XI: 04, 34-46, LNM 581 (1977)
DELLACHERIE, Claude
Les dérivations en théorie descriptive des ensembles et le théorème de la borne (
Descriptive set theory)
At the root of set theory lies Cantor's definition of the ``derived set'' $\delta A$ of a closed set $A$, i.e., the set of its non-isolated points, with the help of which Cantor proved that a closed set can be decomposed into a perfect set and a countable set. One may define the index $j(A)$ to be the smallest ordinal $\alpha$ such that $\delta^\alpha A=\emptyset$, or $\omega_1$ if there is no such ordinal. Considering the set $F$ of all closed sets as a (Polish) topological space, ordered by inclusion, $\delta$ as an increasing mapping from $F$ such that $\delta A\subset A$, let $D$ be the set of all $A$ such that $j(A)<\omega_1$ (thus, the set of all countable closed sets). Then $D$ is coanalytic and non-Borel, while the index is bounded by a countable ordinal on every analytic subset of $D$. These powerful results are stated abstractly and proved under very general conditions. Several examples are given
Comment: See a correction in
1241, and several examples in Hillard
1242. The whole subject has been exposed anew in Chapter~XXIV of Dellacherie-Meyer,
Probabilités et PotentielKeywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Exposition,
Original additions
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XII: 41, 523-523, LNM 649 (1978)
DELLACHERIE, Claude
Erratum et addendum à ``les dérivations en théorie descriptive des ensembles et le théorème de la borne'' (
Descriptive set theory)
A few corrections to
1104Keywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Correction Retrieve article from Numdam
XII: 42, 524-563, LNM 649 (1978)
HILLARD, Gérard
Exemples de normes en théorie descriptive des ensembles (
Descriptive set theory)
The situations described in this paper are special cases of
1104, where a coanalytic set $A$ was represented as the union of an increasing family $A_{\alpha}$ of analytic sets indexed by the countable ordinals, such that every analytic subset of $A$ is contained in some $A_{\alpha}$. The hypotheses of
1104 are not easy to check: they are shown here to include the classical Cantor derivation on the coanalytic space of countable compact sets, and a new example on the coanalytic space of all right continuous functions
Comment: The whole subject has been exposed anew in Chapter XXIV of Dellacherie-Meyer,
Probabilités et PotentielKeywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Original Retrieve article from Numdam
