Simply connected space

A path-connected topological space is simply connected when every loop can be continuously contracted to a point. Equivalently, its fundamental group is trivial.

Path-connected space: A topological space in which any two points can be joined by a continuous path. Path-connectedness is the first condition in the definition.
Covering space: A map of topological spaces that locally looks like a disjoint union of copies of its target. Coverings encode how loops wind and help detect nontrivial fundamental groups.
Euclidean space: The space of coordinate vectors in a fixed finite dimension, equipped with its usual topology. Every Euclidean space is simply connected.
Cauchy's integral theorem: A theorem stating that a holomorphic function has zero integral around suitable closed curves in a simply connected domain. Simple connectivity is a standard domain condition for this contour-integral result.
Locally simply connected space: A space in which every point has arbitrarily small neighborhoods that are simply connected. Local simple connectivity does not require the whole space to have trivial loops.
Loop (topology): A continuous map from a circle into a topological space, with a chosen point serving as its basepoint. Simple connectivity asks whether every loop can be contracted.
Universal cover: A connected, simply connected covering space that covers a given connected, locally path-connected space. It replaces a space’s loops with points in a simply connected covering space.
Sphere: The set of points at a fixed distance from a center in Euclidean space. Spheres of dimension at least two are simply connected, unlike the circle.
Holomorphic logarithm: A holomorphic function whose exponential equals a given nonvanishing holomorphic function. A nonvanishing holomorphic function on a simply connected domain admits a holomorphic logarithm.
Contractible space: A topological space homotopy equivalent to a point. Contractibility implies simple connectivity, but simple connectivity does not imply contractibility.

Linked from 22 pages