simply connected

A topological spaceMathworldPlanetmath is said to be simply connected if it is path connected and the fundamental groupMathworldPlanetmathPlanetmath 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 equivalentPlanetmathPlanetmath 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 2, S2 or the Riemann sphere. Non-examples of a simply connected space are the circle, the annulusMathworldPlanetmath, 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
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