A topological space X is called simply connected if it is pathconnected and any continuous map f : S^{1} > X (where S^{1} denotes the unit circle in Euclidean 2space) can be contracted to a point in the following sense: there exists a continuous map F : D^{2} > X (where D^{2} denotes the unit disk[?] in Euclidean 2space) such that F restricted to S^{1} is f.
An equivalent formulation is this: X is simply connected if and only if it is path connected, and whenever p : [0,1] → X and q : [0,1] → 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 pathconnected and the fundamental group of X is trivial, i.e. consists only of the identity element.
A surface (twodimensional topological manifold) is simply connected if and only if it is connected and its genus is 0. Intuitively, the genus is the number of "holes" or "handles" of the surface.
If a space X is not simply connected, one can often rectify this defect by using its universal cover, a simply connected space which maps to X in a particularly nice way.
If X and Y are homotopy equivalent and X is simply connected, then so is Y.
The notion of simply connectedness is important in complex analysis because of the following facts:
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