Kuratowski's Closure-Complement Problem/Mistake

Source Work

 * Part $\text{II}$: Counterexamples
 * Section $32$: Special Subsets of the Real Line
 * Item $9$: Figure $12$
 * Item $9$: Figure $12$

This mistake can be seen in the second edition (1978) as republished by Dover in 1995: ISBN 0-486-68735-X

Mistake
and present a more complicated $14$-set than is necessary to demonstrate the theorem:
 * $A := \left\{ {\tfrac 1 n: n \in \Z_{>0} }\right\} \cup \left({2 \,.\,.\, 3}\right) \cup \left({3 \,.\,.\, 4}\right) \cup \left\{{4 \tfrac 1 2}\right\} \cup \left[{5 \,.\,.\, 6}\right] \cup \left({\left[{7 \,.\,.\, 8}\right) \cap \Q}\right)$

They present Figure $12$ to illustrate the various generated subsets graphically:


 * 14SetsByClosureAndComplement-Faulty.png

The following mistakes can be identified in the above diagram:


 * $(1): \quad$ The set $A$ as presented expresses the interval of rationals as closed, whereas it is in fact half open.


 * $(2): \quad$ The sets are all presented as subsets of $\R_{\ge 0}$, while this is not stated in the text.


 * $(3): \quad$ $0$ is erroneously excluded from $A^{\prime}$.

A corrected version of this diagram is presented below:


 * 14SetsByClosureAndComplement.png

Also see

 * Kuratowski's Closure-Complement Problem