拓扑空间X称为是连通的。当且仅当以下叙述之一成立：

• X不能表示为两个不相交的非空开集的并集。
• ∀A⊆X，A≠X或∅，A∩(X-A)≠∅。

一个拓扑空间被称为是不连通的，若它不是连通的。

连通性是拓扑空间的一个拓扑不变性质，即两个拓扑空间之间若存在一个同胚映射，其中一个空间是连通的，则另一个空间也是连通的。

一些数学家承认空集(按照它独有的拓扑)是连通空间，不过也有数学家不承认这一点。

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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path.