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 $12$

Mistake
and present a more complicated $14$-set than is necessary to demonstrate the theorem:
 * $A := \set {\tfrac 1 n: n \in \Z_{>0} } \cup \openint 2 3 \cup \openint 3 4 \cup \set {4 \tfrac 1 2} \cup \closedint 5 6 \cup \paren {\hointr 7 8 \cap \Q}$

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