Definition:Simply Connected

Definition

Let $T = \struct{S, \tau}$ be a path-connected topological space.

Definition by fundamental group

$T$ is said to be simply connected if the fundamental group $\map {\pi_1} T$ of $T$ is trivial.

Definition by path-homotopy of loops

$T$ is said to be simply connected if all loops in $T$ with identical base points are path-homotopic.

Definition by path-homotopy of paths

$T$ is said to be simply connected if all paths in $T$ with identical initial points and final points are path-homotopic.

Definition by null-homotopy

$T$ is said to be simply connected if all loops in $T$ are path-homotopic with a constant loop.

Also known as

Some texts use the hyphenated form simply-connected, but it is uncommon.

Also see

• Equivalence of Definitions of Simple Connectedness
• Fundamental Group is Independent of Base Point for Path-Connected Space
• Jordan Curve Characterization of Simply Connected Set, which has a characterization of simply connected open subspaces of the Euclidean plane $\R^2$.

Sources

• 2000: James R. Munkres: Topology (2nd ed.): $9$: The Fundamental Group: $\S 52$: The Fundamental Group