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Concept
Simply connected space
Summary
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) 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.
A topological space is called if it is path-connected and any loop in defined by can be contracted to a point: there exists a continuous map such that restricted to is Here, and denotes the unit circle and closed unit disk in the Euclidean plane respectively.
An equivalent formulation is this: is simply connected if and only if it is path-connected, and whenever and are two paths (that is, continuous maps) with the same start and endpoint ( and ), then can be continuously deformed into while keeping both endpoints fixed. Explicitly, there exists a homotopy such that and
A topological space is simply connected if and only if is path-connected and the fundamental group of at each point is trivial, i.e. consists only of the identity element. Similarly, is simply connected if and only if for all points the set of morphisms in the fundamental groupoid of has only one element.
In complex analysis: an open subset is simply connected if and only if both 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 be connected leads to an interesting exploration of open subsets of the plane with connected extended complement.
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