MathDB

A map

F : P(X) \rightarrow P(X)

, where

P(X)

denotes the set of all subsets of

X

, is called a

<span class='latex-italic'>closure operation</span>

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))\equal{}F(A). The cardinal number \min \{ |A| : \;A \subset X\ ,\;F(A)\equal{}X\ \} is called the

<span class='latex-italic'>density</span>

F

and is denoted by

d(F)

. A set

H \subset X

is called

<span class='latex-italic'>discrete</span>

with respect to

F

if u \not \in F(H\minus{}\{ 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

Problem Statement