# Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected[1]) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.

This shape represents a set that is not simply connected, because any loop that encloses one or more of the holes cannot be contracted to a point without exiting the region.

A topological space X is called simply connected if it is path-connected and any loop in X defined by f : S1X can be contracted to a point: there exists a continuous map F : D2X such that F restricted to S1 is f. Here, S1 and D2 denotes the unit circle and closed unit disk in the Euclidean plane respectively.

Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected.

A sphere is simply connected because every loop can be contracted (on the surface) to a point.

The definition rules out only handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of any dimension, is called contractibility.

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.

A surface (two-dimensional topological manifold) is simply connected if and only if it is connected and its genus (the number of handles of the surface) is 0.

A universal cover of any (suitable) space X is a simply connected space which maps to X via a covering map.

If X and Y are homotopy equivalent and X is simply connected, then so is Y.

The notion of simple connectedness is important in complex analysis because of the following facts:

The notion of simple connectedness is also a crucial condition in the Poincaré conjecture.