Topology (from the Greek τόπος, "place", and λόγος, "study") is the mathematical study of shapes and spaces. It is a major area of mathematics concerned with the most basic properties of space, such as connectedness, continuity and boundary. It is the study of properties that are preserved under continuous deformations including stretching and bending, but not tearing or gluing.
There is a common joke among mathematicians that a topologist might accidentally mix up his mug and doughnut because two spaces are homeomorphic if one can be deformed into the other without cutting or gluing, as is the case.
See http://en.wikipedia.org/wiki/Topology
and http://www.georgehart.com/bagel/bagel.html.
For more about topology, try reading http://www.math.wayne.edu/~rrb/topology.html.
