Analytic set

A subset of a complete separable metric space that is a continuous image of the space of irrational numbers. The concept of an analytic set was introduced by N.N. Luzin [1]. His classical definition has been generalized to general metric and topological spaces.

1) An analytic set in an arbitrary topological space $ X $ is a subset of that space that is the image of a closed subset of the space of irrational numbers under an upper semi-continuous multi-valued mapping with compact images of points and a closed graph [2]. If $ X $ is a Hausdorff space, the last-mentioned condition is automatically satisfied. If $ X $ is metrizable, this definition is equivalent to the classical one.

2) In a complete separable metric space the classical analytic sets are identical with $ {\mathcal A} $- sets (cf. $ {\mathcal A} $- set). This fact forms the base of another definition of analytic sets (in their capacity as $ {\mathcal A} $- sets) in general metric and topological spaces [3], [4], [5]. In the class of completely-regular spaces, analytic sets in the sense of 1) are absolute analytic sets in the sense of 2). In the class of non-separable metrizable spaces definition 2) is employed, since definition 1) yields separable analytic sets.

3) An analytic set in a Hausdorff space [6], [7], is a continuous image of a subset of a compact space of type $ F _ {\sigma \delta } $.

4) An analytic set is a continuous image of a set belonging to the family $ K _ {\sigma \delta } $, where $ K $ is the family of all closed compact subsets of some topological space [8]. Definitions 1), 3) and 4) equivalent in the class of Hausdorff spaces.

5) For a generalization in another direction see [4]; $ k $- analytic sets are obtained from closed sets of a topological space by generalized $ {\mathcal A} $- operations (cf. $ {\mathcal A} $- operation; the Baire space of countable weight is replaced by a Baire space of weight $ k $) and are a generalization of analytic sets in the sense of 2).

References

[1]	N.N. Luzin, "Sur la classification de M. Baire" C.R. Acad. Sci. Paris Sér. I Math. , 164 (1917) pp. 91–94
[2]	Z. Frolík, "Respectability of the graphs of composites" Mathematika , 16 (1969) pp. 153–157
[3]	W. Sierpiński, "General topology" , Univ. Toronto Press (1956) (Translated from Polish)
[4]	A.H. Stone, "Non-separable Borel sets II" Gen. Topol. and Appl. , 2 (1972) pp. 249–270
[5]	K. Kuratowski, A. Mostowski, "Set theory" , North-Holland (1968)
[6]	V.E. Schneider, "Descriptive theory of sets in topological spaces" Uchen. Zap. Moskov. Gos. Univ. , 135 (1948) pp. 37–85
[7]	G. Choquet, "Theory of capacities" Ann. Inst. Fourier (Grenoble) , 5 (1953–1954) pp. 131–295
[8]	M. Sion, "On analytic sets in topological spaces" Trans. Amer. Math. Soc. , 96 (1960) pp. 341–354
[9]	J.E. Jayne, "Structure of analytic Hausdorff spaces" Mathematika , 23 (1976) pp. 208–211
[10]	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)

A.G. El'kin

6) An analytic set in the theory of analytic functions is a set that can locally be defined as the set of common zeros of a finite number of holomorphic functions. If $ S $ is an analytic set in an open subset $ U $ of the complex $ n $- dimensional space $ \mathbf C ^ {n} $, this means that for each point $ a \in U $ there exist a neighbourhood $ V \subset U $ and a finite tuple of functions $ f _ {1} \dots f _ {r} $, holomorphic in $ V $, such that $ S \cap V = \{ {z \in V } : { {f _ {1} } (z) = \dots = f _ {r} (z) = 0 } \} $. If the functions $ f _ {i} $ can be selected (in some neighbourhood $ V $) so that the rank of the Jacobi matrix $ | \partial f _ {i} / \partial z _ {j} | $ at the point $ a $ is $ r $, $ a $ is called a regular point of the analytic set $ S $; the number $ n - r $ is called the (complex) dimension of $ S $ at $ a $ and is denoted by $ { \mathop{\rm dim} _ {a} } S $. The set $ S ^ {*} $ of all regular points of an analytic set $ S $ is an open everywhere-dense subset of $ S $( in the induced topology on $ S $ as a subset of $ U $). Its complement $ S \setminus S ^ {*} $— the set of singular points of $ S $— is an analytic set in $ U $ that is nowhere dense in $ S $.

