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.