Before the philosopher Bertrand Russell began really thinking about the nature of sets, sets were defined informally. Set theory before various axiomatic definitions is now referred to as naïve set theory. Russell’s paradox, discovered by Bertrand Russell in 1901 was also discovered a year earlier by Ernst Zermelo but never published. This paradox shows that […]

### February Roundup (Maths, Tech, Computing)

Hi everyone, I’m back with a couple of short summaries on topics that I’ve found significant and/or important in February. As before, links to articles that I used on each of the topics will be included at the bottom of the article in case you want to read more into the subjects. Telugu Symbol Crashes […]

### January Roundup (Maths, Tech, Computing)

Welcome to Plancktime’s first ever monthly roundup featuring short summaries on a couple of maths, tech or computing news that I thought were interesting and/or important. Links to articles that I used on each of the topics will be included at the bottom of the article in case you want to read more into the […]

### 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 […]

### 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 […]

### 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 […]

### 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 […]