The Cantor set provides some of the most pathological examples in real analysis. Introduced by G. Cantor in 1883, the Cantor set (or Cantor dust) can be thought of as the remainder of the unit interval after removing open middle thirds *ad infinitum*. In the following, we discuss the pathology relating to the “size” of the Cantor set where, depending on how you define it, the Cantor set has the size of a point, the entire real line, or somewhere in-between.

# Analysis

## The Contraction Mapping Theorem and Your New Favorite Math Tale

The contraction mapping theorem, also known as the Banach-Caccioppoli theorem, guarantees the existence of a fixed point for maps which “shrink”. In the following we’ll discuss the theorem, provide a sketch of the proof, and describe a neat application that will be sure to impress your math and non-math friends.

Continue reading