These two branches of mathematics are often mentioned together because they both involve the study of properties of space. But whereas geometry focuses on properties of space that involve size, shape, and measurement, topology concerns itself with the less tangible properties of relative position and connectedness.

Nearly every high school student has had some contact with Euclidean geometry. This subject remained virtually unchanged for about 2000 years, during which time it was the jewel in the crown of mathematics, the archetype of logical exactitude and mathematical certainty.

And then in the seventeenth century things changed in a number of ways.

Building on the centuries old computational methods devised by astronomers, astrologers, mariners, and mechanics in their practical pursuits, Descartes systematically introduced the theory of equations into the study of geometry. Newton and others studied properties of curves and surfaces described by equations using the new methods of calculus, just as students now do in current calculus courses. These methods and ideas led eventually to what we call today differential geometry, a basic tool of theoretical physics. For example, differential geometry was the key mathematical ingredient used by Einstein in his development of relativity theory.

Another development culminated in the nineteenth century in the dethroning of Euclidean geometry as the undisputed framework for studying space. Other geometries were also seen to be possible. This axiomatic study of non-Euclidean geometries meshes perfectly with differential geometry, since the latter allows non-Euclidean models for space. Currently there is no consensus as to what kind of geometry best describes the universe in which we live.

Finally, the eighteenth and nineteenth century saw the birth of topology (or, as it was then known, analysis situs), the so-called geometry of position. Topology studies geometric properties that remain invariant under continuous deformation. For example, no matter how a circle changes under a continuous deformation of the plane, points that are within its perimeter remain within the new curve, and points outside remain outside. For another example, no continuous deformation can change a sphere into a plane. So they are topologically distinct.

Topology can be seen as a natural accompaniment to the revolutionary changes in geometry already described. For, once one recognizes that there is more than one possible way of geometrizing the world, i.e., more than just the Euclidean way of measuring sizes and shapes, then it becomes important to inquire which properties of space are independent of such measurement. Topology, which finally came into its own in the twentieth century, is the foundational subject that provides answers to questions such as these. It is a basic tool for physicists and astronomers who are trying to understand the structure and evolution of the universe. Indeed, recent astronomical observations, together with basic results of topology, offer the exciting prospect that we will soon be in possession of the global topological structure of the cosmos.