直觀的 圖形의 位相學的 考찰 = The Topological observation on Intuitive Figure

다국어 초록 (Multilingual Abstract)

It was not long before "topology," orignated from the theory by Euler in 1736, set up its intrinsic nature as "Science", but, all down the ages, in fact, the human beings have arrived at the intuitive conceptions concerning "figure" in geometry in the course of their lives.
While Mathematics has developed more than two thousand years, mankind has made numerals, and done sums, and, as the results of the discovery of numerals and the systems of calculations, Euclidian geometry, inclusive of "analysis," has come into being.
During the nineteenth century of the great reorganization of Mathematics, the set theory was discovered by Cantor. "Construction" in geometry has been created by the set theory which extends to the whole fields of "Science" as well as "figure" and "operation" has been introduced into the theory and led to the algebraic system, so that modern algebra has been completed, and then "topological space" is to be formed by bringing the other new conceptions into the set theory.
Our human beings are to observe, through the simple, basic intuition that has been carried with from time immemorial, how the processes of such topological space are being formed.
And on the basis of figure and thought to it, we conclude as follows;
(A) When any positive number ε is given, there exists the point a that ρ(a, x??), the distance from x?? in A, is smaller than ε.
And so x?? to meet (A) is considered to be adhered to A.
The conception of connection is that:
(B) Even when X is divided into some two subsets A and B and they are not null sets, they are said to stick to each other.
(C₁) When any subset A in X and x?? comes into contact with A, f(x??)∈Y is sure to adhere to f(A)⊂Y.
(C₂) When any adequate positive number δ is taken towards the positive number ε given at will, ρ(f(x)f(x??))<ε is made toward x∈X, which is ρ(x,x??)<δ.
In view of (C₁) and (C₂), the result is that:
(Q) When any subset A in X, and A is an infinite set, there exists the point in itself to A in the meeting-point on X.
The general idea of limitation and compactness has been introduced from the thought that an infinite number of points are stuck to this. We can, therefore, observe whether the general notion of adhesion can be applied to all the conditions of figure, and give the definition of topological space by thinking that the word 'adhesion' and the intuition of 'figure' are to be linked with each other.
Terms of meeting (Aφ), (AI), (AII), (AIII), (BII) are as follows:
(Aφ)...There is not a point which sticks to the null-set.
(AI)...A point which belongs to A is joined to A.
(AII)...In order that X?? may adhere to A∪B, it is necessary that X?? should be put at least on either A or B.
(AIII)...When all the points on A₁are on A, the point to A₁belongs to A.
(BII)...Toward the two different points a and b on X, there exists such subsets as C and D on X.
(i) C∪D=X C∩D=φ
(ii) a is not attached to C, nor is b to D.
And we have so far taken into intuitive consideration that the topological space can be defined by the fundamental neighbourhood or neighbourhood system.

목차 (Table of Contents)

• Ⅰ. 序言
• Ⅱ. 直觀과 數學
• 1. 圖形의 連結
• 2. 寫像
• Ⅲ. 連結과 Compact性
• 1. 連續寫像과 連結性
• 2. Compact性
• Ⅳ. 位相空間
• 1. 位相空間의 定義
• Ⅴ. 結言