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''(in descriptive set theory)''
 
''(in descriptive set theory)''
  
There exists an [[A-set|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914801.png" />-set]] (of the number axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914802.png" />) that is not a [[Borel set|Borel set]].
+
There exists an [[A-set|$\mathcal{A}$-set]] (of the number axis $\mathbb{R}$) that is not a [[Borel set|Borel set]].
  
In order that a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914803.png" />-set be a Borel set it is necessary and sufficient that its complement also be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914804.png" />-set.
+
In order that a given $\mathcal{A}$-set be a Borel set it is necessary and sufficient that its complement also be an $\mathcal{A}$-set.
  
Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914805.png" />-set in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914806.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914807.png" /> is the (orthogonal) projection of a Borel set (even of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914808.png" />) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s0914809.png" /> (and consequently, a plane Borel set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148010.png" /> exists whose projection is not a Borel set); the projection of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148011.png" />-set is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148012.png" />-set.
+
Every $\mathcal{A}$-set in the $n$-dimensional space $\mathbb{R}^n$ is the (orthogonal) projection of a Borel set (even of type $G_\delta$) in $\mathbb{R}^{n+1}$ (and consequently, a plane Borel set of type $G_\delta$ exists whose projection is not a Borel set); the projection of an $\mathcal{A}$-set is an $\mathcal{A}$-set.
  
