This article is written like a personal reflection or opinion essay that states a Wikipedia editor’s personal feelings about a topic. Please help improve it by rewriting it in an encyclopedic style. Informally, a thick object in our space is simply-connected if it consists of one piece complex analysis gamelin pdf download does not have any “holes” that pass all the way through it. In two dimensions, a circle is not simply-connected, but a disk and a line are.
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.