A Note on the T₁-Space and Derived Set

Abstract

位相空間 X가 T₁-space라 함은 X의 각 한 점으로 구성된 집합들이 모두 closed일 때를 말하고. 한 set의 모든 集積點들의 집합을 導集合이라 부르기로 한다.

우리는 여기서 다음과 같은 사실을 밝힌다. 즉 만약 X가 T₁-空間이면 X의 각 부분집합의 導集合은 closed이지만 그 逆은 성립하지 않으며 더욱 나아가서 어떤 空間 X의 각 부분집합의 3導集合이 closed일 필요하고도 충분한 조건은 X의 모든 singleton{x}의 導集合이 각각 closed인 것이다.

A topological space is a T₁-space iff each set which consists of a single point is closed, and the set of all limit points of a set is called the derived set of it.

We show the following: If X is a T₁-space then the derived set of each subset of X is closed, but the converse is not always true. Furthermore, as a sharper result, a necessary and sufficient condition that the derived set of each subset be closed is that the derived set of the singleton{x} be closed for each point x in a topological space X.

A topological space is a T₁-space iff each set which consists of a single point is closed, and the set of all limit points of a set is called the derived set of it.

We show the following: If X is a T₁-space then the derived set of each subset of X is closed, but the converse is not always true. Furthermore, as a sharper result, a necessary and sufficient condition that the derived set of each subset be closed is that the derived set of the singleton{x} be closed for each point x in a topological space X.