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Sites for parents Hausdorff Distance We talk about points in a space , like in the definition of a circle as a set of all points equidistant from a given point. But we have already pointed to an example of a distance defined between two functions . Functions can also be added and multiplied , and in mathematics sets whose elements are functions are called space s (sometimes, of course, functional spaces .) as many other sets . The advantage is in that, once some common properties of various sets have been isolated, their study will apply to all the particular cases regardless of the nature of elements the sets comprise. It may be confusing sometimes , for example, when we consider spaces of functions or curves or matrices. A point in a space is something elementary, simple and, like an atom (of many years ago), indivisible. But here exactly lies one of the sources from which mathematics draws its power. Going to a level of abstraction that knows nothing of the nature of the objects it deals with spreads the results over vast territory strewn with apparently unrelated objects pointing to unexpected similarities and, by doing so, outlines also the limits of analogy. We not only learn what is common but better understand the differences. Here I wish to consider spaces whose elements - points - are sets themselves. Proving a result on separating points in the plane with circles | |
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