Connected set

Let $(X,d)$ be a metric space and let $E\subseteq X$. A subset $U\subseteq E$ is open in the subspace $E$ (also called relatively open) if there exists an open set $O\subseteq X$ such that

The set $E$ is connected if there do not exist nonempty disjoint sets $U,V\subseteq E$ that are open in $E$ and satisfy

Equivalently, the only subsets of $E$ that are both open in $E$ and closed in $E$ (i.e. clopen in $E$) are $\varnothing$ and $E$.

Connectedness is the topological abstraction of “being in one piece.” It is fundamental for the intermediate value property and for global qualitative behavior of continuous functions .

Examples:

• Any interval $[a,b]\subset\mathbb{R}$ is connected.
• The set $(0,1)\cup(2,3)\subset\mathbb{R}$ is not connected.
• The unit circle $S^1=\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\}$ is connected.