Definition:Connected (Topology)

Topological Space
Let $T$ be a topological space.

Then $T$ is connected iff there does not exist any continuous surjection from $T$ onto a discrete two-point space.

Equivalently, $T$ is connected iff it admits no partition.

Set in Topological Space
Let $T$ be a topological space.

Let $A \subseteq T$.

Then $A$ is connected iff it cannot be expressed as the union of two separated sets.

Points in Topological Space
Let $T$ be a topological space.

Let $a, b \in T$.

Then $a$ and $b$ are connected (in $T$) iff there exists a connected set in $T$ containing both $a$ and $b$.

Disconnected
If: are not connected, then they are disconnected.
 * a topological space $T$
 * a subset $A$ of a topological space $T$
 * two points $a$ and $b$ of a topological space $T$

Also see

 * Equivalence of Connectedness Definitions for a series of equivalent definitions for connectedness.