Definition:Connected (Topology)

This page is about connectedness in topology. For other uses, see Definition:Connected.

Definition

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 if 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 if there exists a connected set in $T$ containing both $a$ and $b$.

Disconnected

If:

• a topological space $T$
• a subset $A$ of a topological space $T$
• two points $a$ and $b$ of a topological space $T$

are not connected, then they are disconnected.

Also see

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

• Results about connectedness can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Closures and Interiors
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 4$