Editing Simply connected space (section)

==Examples==
[[Image:Torus cycles.png|thumb|right|150px|A torus is not a simply connected surface. Neither of the two colored loops shown here can be contracted to a point without leaving the surface. A [[solid torus]] is also not simply connected because the purple loop cannot contract to a point without leaving the solid.]]
* The [[Euclidean space|Euclidean plane]] <math>\R^2</math> is simply connected, but <math>\R^2</math> minus the origin <math>(0, 0)</math> is not. If <math>n > 2,</math> then both <math>\R^n</math> and <math>\R^n</math> minus the origin are simply connected.
* Analogously: the [[n-sphere|''n''-dimensional sphere]] <math>S^n</math> is simply connected if and only if <math>n \geq 2.</math>
* Every [[convex subset]] of <math>\R^n</math> is simply connected.
* A [[torus]], the (elliptic) [[cylinder (geometry)|cylinder]], the [[Möbius strip]], the [[projective plane]] and the [[Klein bottle]] are not simply connected.
* Every [[topological vector space]] is simply connected; this includes [[Banach space]]s and [[Hilbert space]]s.
* For <math>n \geq 2,</math> the [[special orthogonal group]] <math>\operatorname{SO}(n, \R)</math> is not simply connected and the [[special unitary group]] <math>\operatorname{SU}(n)</math> is simply connected.
* The one-point compactification of <math>\R</math> is not simply connected (even though <math>\R</math> is simply connected).
* The [[Long line (topology)|long line]] <math>L</math> is simply connected, but its compactification, the extended long line <math>L^*</math> is not (since it is not even path connected).