Connected set

A connected set is a subset $C\subseteq X$ of a topological space $X$ such that there do not exist disjoint nonempty sets $U,V\subseteq C$ that are open in the subspace topology on $C$ and satisfy $C=U\cup V$. Equivalently, the only subsets of $C$ that are both open and closed in the subspace topology are $\varnothing$ and $C$.

Connectedness is a basic qualitative invariant of spaces, and it is preserved by continuous maps (see continuous images of connected sets ). Maximal connected pieces are called connected components .

Examples:

• Any interval in $\mathbb{R}$ (with the usual topology) is connected.
• The set $(-1,0)\cup(0,1)\subseteq\mathbb{R}$ is not connected.