Wandering Around

Introduction This is a collection of thoughts that I have been carrying with me for some time. When I was born, in 2001, there was already some technology and also Moore’s Law, which says that the number of transistors on a chip will double every 18–24 months. He was right; chips have doubled over the years. Intel’s first processor, the Intel 4004, released and manufactured in 1971, had only 2,300 transistors. And not so surprisingly, Apple’s A17 chip, released and manufactured in 2023, has 19 billion transistors. Can you imagine? From my side, I cannot! ...

The motivation of this post is to enlarge the steps of theorems and the definition of relativeness, compactness and more. Since the famous and brilliant book Principles of Mathematical Analysis by Walter Rudin is so terse so to say. I love “Baby Rudin”, but unfortunately it is hugely incomprehensible if you do not know what you are doing. Rudin, of course, had deliberately skipped over the details. He probably guessed that the reader was quite gifted in mathematics or the reader is taught by a brilliant teacher. ...

Prerequisites: metric spaces $\textbf{\small Definition 1.1.:}$ (Neighbourhood) Let $X$ be a metric space. A neighbourhood of a point $p$ is a set $N_r(p)$ consisting of all points $q$ such that $d(p,q) < r$. The number $r$ is called the radius of $N_r (p)$. $$ \begin{aligned} N_r(p) = \{q : d(p,q) < r \text{ where } p,q \in X\} \end{aligned} $$ $$ \begin{aligned} N_r(p) = \{q : d(p,q) < r \text{ where } p,q \in X\} \end{aligned} $$ If we are in an Euclidean metric $(\mathbb{R}^k)$ then, ...

I was freshman when I was first introduced supremum and infimum. I could not understand a thing, yet everybody seemed to understand. I felt like stupid, inadequate and also insufficient person to learn mathematics. However, reality is not like that. In truth, I was the one to blame; I hadn’t dived deep into these topics deeply enough. I do not know if I am alone or not, but I think supremum and infimum are the topics which cannot by bypassed without learning and embracing. Today I want to show an interesting fact; if a set has least-upper-bound property and bounded below, then it has infimum. But first let us define supremum and infimum. ...

First Initiation When we first started to learn mathematics, our concern is only to solve problems, understand what equation means, comprehend what teacher says during class and etc. At high school we learn more about sets and functions. However, if we are not curious enough, then the philosophical meaning of the word “mathematical space” is not emphasized. When $f: \mathbb{R} \rightarrow \mathbb{Q}$ is written, everybody who has learned high school math can say “the function takes values from $\mathbb{R}$ and maps it to $\mathbb{Q}$. But have people reading this ever thought about the background? ...