UBC Theses and Dissertations
Non-standard analysis Cooper, Glen Russell
In this thesis some classical theorems of analysis are provided with non-standard proofs. In Chapter 1 some compactness theorems are examined. In 1.1 the monad μ(p) of any point p contained in a set X (and relative to a family H of subsets of X ) is defined. Using monads, a nonstandard characterization of compact families of subsets of X is given. In 1.2 it is shown that the monad μ(p) of any point p ε X (relative to H ) remains unchanged if H is extended to the smallest topology τ(H) on X containing H . Then, as an immediate consequence, the Alexander Subbase theorem is proved. In 1.3 monads are examined in topological products of topological spaces. Then, in 1.4 and 1.5 respectively, both Tychonoff's theorem and Alaoglu's theorem are easily proved. In Chapter 2 various extension results of Tarski and Nikodým (in the theory of Boolean algebras) are presented with rather short proofs. Also, a result about Boolean covers is proved. The techniques of non-standard analysis contained in Abraham Robinson's book (Robinson 1974) are used throughout. The remarks at the end of each chapter set forth pertinent references.
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