Simply connected space

A geometrical object is called simply connected if it consists of one piece and doesn't have any circle-shaped "holes" or "handles". Higher-dimensional holes are allowed. For instance, a doughnut (with hole) is not simply connected, but a ball (even a hollow one) is. A circle is not simply connected but a disk and a line are. The opposite is non-simply connected or, in a somewhat old-fashioned term, multiply connected.

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Informally, suppose someone hands you an object made out of a strong, inflexible material, one that won't bend or break under any condition. Shake the object and turn it every direction you can think of. If anything falls off, rattles, spins, or otherwise moves separately of the object, it's not a "simple" object. Formally, such a simple object is called a connected space, but for our informal definition, we can just think of a simple object as being an object that's all one piece.

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Once you have a simple object, take a piece of string and insert one end into the object at any point. Let that end of the string follow any path, leaving behind string everywhere it goes, and then emerge at the same spot it went in, so that you have a loop going through the object.

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Now hold on to both ends of the string and maneuver the string inside the object until you are able to pull the loop out through the hole. You may need to feed in some extra string, but that's not a problem. If you can find any path inside the object that makes it impossible to get the loop of string out, the object is not simply connected. If no path from any point of entry gets the loop caught in the object, then it is simply connected.

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Notice how this definition does not rule out higher-dimensional holes. For example, while a hollow ball has a 2-dimensional hole in its middle, any loop you tie around the ball you can shrink to a point. The stronger condition, that the object have no holes of any dimension, is called contractibility.

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In algebraic topology this idea is made into a formal tool.

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