Kuratowski's Closure-Complement Problem/Mistake

Source Work

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.):

Part $\text {II}$: Counterexamples

Section $32$: Special Subsets of the Real Line

Item $9$: Figure $12$

Mistake

Steen and Seebach 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:

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:

Also see

• Kuratowski's Closure-Complement Problem

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $9$