## Construction of real numbers

This is a note for me who forget again in the future. There are several way to construct real numbers. Dedekind cuts https://en.wikipedia.org/wiki/Dedekind_cut A Dedekind cut is a partition of the rationals $\mathbb{Q}$ into two subsets $A$ and $B$ such that: $A$ is non emply. $A\neq\mathbb{O}$ If $x,y\in\mathbb{Q}$, $x<y$, and $y\in A$, then $x\in A$. ($A$ is “closed downwards”.) If $x\in A$, then therre exists a $y\in A$ such that $x<y$ ($A$ does not contain a greatest element.

## Code coverage - tutorial

Code coverage Definition from Wikipedia: In computer science, test coverage is a measure (in percent) of the degree to which the source code of a program is executed when a particular test suite is run. Types: Function coverage – has each function (or subroutine) in the program been called? Statement coverage – has each statement in the program been executed? Edge coverage – has every edge in the control-flow graph been executed?

## Tokio - Rust Concurrency (draft)

I started to discover Tokio framework. https://tokio.rs/tokio/tutorial You can find all official tutorial code at here. What is Tokio ultimately Asynchronous Rust code does not run on its own, so you must choose a runtime to execute it. Tokio is an asynchronous runtime for the Rust programming language. Handson tutorial with Redis-clone We’ll run a clone of Redis server. While running the redis server, your prompt will be blocked. For someone who doesn’t know about Redis, it is a key-value store server, and you need to follow a redis protocol on TCP/IP.

## Topological space (draft)

This is my note of the video https://youtu.be/1wyOoLUjUeI. It’s not a full guide of topology. I wanted to know what is continuity the meaning of “topology” Topological space Let $M$ be a set. Then, a choice $\mathcal{O}\subseteq\mathcal{P}(M)$ is called a topology on $M$ if $\phi\in\mathcal{O}$ and $M\in\mathcal{O}$, $O_i, O_j \in\mathcal{O} \Rightarrow\bigcap\{O_i, O_j\}\in\mathcal{O}$, and $\displaystyle O_i\in\mathcal{O} \Rightarrow \bigcup_i O_i\in\mathcal{O}$. The pair $(M, \mathcal{O})$ is then called a topological space. An elements of a topology is called an open set.

## Classification and construction of sets

This is my note of the video https://youtu.be/6EIWRg-6ftQ. It’s not a full guide how to construct “numbers” from the empty set. Classification of sets In Mathematics, we usuallly investigate the “structure” of “spaces” in terms of “maps”. Space and map The terminology space is usually meant to be some set equipped with some “structure”. A map $\phi: A\rightarrow B$ is a relation such that for every $a\in A$ there exists exactly one $b\in B$ such that $\phi(a,b)$.

## Axioms of set theory (ZFC)

This is my note of the video https://youtu.be/AAJB9l-HAZs. It’s not a full guide of axiomatic set theory. Relation Relation is a predicate of two variables. $\in$-relation Set theory is built on the postulate that there is a fundamental relation called $\in$ (epsilon relation). There will be no definition of what $\in$ is, or of what a set is. Instead, we’ll define 9 axioms, that spesk of $\in$ and sets. We call $x$ as an element of $y$ if $x\in y$ is a proposition.

## Propositional and predicates logic

This is my note of the video https://youtu.be/V49i_LM8B0E. It’s not a full guide of propositional and predicates logic. Mathmatics as a language of Physics Physics “casts” concepts about the real world into rigorous mathematical forms. Mathematics is just a language for the Physics. Physics interprets the meaning of mathematical statements. But to describe Mathematics, we need logic (especially, standard logic). Psycologically, an equivalent but a different form will help us to understand an interpretation.

## Lebesgue measure (updating)

My personal motivation(s): want to know when it is allowed to swap limit and sum. want to know when it is allowed to swap limit and integral. want to know what the Lebesgue integral is. want to make a base for probability theory. $\sigma$-algebra Suppose a set $X$ and its power set $\mathcal{P}(X)$. The “measure” will be defined for some elements of $\mathcal{P}(X)$ later. $\Sigma\subseteq\mathcal{P}(X)$ is $\sigma$-algebra if $\phi, X \in \Sigma$, $A\in\Sigma \implies A^c := X\backslash A \in\Sigma$, and $\displaystyle\{A_i\}_{i\in\mathbb{N}} \implies \bigcup_i A_i \in \Sigma$.