Connected space

Equivalent definitions in tabular format
A topological space is said to be connected if it satisfies the following equivalent conditions.

The term is typically used for non-empty topological spaces. Whether the empty space can be considered connected is a moot point.

Basic examples

 * The one-point space is a connected space.
 * Euclidean space is connected. More generally, any path-connected space, i.e., a space where you can draw a line from one point to another, is connected. In particular, connected manifolds are connected.
 * In algebraic geometry, the Zariski topology is connected.

Non-examples

 * Any discrete space of size more than one is not connected.
 * Totally disconnected spaces, like the set of rational numbers, is not connected, despite points being "close" to one another.

Facts
Any topological space (not necessarily connected) can be partitioned into its connected components. The space is connected iff it has a single connected component, namely the whole space itself.

Relation with size of space
Combining connectedness with a separation axiom usually yields a lower bound on the cardinality of the space as long as it has at least two points. Below are some examples of such facts:

Opposite properties

 * Totally disconnected space

Textbook references

 * , Page 148 (formal definition)
 * , Page 11 (formal definition)