## Analysis – HW 4 – Fractals

October 29, 2013

This set is due Friday, November 15, at the beginning of lecture.

We discuss the Hausdorff metric on the collection of compact subsets of a complete metric space, and fractals obtained by contractions. (But note that not all fractals one may want to study come from this procedure.)

[Edit, Nov. 12, 2013: Problem 4 has been changed to: Provide an example showing that, in general, if $e\in\mathbb R^n$ and $L$ is compact, then $d_H(\{e\},L)\ne d(e,L)$. (There are cases where the equality holds, and it may be useful to provide some examples of this as well.)]

## 502 – The Banach-Tarski paradox

December 17, 2009

1. Non-measurable sets

In these notes I want to present a proof of the Banach-Tarski paradox, a consequence of the axiom of choice that shows us that a naive understanding of the concept of volume can lead to contradictions. A good reference for this topic is the very nice book The Banach-Tarski paradox by Stan Wagon.