Simply Connected Space - Formal Definition and Equivalent Formulations

Formal Definition and Equivalent Formulations

A topological space X is called simply connected if it is path-connected and any continuous map f : S1 → X (where S1 denotes the unit circle in Euclidean 2-space) can be contracted to a point in the following sense: there exists a continuous map F : D2 → X (where D2 denotes the unit disk in Euclidean 2-space) such that F restricted to S1 is f.

An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenever p : → X and q : → X are two paths (i.e.: continuous maps) with the same start and endpoint (p(0) = q(0) and p(1) = q(1)), then p and q are homotopic relative {0,1}. Intuitively, this means that p can be "continuously deformed" to get q while keeping the endpoints fixed. Hence the term simply connected: for any two given points in X, there is one and "essentially" only one path connecting them.

A third way to express the same: X is simply connected if and only if X is path-connected and the fundamental group of X at each of its points is trivial, i.e. consists only of the identity element.

Yet another formulation is often used in complex analysis: an open subset X of C is simply connected if and only if both X and its complement in the Riemann sphere are connected.

The set of complex numbers with imaginary part strictly greater than zero and less than one, furnishes a nice example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. It might also be worth pointing out that a relaxation of the requirement that X be connected, leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has connected extended complement exactly when each of its connected components are simply connected.