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Crazy Things in R

Beek, M. van (2007) Crazy Things in R. Bachelor's Thesis, Mathematics.

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One might be tempted to think that the real line R is a rather simple set, and one about which not much remains to be discovered. However, we soon find out that matters are not as simple as that. The continuum remains a mysterious entity, even to this day. What does one do, for example, when confronted with two subsets of R and asked which of them is larger? There are many possibilities, for a start one might count the elements in each and then see which contains more elements. This even goes quite well when considering sets with an innite number of points, using an injective function we are able to compare the number of elements in one set with the number in the other. On the other hand, we might find this method a bit restrictive, many sets are put on an equal footing with each other that intuitively seem very different. For example, we find that the set of integers Z and the set of rational numbers Q contain exactly the same number of elements. In some cases such as this one, looking at the distribution of the sets in R gives us a better idea of how the sets are different. Of course, one could do some simple measuring, one could simply put a ruler next to R and see how long a set is if you squish all the points in it together. A function which allows us to determine the length of a set in such a manner is called a measure. In this way we are able to distinguish between, for example, intervals of different length, something nothing up till now has done. Thus all sets we want to describe must be looked at from many points of view before anything can be said concerning how `large' that set is. We have numerous examples of a set being immense from the point of view of topology, yet tiny with respect to measure, or vice versa. We need not look far for interesting sets, even some very simple ones have many surprising things happening in them, and are well worth our time studying. After having been loaded with several different ways of looking at sets, we might start wondering where it will all end. Do any of these definitions have anything to do with each other? We certainly cannot equate any of them, as numerous examples show. This question brings us the essence of Erdos's Duality theorem. This metatheorem manages to give a surprising result concerning two completely different definitions of `smallness'.

Item Type: Thesis (Bachelor's Thesis)
Supervisor name: Broer, H.
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 07:28
Last Modified: 17 Apr 2019 12:12

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