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References

What is Topology?

In the branch of mathematics called topology, concepts such as compactness, connectedness, continuity, convergence, distance, and neighborhood all involve some notion of nearness. On the real line R for example, we can measure how near two points are by the absolute value of their difference. In the plane, we can measure how near two points are using the Pythagorean Theorem. These are examples of the general notion of distance or metric.

Here are some of the foundational ideas of topology. A sequence of real numbers converges to a limit provided its terms eventually are as near as you please to that limit, and a function  f : R -> R  is continuous if it preserves nearness or equivalently if it preserves limits. In particular, a continuous function must map a convergent sequence onto a convergent sequence. An interval in R consists of just one piece with no gaps. From this perspective, the Intermediate Value Theorem says that a continuous function  f : R -> R  must map an interval onto an interval and cannot cut it into pieces. The Extreme Value Theorem says that a continuous function  f : R -> R  must map a closed bounded interval onto a closed bounded interval. Topology is the study of sets with a notion of nearness, functions that preserve that nearness (that is, functions that are continuous relative to that notion of nearness), and properties that are preserved by those functions. For example, the range of a continuous function cannot have more pieces than its domain.

A deformation is a continuous distortion of a space in time. Dilations, gluings, identifications, merges, shrinks, stretches, and twists are examples of deformations; cuts, rips, and tears are not. Imagine a solid ball made of an pliable material such as Play-Doh ®. Using your hands you can roll the ball into a rope and then pull its ends around and press them together to form a doughnut. The doughnut will occupy a finite volume no matter how you deform it further, and you cannot deform the doughnut into two doughnuts. However, you can deform the doughnut back into a ball like this. First flatten the doughnut into a washer, and then deform the washer into a disk by dilating the washer and merging its inner cylindrical surface to its axis of revolution. Finally, compress and shape the disk into a ball. A doughnut (with a hole) can be deformed into a coffee cup (with a handle) without merging points (as illustrated on page 178 in Mathematics, published by Time, Inc. in 1963). Objects that can be deformed into each other without merging points are said to be isotopic and are considered to be identical in topology. So a topologist is one who cannot tell the difference between a coffee cup and a doughnut. However, a topologist can tell the difference between a ball and a doughnut and much more. See, for example, Solving the Poincare Conjecture Wins Science's Breakthrough of the Year. Because of ideas like these, topology is sometimes called rubber sheet geometry.