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connected




A space S is said to be connected if any two points in S can be connected by a curve lying wholly within S. Two spaces can be added by what is called a connected sum. Roughly speaking, this involves pulling out a disk from each surface, creating holes, and then sewing the two surfaces together along the boundaries of the holes. In this way, a one-holed torus can be added to a two-holed torus to give a three-holed torus; alternatively, a projective plane can be added to a projective plane to give a Klein bottle. The operation is commutative and associative and there is even an identity element: for example, adding a sphere to any surface simply returns the same surface.


Related entry

   • simply-connected


Related category

   • TOPOLOGY