A \cap B \cap C \neq \emptyset$.