An Injective Tango! (Schroeder-Bernstein Theorem)

Now there’s a simple theorem in set theory whose proof has always appeared a bit cloudy to me, since I’ve never been able to find it written in a straightforward manner. This theorem is the Schroeder-Bernstein Theorem, whose statement is utterly intuitive: Schroeder-Bernstein Theorem: Let and be sets. If there exists an injection , and an […]

MichelFest: Minkowski & Hausdorff Dimensions

I left off the last post on An Introduction to Fractal Geometry discussing the issue of dimension for fractals. Namely, using the defined box dimension, we saw that the Cantor Set and the Sierpinski Carpet have non-integer dimension. We also saw some issues with the Box Dimension, specifically with density, and thus will seek different notions of […]

MichelFest: An Introduction to Fractal Geometry

So I’m currently at The Summer School on Fractal Geometry and Complex Dimensions at Cal Poly San Luis Obispo. The event is a conference/summer school held in honour of the 60th birthday of Michel Lapidus of U.C., Riverside, a French Mathematician, my mentor, editor of The Journal of Fractal Geometry, and one of the world’s leading fractal […]

Measure Theory + Algebra + AOC = Vitali Set

Perhaps the least intuitive concept one first encounters in the study of Abstract Measure Spaces, or even the Lebesgue Measure on , is the concept of an unmeasurable set. What the heck does this even mean? Well it really just means that there’s no good way to assign a “size” to this set. A -algebra is really just trying […]

Étale Cohomology

This is a post from here. I have become very interested in algebraic geometry (only the terminology makes it kind of confusing), and I revised what little I knew (or used to know when I wrote the original post) about étale cohomology. Here I’ll try to give a brief introduction and some motivation as to why étale […]

The Language of Category Theory (Part II)

We sort of paused after our first category theory post. We left off with the definition of a functor. What should a functor be? Like group homomorphisms, ring homomorphisms, linear transformations, continuous maps, etc., it should be something that “preserves the structure of the category”. This is reflected in the following formal definition: Definition: Let and be categories. Then a […]

Introduction to Topology- Part 2

So our last post familiarized us with the notion of a topology on a set. We know how to define a topology and a few basic examples of topological spaces. A natural problem arises when discussing topologies on large spaces. Suppose I asked you to list all open subsets of . This would take quite […]

Compactness, Completeness, Connectedness, Oh My! (Part II)

Here’s a summary from the last post regarding completeness: Completeness: A complete space is one in which all sequences that want to converge do converge. Intuitively, this means that if “points” (I use this term loosely, because points could represent continuous functions, or categories, etc) in a sequence eventually become arbitrarily close with respect to some sort of metric, then there […]