|Topology is a branch of mathematics that describes
properties which remain unchanged under smooth deformations. If we
imagine surfaces to be made of clay, a smooth deformation is any bending,
mushing and shaping which does not require the discontinuous action of
tearing, punching of holes, or gluing. The number of holes in an
object is a property that is preserved in a continuous
transformation. A favorite example is a clay doughnut which can be
smoothly manipulated into the shape of a coffee mug. The doughnut hole
becomes the coffee cup handle so that the number of holes is preserved.
The doughnut and the coffee mug therefore have the same topology.
Imagine what it would be like if you were an ant on the surface of a torus, that is, a doughnut shape. Maybe all your friends believed that your "universe" (the surface of the torus) is flat and stretches forever in every direction. Yet you have doubts. So you might set out in a very straight line, dropping breadcrumbs as you go. Soon you come to the trail of breadcrumbs, which proves the "universe" is curved. In topological terms, A torus is finite and compact.
You can imagine making a topologically connected surfaces like a torus, which has one hole, by starting with a flat rectangular sheet of paper. You can glue the left edge to the right.
Then you can glue the top edge to the bottom edge. Notice that you have glued every point on the edge of the square paper to a point exactly across from it, either sideways or top-to-bottom. Now you have the topological equivalent of a torus, or a doughnut, or a coffee cup, or, in fact, any surface that has one hole.
A web site, which is the source of information in this page, and provides much more information, is
"In Space, do all roads lead to home?", by Janna Levin
The webmaster and author of this Math Help site is Graeme McRae.