Simply connected

In topology, a topological space is called simply-connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other such path while preserving the two endpoints in question (see below for an informal discussion).

If a space is not simply-connected, it is convenient to measure the extent to which it fails to be simply-connected; this is done by the fundamental group. Intuitively, the fundamental group measures how the holes behave on a space; if there are no holes, the fundamental group is trivial — equivalently, the space is simply connected.

Informally, a thick object in our space is simply-connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply-connected, but a disk and a line are. Spaces that are connected but not simply connected are called non–simply connected or, in a somewhat old-fashioned term, multiply connected.

To illustrate the notion of simple connectedness, 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 has no holes of any dimension, is called contractibility.

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