Kuratowski's Closure-Complement Problem/Mistake

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Source Work

1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology:

Part $\text{II}$: Counterexamples
Section $32$: Special Subsets of the Real Line
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

Steen and Seebach 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