All these results were obtained by M.Ya. Suslin [[#References|[1]]]. In order to define an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148013.png" />-set, he used the [[A-operation|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148014.png" />-operation]], while other methods of defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148015.png" />-sets were discovered subsequently. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148016.png" />-operation was in fact first discovered by P.S. Aleksandrov [[#References|[2]]], who demonstrated (although he did not explicitly formulate it) that every Borel set can be obtained as the result of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148017.png" />-operation over closed sets (and is consequently an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148018.png" />-set), and used this to prove a theorem on the cardinality of Borel sets (in fact, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148019.png" />-sets) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148020.png" />. N.N. Luzin subsequently posed the question of the existence of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148021.png" />-set that is not a Borel set. Theorem 1) answered this question. Theorems 1) and 2) were put forward by Suslin without proof [[#References|[1]]]. Suslin did subsequently prove them, but it was not until Luzin simplified the proofs that they were published. In order to prove 1), Suslin constructed a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148022.png" />-set that was universal for all Borel sets and examined the set of its points that lie on the diagonal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148023.png" /> (see [[#References|[3]]], p. 94). Theorem 2) is now often called the Suslin criterion for Borel sets. Suslin's proof of this theorem was based on a decomposition of a [[CA-set|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148024.png" />-set]] into the sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148025.png" /> Borel sets (see [[#References|[4]]], [[#References|[5]]]).
+
All these results were obtained by M.Ya. Suslin [[#References|[1]]]. In order to define an $\mathcal{A}$-set, he used the [[A-operation|$\mathcal{A}$-operation]], while other methods of defining $\mathcal{A}$-sets were discovered subsequently. The $\mathcal{A}$-operation was in fact first discovered by P.S. Aleksandrov [[#References|[2]]], who demonstrated (although he did not explicitly formulate it) that every Borel set can be obtained as the result of the $\mathcal{A}$-operation over closed sets (and is consequently an $\mathcal{A}$-set), and used this to prove a theorem on the cardinality of Borel sets (in fact, of $\mathcal{A}$-sets) in $\mathbb{R}$. N.N. Luzin subsequently posed the question of the existence of an $\mathcal{A}$-set that is not a Borel set. Theorem 1) answered this question. Theorems 1) and 2) were put forward by Suslin without proof [[#References|[1]]]. Suslin did subsequently prove them, but it was not until Luzin simplified the proofs that they were published. In order to prove 1), Suslin constructed a plane $\mathcal{A}$-set that was universal for all Borel sets and examined the set of its points that lie on the diagonal $x=y$ (see [[#References|[3]]], p. 94). Theorem 2) is now often called the Suslin criterion for Borel sets. Suslin's proof of this theorem was based on a decomposition of a [[CA-set|$\mathcal{CA}$-set]] into the sum of $\aleph_1$ Borel sets (see [[#References|[4]]], [[#References|[5]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. [M.Ya. Suslin] Souslin,  "Sur une définition des ensembles mesurables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148026.png" /> sans nombres transfinis"  ''C.R. Acad. Sci. Paris'' , '''164'''  (1917)  pp. 88–91</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Alexandroff,  "Sur la puissance des ensembles mesurables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148027.png" />"  ''C.R. Acad. Sci. Paris'' , '''162'''  (1916)  pp. 323–325</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Keldysh,  P.S. Novikov,  "The work of N.N. Luzin in the domain of the descriptive theory of sets"  ''Uspekhi Mat. Nauk'' , '''8''' :  2  (1953)  pp. 93–104  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.N. Luzin,  "Collected works" , '''2''' , Moscow  (1958)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  M. [M.Ya. Suslin] Souslin,  "Sur une définition des ensembles mesurables $B$ sans nombres transfinis"  ''C.R. Acad. Sci. Paris'' , '''164'''  (1917)  pp. 88–91</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Alexandroff,  "Sur la puissance des ensembles mesurables $B$"  ''C.R. Acad. Sci. Paris'' , '''162'''  (1916)  pp. 323–325</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Keldysh,  P.S. Novikov,  "The work of N.N. Luzin in the domain of the descriptive theory of sets"  ''Uspekhi Mat. Nauk'' , '''8''' :  2  (1953)  pp. 93–104  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  N.N. Luzin,  "Collected works" , '''2''' , Moscow  (1958)  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
For a more comprehensive historical note on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148028.png" />-sets (also called analytic sets) see Rogers' contribution in [[#References|[a1]]]. In particular, Suslin's work began with the discovery of a mistake in a famous paper of H. Lebesgue (1905), which also had a big positive influence on the construction of the first tools of descriptive set theory (universal sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148029.png" />-operation, etc.). The theorem on the cardinality of Borel sets proved by Aleksandrov, was independently proved by F. Hausdorff [[#References|[a2]]] (in a similar manner). Suslin's theorem 2) is now considered to be a corollary of the first separation theorem (see [[Luzin separability principles|Luzin separability principles]]). It has a more powerful version in effective descriptive set theory (see (the comments to) [[Descriptive set theory|Descriptive set theory]]), called the Suslin–Kleene theorem: A set is hyper-arithmetic (roughly speaking, is an effective Borel set) if and only if it belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091480/s09148030.png" /> (roughly speaking, is an effective analytic and co-analytic set).
+
For a more comprehensive historical note on $\mathcal{A}$-sets (also called analytic sets) see Rogers' contribution in [[#References|[a1]]]. In particular, Suslin's work began with the discovery of a mistake in a famous paper of H. Lebesgue (1905), which also had a big positive influence on the construction of the first tools of descriptive set theory (universal sets, $\mathcal{A}$-operation, etc.). The theorem on the cardinality of Borel sets proved by Aleksandrov, was independently proved by F. Hausdorff [[#References|[a2]]] (in a similar manner). Suslin's theorem 2) is now considered to be a corollary of the first separation theorem (see [[Luzin separability principles|Luzin separability principles]]). It has a more powerful version in effective descriptive set theory (see (the comments to) [[Descriptive set theory|Descriptive set theory]]), called the Suslin–Kleene theorem: A set is hyper-arithmetic (roughly speaking, is an effective Borel set) if and only if it belongs to $\mathbf{D}_1^1 = \mathbf{S}_1^1 \cap \mathbf{P}_1^1$ (roughly speaking, is an effective analytic and co-analytic set).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.A. Rogers,  J.E. Jayne,  C. Dellacherie,  F. Tøpsoe,  J. Hoffman-Jørgensen,  D.A. Martin,  A.S. Kechris,  A.H. Stone,  "Analytic sets" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Hausdorff,  "Die Mächtigkeit der Borelschen Mengen"  ''Math. Ann.'' , '''77'''  (1916)  pp. 430–437</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y.N. Moschovakis,  "Descriptive set theory" , North-Holland  (1980)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C.A. Rogers,  J.E. Jayne,  C. Dellacherie,  F. Tøpsoe,  J. Hoffman-Jørgensen,  D.A. Martin,  A.S. Kechris,  A.H. Stone,  "Analytic sets" , Acad. Press  (1980)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Hausdorff,  "Die Mächtigkeit der Borelschen Mengen"  ''Math. Ann.'' , '''77'''  (1916)  pp. 430–437</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  Y.N. Moschovakis,  "Descriptive set theory" , North-Holland  (1980)</TD></TR>
 +
</table>

Latest revision as of 19:53, 15 October 2014

(in descriptive set theory)

There exists an $\mathcal{A}$-set (of the number axis $\mathbb{R}$) that is not a Borel set.

