User:Anghel/Sandbox

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

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

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

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

$T$ is said to be simply connected if all loops $\gamma : \closedint 0 1 \to T$ are homotopic to a constant mapping $\sigma : \closedint 0 1 \to T$, where the homotopy between $\gamma$ and $\sigma$ is both a path-homotopy and a nulhomotopy.

Also see

 * Simple Connectedness is Preserved under Homeomorphism, which shows that simple connectedness is a topological property.
 * Fundamental Group is Independent of Base Point for Path-Connected Space

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

1 => 2
Let $x \in S$.

From Fundamental Group is Independent of Base Point for Path-Connected Space, it follows that all fundamental groups $\map {\pi_1}{T, x}$ are isomorphic to $\map {\pi_1}{ T }$.

By assumption, it follows that all fundamental groups $\map {\pi_1}{T, x}$ are trivial.

As the single element of $\map {\pi_1}{T, x}$ is a homotopy class of paths, it follows that all loops in $T$ with base point $x$ belong to the same homotopy class.

It follows that these loops are path-homotopic.

Links

 * Fundamental Group is Independent of Base Point for Path-Connected Space
 * Homotopy Characterisation of Simply Connected Sets

Category:Definitions/Simply Connected Spaces]] Category:Definitions/Topology]] Category:Definitions/Algebraic Topology]]