# Definition:Simply Connected

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## 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