First, we define what is meant by a closed set. Definition 4.1 Closed Set We say a subset of a Topological space is closed if is open. Now, the intuition behind closed sets is that these sets include their boundary, so that when you remove them from the entire set, they don’t include their boundary […]

### Topology 3: Common Topologies

I will first introduce the Order Topology. First, in order to describe what orders are, we must first define what relations are. Definition 3.1 Relation We say is a relation on if , where if , we write . Now, there are many uses for relations, but for our purposes, we only need one form […]

### Topology 2: Bases

We start with the definition of a basis. Definition 2.1 Basis Suppose is a set. We call a basis if (1) For every , there exists such that (2) For every , and , there exists such that . Figure 2.1 It is very clear to see that how the standard topology defined on […]

### Topology 1: Introduction

Consider the open interval in . This will be a basis for what we consider to be open. Notice that given any point , you may find another interval around it completely contained in the original interval. Figure 1.1 Note that this is not true of the closed interval , as the endpoints and do […]

### Trans-Finite Ordinals

Axioms of Set Theory Now, in my other articles, I have introduced rigorous mathematics to people who may not be familiar with it. Today, the rigor will be stepped up 10 fold. So, now, let me introduce you to the idea of axioms: We have already discussed axioms in the topic of Topology and VectorSpaces, but […]

### Partitions and Generating Functions

Introduction to Partitions A partition is the number of ways a positive integer can be summed to by other positive integers. So, to find the partition of , we look at all the ways we can sum up to four: There are no other ways to add positive integers to , because then it would just […]

### The Matrix Form of the Derivative

Introduction to General Vector Spaces You may know of the Vector Spaces , with an addition and a real scalar multiplication . Since this is a nice intuition, let’s look at some of the properties of this: Addition (assume ) (i) (ii) […]

### The Importance of Primes in Group Theory

You may be wondering what Group Theory is even about, let alone the importance of primes in it. But, in this article, I intend to inform what group theory is and what the importance of primes are in it. Introduction Say we have a set , and an operation , that takes in […]

### Plasma Oscillations – “Langmuir Wave” approximation derivation

In a previous article we established an approximate estimate for the frequency of oscillations in a “cold plasma” because of small disturbances in the arrangement of the electrons and ions. In this article we will establish a much better model for oscillations in a plasma, which takes into consideration the fact that Plasmas tend to be […]

### Inside the CPU – Instruction cycles

One of the most widely taught processes of a computer is the instruction cycle (fetch-decode-execute cycle), yet it seems to be one that causes a lot of headache with students. Instead of finding a way to memorise all the acronyms and orders, how about we clearly explain why each process happens, rather than simply what […]