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1	From the Axiom of Choice to Tychono ’s Theorem Hörngren, Gustav January 2015 (has links) A topological space X, is shown to be compact if and only if every net in X has a cluster point. If s is a net in a product Q 2A X, where each Xis a compact topological space, then, for every subset B of A, such that the restriction of s to B has a cluster point in the partial product Q 2B X, it is found that the restriction of s to B [ fg – extending B by one element 2 A n B – has a cluster point in its respective partial product Q 2B[fg X, as well. By invoking Zorn’s lemma, the whole of s can be shown to have a cluster point. It follows that the product of any family of compact topological spaces is compact with respect to the product topology. This is Tychono’s theorem. The aim of this text is to set forth a self contained presentation of this proof. Extra attention is given to highlight the deep dependency on the axiom of choice. Urvalsaxiomet Tychonoffs sats Zorns lemma topologiskt rum kompakthet
2	The Banach-Tarski Paradox : How I Learned to Stop Worrying and Love the Axiom of Choice Wahlberg, Mats Karl Anders January 2022 (has links) This thesis presents the strong and weak forms of the Banach-Tarski paradox based on the Hausdorff paradox. It provides modernized proofs of the paradoxes and necessary properties of equidecomposable and paradoxical sets. The historical significance of the paradox for measure theory is covered, along with its incorrect attribution to Banach and Tarski. Finally, the necessity of the axiom of choice is discussed and contrasted with other axiomatic and topological assumptions that enable the paradoxes. / Den här uppsatsen presenterar den starka och svaga formen av Banach-Tarskis paradox baserade på Hausdorffs paradox. Den tillhandahåller moderniserade bevis av paradoxerna och nödvändiga egenskaper av likuppdelningsbara och paradoxala mängder. Den historiska betydelsen av paradoxen på måtteori tas upp samt dess felaktiga tillskrivning till Banach och Tarski. Till sist diskuteras behovet av urvalsaxiomet som ställs i kontrast mot andra axiomatiska och topologiska antaganden som möjliggör paradoxerna. Banach Tarski Banach-Tarski Banach-Tarski paradox Hausdorff Hausdorff paradox paradox measure measure theory axiom of choice Banach Tarski Banach-Tarski Banach-Tarskis paradox Hausdorff Hausdorffs paradox paradox mått måtteori urvalsaxiomet valaxiomet Mathematics Matematik Mathematical Analysis Matematisk analys Geometry Geometri

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From the Axiom of Choice to Tychono ’s Theorem

The Banach-Tarski Paradox : How I Learned to Stop Worrying and Love the Axiom of Choice