Inequality for derived set

Question

Given a topological space $X$ and a subset $S$ of $X$, in Kuratowsky 1966 (volume 1, page 76) there is the inequality $$ S''\subseteq S'\tag{*} $$ in wikipedia there is the inequality $$ S''\subseteq S'\cup S\tag{**} $$ (which does not exclude (*)) and in the talk behind the web page there is this example proving (*) wrong. $X=\{a,b\}$, $S=\{a\}$, $S'=\{b\}$, $S''=\{a\}$. $a\notin S'$ because all the intersections with a neighborhood of $a$ (only $X$) contain no other point than $x$ itself.

Could someone shed some light on this inconsistency?

Answer 1

Kuratowski assumes all spaces are $T_1$. The counterexample is not $T_1$.

My French edition has (first chapter, paragraph 4, système d'axiomes)

II) si $X$ ne contient qu'un seul point ou n'en contient aucun, on a $\overline{X}=X$.

(If $X$ has only one point, or none at all, we have $\overline{X} =X$), so a space has all singletons closed, so the space is $T_1$ in modern terms.