# 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 nulhomotopy

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