*Insert Star Wars joke here* All forces in the universe are either attractive or repulsive. We know that gravity is a force that is always attractive. Unlike the electromagnetic force, a particle cannot be repelled by gravity *. But what if I told you that from the point of view of a comet or a planet orbiting […]

### The Halting Problem

You’re sat at your desk, watching a progress bar stuck at 99%, wondering if it will ever finish. This is exactly the question that the halting problem asks. Given an input, will a computer program run forever, or eventually halt? For example, the simple program while (true) { print(“This one goes forever.”); } would loop […]

### Topology 5 : Continuous Functions

Continuity is a very important concept in Analysis, and even in General Topology, so it is very important that an intuition for these definitions are had. Say a function were “discontinuous,” like the one below. Figure 5.1 Now, we select an open set in that “separates” this function into two different sections. This will intuitively […]

### The Future of Space Exploration

Space Travel, is the best of it over? Ever since we have seen the stars, mankind has been striving to discover how not only the world works, but how the universe functions as well. One of the most important ways of doing this is via space exploration, which as time has gone by has become […]

### Manifolds 1 : Introduction

Have you ever wondered how we can tell if a topological space looks thin? Something that is like a surface; paper-like in nature. Something like the surface of a sphere (called the -sphere or ) or a blanket. Could such a concept be rigorously defined? We can’t say that it would have to be homeomorphic […]

### Can a wave be represented by infinite discrete oscillators?

In this article I will find the equation of motion for an arbitrary number of discrete coupled oscillators (I will call “beads”) on a string. Then we will take the limit as the number of oscillators tends towards infinity, and see if the wave equation can be reproduced. If so it would be a good […]

### Does Neutron Star Collapse Violate Quantum Theory?

With the death of a star meaning that radiation pressure can no longer support the gravitational crush due to its large mass, Pauli’s exclusion principle maintains an equilibrium in what has now become a neutron star. Heisenberg uncertainty allows for a neutron star to undergo ultimate collapse, as a star becomes more massive momentum space […]

### Topology 4: Hausdorff Topologies and Closed Sets

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 […]