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.

Disconnected
If $T$ is not connected, then it is disconnected.

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

Let $A \subseteq T$.

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

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

Let $x, y \in T$.

Then $x$ and $y$ are connected if there exists a connected set in $T$ containing both $x$ and $y$.