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is simply connected because every loop can be contracted (on the surface) to a point.]] To illustrate the notion of simply connected, suppose we are considering an object in three dimensions; for example, an object in the shape of a box, a doughnut, or a corkscrew. Think of the object as a strangely shaped Aquarium full of water, with rigid sides. Now think of a diver who takes a long piece of string and trails it through the water inside the aquarium, in whatever way he pleases, and then joins the two ends of the string to form a closed loop. Now the loop begins to contract on itself, getting smaller and smaller. (Assume that the loop magically knows the best way to contract, and won't get snagged on jagged edges if it can possibly avoid them.) If the loop can always shrink all the way to a point, then the aquarium's interior ''is'' simply connected. If sometimes the loop gets caught — for example, around the central hole in the doughnut — then the object is ''not'' simply connected. Notice that the definition only rules out "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 have no holes of ''any'' dimension, is called Contractibility . FORMAL DEFINITION AND EQUIVALENT FORMULATIONS A 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'' : [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 path-connected and the Fundamental Group of ''X'' is trivial, i.e. consists only of the Identity Element . Yet another formulation is often used in are connected. EXAMPLES
PROPERTIES A surface (two-dimensional topological Manifold ) is simply connected if and only if it is connected and its Genus is 0. Intuitively, the genus is the number of "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''. Note that the image of a simply connected set under a continuous function need not to be simply connected. Take for example the complex plane under the exponential map, the image is C - {0}, which clearly is not simply connected. The notion of simply connectedness is important in Complex Analysis because of the following facts:
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