In order that a given $\mathcal{A}$-set be a Borel set it is necessary and sufficient that its complement also be an $\mathcal{A}$-set.

Every $\mathcal{A}$-set in the $n$-dimensional space $\mathbb{R}^n$ is the (orthogonal) projection of a Borel set (even of type $G_\delta$) in $\mathbb{R}^{n+1}$ (and consequently, a plane Borel set of type $G_\delta$ exists whose projection is not a Borel set); the projection of an $\mathcal{A}$-set is an $\mathcal{A}$-set.

All these results were obtained by M.Ya. Suslin [1]. In order to define an $\mathcal{A}$-set, he used the $\mathcal{A}$-operation, while other methods of defining $\mathcal{A}$-sets were discovered subsequently. The $\mathcal{A}$-operation was in fact first discovered by P.S. Aleksandrov [2], who demonstrated (although he did not explicitly formulate it) that every Borel set can be obtained as the result of the $\mathcal{A}$-operation over closed sets (and is consequently an $\mathcal{A}$-set), and used this to prove a theorem on the cardinality of Borel sets (in fact, of $\mathcal{A}$-sets) in $\mathbb{R}$. N.N. Luzin subsequently posed the question of the existence of an $\mathcal{A}$-set that is not a Borel set. Theorem 1) answered this question. Theorems 1) and 2) were put forward by Suslin without proof [1]. Suslin did subsequently prove them, but it was not until Luzin simplified the proofs that they were published. In order to prove 1), Suslin constructed a plane $\mathcal{A}$-set that was universal for all Borel sets and examined the set of its points that lie on the diagonal $x=y$ (see [3], p. 94). Theorem 2) is now often called the Suslin criterion for Borel sets. Suslin's proof of this theorem was based on a decomposition of a $\mathcal{CA}$-set into the sum of $\aleph_1$ Borel sets (see [4], [5]).

References

[1] M. [M.Ya. Suslin] Souslin, "Sur une définition des ensembles mesurables $B$ sans nombres transfinis" C.R. Acad. Sci. Paris , 164 (1917) pp. 88–91
[2] P.S. [P.S. Aleksandrov] Alexandroff, "Sur la puissance des ensembles mesurables $B$" C.R. Acad. Sci. Paris , 162 (1916) pp. 323–325
[3] L.V. Keldysh, P.S. Novikov, "The work of N.N. Luzin in the domain of the descriptive theory of sets" Uspekhi Mat. Nauk , 8 : 2 (1953) pp. 93–104 (In Russian)
[4] N.N. Luzin, "Collected works" , 2 , Moscow (1958) (In Russian)
[5] F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))


Comments

For a more comprehensive historical note on $\mathcal{A}$-sets (also called analytic sets) see Rogers' contribution in [a1]. In particular, Suslin's work began with the discovery of a mistake in a famous paper of H. Lebesgue (1905), which also had a big positive influence on the construction of the first tools of descriptive set theory (universal sets, $\mathcal{A}$-operation, etc.). The theorem on the cardinality of Borel sets proved by Aleksandrov, was independently proved by F. Hausdorff [a2] (in a similar manner). Suslin's theorem 2) is now considered to be a corollary of the first separation theorem (see Luzin separability principles). It has a more powerful version in effective descriptive set theory (see (the comments to) Descriptive set theory), called the Suslin–Kleene theorem: A set is hyper-arithmetic (roughly speaking, is an effective Borel set) if and only if it belongs to $\mathbf{D}_1^1 = \mathbf{S}_1^1 \cap \mathbf{P}_1^1$ (roughly speaking, is an effective analytic and co-analytic set).

References

[a1] C.A. Rogers, J.E. Jayne, C. Dellacherie, F. Tøpsoe, J. Hoffman-Jørgensen, D.A. Martin, A.S. Kechris, A.H. Stone, "Analytic sets" , Acad. Press (1980)
[a2] F. Hausdorff, "Die Mächtigkeit der Borelschen Mengen" Math. Ann. , 77 (1916) pp. 430–437
[a3] Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)
How to Cite This Entry:
Suslin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Suslin_theorem&oldid=33668
This article was adapted from an original article by A.G. El'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article