In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces.

A subset of a topological space ${\displaystyle X}$ is a connected set if it is a connected space when viewed as a subspace of ${\displaystyle X}$.

Some related but stronger conditions are path connected, simply connected, and ${\displaystyle n}$-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.

A topological space ${\displaystyle X}$ is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, ${\displaystyle X}$ is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.

For a topological space ${\displaystyle X}$ the following conditions are equivalent:

Historically this modern formulation of the notion of connectedness (in terms of no partition of ${\displaystyle X}$ into two separated sets) first appeared (independently) with N.J. Lennes, Frigyes Riesz, and Felix Hausdorff at the beginning of the 20th century. See (Wilder 1978) for details.

Given some point ${\displaystyle x}$ in a topological space ${\displaystyle X,}$ the union of any collection of connected subsets such that each contains ${\displaystyle x}$ will once again be a connected subset. The connected component of a point ${\displaystyle x}$ in ${\displaystyle X}$ is the union of all connected subsets of ${\displaystyle X}$ that contain ${\displaystyle x;}$ it is the unique largest (with respect to ${\displaystyle \subseteq }$) connected subset of ${\displaystyle X}$ that contains ${\displaystyle x.}$ The maximal connected subsets (ordered by inclusion ${\displaystyle \subseteq }$) of a non-empty topological space are called the connected components of the space. The components of any topological space ${\displaystyle X}$ form a partition of ${\displaystyle X}$: they are disjoint, non-empty and their union is the whole space. Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the rational numbers are the one-point sets (singletons), which are not open. Proof: Any two distinct rational numbers ${\displaystyle q_{1}<q_{2}}$ are in different components. Take an irrational number ${\displaystyle q_{1}<r<q_{2},}$ and then set ${\displaystyle A=\{q\in \mathbb {Q} :q<r\}}$ and ${\displaystyle B=\{q\in \mathbb {Q} :q>r\}.}$ Then ${\displaystyle (A,B)}$ is a separation of ${\displaystyle \mathbb {Q} ,}$ and ${\displaystyle q_{1}\in A,q_{2}\in B}$. Thus each component is a one-point set.

Let ${\displaystyle \Gamma _{x}}$ be the connected component of ${\displaystyle x}$ in a topological space ${\displaystyle X,}$ and ${\displaystyle \Gamma _{x}'}$ be the intersection of all clopen sets containing ${\displaystyle x}$ (called quasi-component of ${\displaystyle x}$). Then ${\displaystyle \Gamma _{x}\subset \Gamma '_{x}}$ where the equality holds if ${\displaystyle X}$ is compact Hausdorff or locally connected.