journey of Froakie (ケロマツの旅路)

Introduc tion 日本語 ver Today, I will write definition of continuity of function by definition of an open set. I wrote  this post about the definition of Topology space, open set, and a thing that open set satisfy the axiom of Topology, but I did not write about continuity of function by definition of an open set. Actually, It is very important. Overview  Open set $\epsilon-\delta$ reasoning definition of continuity of function by an open set Equivalence Open set Let (X,d) is distance space. $A \subset X$ is open set $$\iff$$ $$\forall x \in A,~~\exists \epsilon > 0, ~~s.t.~~ B(x,\epsilon) \subset A$$ here,$$B(x,\epsilon):= \{y\in A| d(x,y) < \epsilon\}$$ This definition of open set satisfies Axim of Topology. It written the last time post. $\epsilon-\delta$ reasoning I will explain the $\epsilon-\delta$ reasoning. This reasoning is learned in bachelor third student at Univ. Let f:X-> Y: map f is countinou...

Introduction 日本語 ver Today, I will write about General Topology. General topology is very impossible for studying mathematics. General topology defines phase to define continuity of function. Today, I will explain phase. Also, This post is my review of General Topology. Overview  Distance space Axiom of phase Topology space Open set Distance space Firstly, Distance space is defined. Let X is set, $d:X\times X ->\mathbb{R}$, (X,d) is distance space $\iff$ $\forall x,y \in X,~~~~~d(x,y) \geq 0$ $\forall x,y \in X,~~~~~x = y \implies d(x,y) = 0$ $\forall x,y \in X,~~~~~d(x,y) = d(y,x)$ $\forall x,y,z \in X, ~~~~d(x,y) + d(y,z) \geq d(x,z)$ this condition is called axiom of distance. d is called distance function. Topology  space Let (X,d) is distance space. $\mathbb{O} \in 2^X$ is phase in (X,d) $\iff$ $\phi,X \in \mathbb{O}$ $\forall O_1,O_2 \in \mathbb{O} \implies O_1 \cap O_2 \in \mathbb{O}$ $\forall \Lambda ,~~\forall \{O_\l...