SCHWEITZER 1982 Problem 1

A map $F : P(X) \rightarrow P(X)$ , where $P(X)$ denotes the set of all subsets of $X$ , is called a $\textit{closure operation}$ on $X$ if for arbitrary $A,B \subset X$ , the following conditions hold:

(i) $A \subset F(A);$

(ii) $A \subset B \Rightarrow F(A) \subset F(B);$

(iii) $F(F(A))=F(A)$ .

The cardinal number $\min \{ |A| : A \subset X\ , F(A)=X\ \}$ is called the $\textit{density}$ of $F$ and is denoted by $d(F)$ . A set $H \subset X$ is called $\textit{discrete}$ with respect to $F$ if $u \not \in F(H-\{ u \})$ holds for all $u \in H$ . Prove that if the density of the closure operation $F$ is a singular cardinal number, then for any nonnegative integer $n$ , there exists a set of size $n$ that is discrete with respect to $F$ . Show that the statement is not true when the existence of an infinite discrete subset is required, even if $F$ is the closure operation of a topological space satisfying the $T_1$ separation axiom.

A. Hajnal