Separable Topological Space of Hereditary

A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.

On Countability of Point-Finite Families of Sets

It is well known that in a separable topological space every point-finite family of open subsets is countable. In the following we are going to show that both in σ-finite measure-spaces and in topological spaces satisfying the countable chain condition, point-finite families consisting of “large” subsets are countable.Notation and terminology. Let A be a set. The family consisting of all (finite) subsets of A is denoted by . Let be a family of subsets of A. The sets and are denoted by and , respectively. We say that the family is point-finite (disjoint) if for each a ∈ A , the family has at most finitely many members (at most one member).