# simply connected

A topological space^{} is said to be *simply connected* if it is path connected and the fundamental group^{} of the space is trivial (i.e. the one element group). What this means, basically, is that every path on the space can be shrunk to a point. This is equivalent^{} to saying that every path is contractible. A simply connected space can be visualized as a space with no “holes”.

Some basic examples of a simply connected space are the unit disc in ${\mathbb{R}}^{2}$, ${S}^{2}$ or the Riemann sphere. Non-examples of a simply connected space are the circle, the annulus^{}, and a punctured plane (a plane with a point removed). In each of the non-examples, any closed curve around the “hole” is a path that can not be shrunk to a point.

Title | simply connected |
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Canonical name | SimplyConnected |

Date of creation | 2013-03-22 11:59:33 |

Last modified on | 2013-03-22 11:59:33 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 9 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 55P15 |

Related topic | SemilocallySimplyConnected |