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Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many ...
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Is $SO_2$ an amenable group?

In S. Wagon's "The Banach-Tarski Paradox," amenable groups are defined on p. 12 as follows: [amenable] groups bear a left-invariant, finitely additive measure of total measure one that is defined ...
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A soft Question Suggestions for good introductory books

This question has been edited in view of comments received.Hope it is better now. Consider the set of all points on the circumference of a circle and the set of all points inside a circle. what is ...