A Compact Introduction to a Generalized Extreme Value Theorem

In a short paper published just one year prior to his thesis, Maurice Frechet gives a simple generalization one what we might today call the Extreme value theorem. This generalization is a simple matter of coming up with ``the right definitions in order to make this work. In this mini PSP, we work through Frechet\u27s entire 1.5 page paper to give an extreme value theorem in more general topological spaces, ones which, to use Frechet\u27s newly coined term, are compact

The Cantor Set Before Cantor

A special construction used in both analysis and topology today is known as the Cantor set. Cantor used this set in a paper in the 1880s. Yet it appeared as early as 1875 in a paper by the Irish mathematician Henry John Stephen Smith (1826 - 1883). Smith, who is best known for the Smith normal form of a matrix, was a professor at Oxford who made great contributions in matrix theory and number theory. In this project, we will explore parts of a paper he wrote titled On the Integration of Discontinuous Functions

Topology From Analysis

Topology is often described as having no notion of distance, but a notion of nearness. How can such a thing be possible? Isn\u27t this just a distinction without a difference? In this project, we will discover the notion of nearness without distance by studying the work of Georg Cantor and a problem he was investigating involving Fourier series. We will see that it is the relationship of points to each other, and not their distances per se, that is a proper view. We will see the roots of topology organically springing from analysis