Connected set - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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 $C=U\cup V$ .

Equivalent characterizations

Equivalently, the only subsets of $C$ that are both open and closed in the subspace topology are $\varnothing$ and $C$ .

Remarks

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.