By definition

$$ \mathop{\rm dim} _ {a} S = \ \overline{\lim\limits}\; _ {\begin{array}{c} z \in S ^ {*} , \\ z \rightarrow a \end{array} } \mathop{\rm dim} _ {z} S , \ a \in S ; $$

the dimension of the analytic set $ S $ is the number

$$ \mathop{\rm dim} S = \sup _ {a \in S } \mathop{\rm dim} _ {a} S . $$

An analytic set $ S $ is called pure $ k $- dimensional if $ { \mathop{\rm dim} _ {a} } S = k $ for all $ a \in S $. For each $ 0 \leq k \leq \mathop{\rm dim} S $, the set $ S _ {k} = \{ {a \in S } : { \mathop{\rm dim} _ {a} S = k } \} $ is a pure $ k $- dimensional analytic set in $ U \setminus \cup _ {j > k } S _ {j} $. Thus, any analytic set in $ U $ can be represented as a finite union of pure analytic sets, $ S = \cup {S _ {k} } $. At the singular points $ { \mathop{\rm dim} _ {a} } ( S \setminus {S ^ {*} } ) < { \mathop{\rm dim} _ {a} } S $, so that the dimension of the analytic set of singular points of a pure $ k $- dimensional analytic set in $ U $ is smaller than $ k $. The connected components of $ S ^ {*} $ are complex manifolds. Since this is also true for the analytic set $ S \setminus S ^ {*} $, one obtains the decomposition:

$$ S = S ^ {*} \cup ( S \setminus S ^ {*} ) ^ {*} \cup \dots $$

of the analytic set into complex manifolds. The decomposition

$$ S = S _ {d} ^ {*} \cup ( S \setminus S _ {d} ^ {*} ) _ {d-1} ^ {*} \cup \dots $$

is more convenient (the dimensions of the summands strictly diminish, $ d = \mathop{\rm dim} S $); it is called the stratification of $ S $; the connected components of the $ k $- th summand of this sum are called the $ k $- dimensional strata of the analytic set $ S $.

An analytic set $ S $ is called reducible (in $ U $) if it is the union of two analytic sets in $ U $ other than itself; otherwise it is called irreducible (in $ U $). All irreducible analytic sets in $ U $ are connected and pure. An analytic set $ S $ in $ U $ is irreducible if and only if the set $ S ^ {*} $ of its regular points is connected. The closure of each connected component of $ S ^ {*} $ is an irreducible analytic set in $ U $; such analytic sets are called the irreducible components of $ S $. All analytic sets in $ U $ are locally finite unions of their irreducible components. If two analytic sets have no common irreducible components, the dimension of their intersection is strictly smaller than the dimension of each set. If the intersection of two irreducible sets in $ U $ contains a set that is open in each one, these analytic sets are identical (the identity theorem).

An analytic set $ S $ in $ U $ is called irreducible at a point $ a \in S $ if there exists a fundamental system of neighbourhoods $ V _ {i} $ of the point $ a $ in $ U $ such that all analytic sets $ S \cap V _ {i} $ in $ V _ {i} $ are irreducible; $ a $ is then called an irreducibility point of the analytic set $ S $. In a neighbourhood of each irreducibility point the analytic set has the structure of an analytic cover, i.e. for each such point $ a \in S $, $ { \mathop{\rm dim} _ {a} } S = k $, there exist a connected neighbourhood $ V \subset U $, a linear mapping $ \lambda : \mathbf C ^ {n} \rightarrow \mathbf C ^ {k} $ and an analytic set $ \sigma \subset \lambda ( V ) $ such that the restriction of $ \lambda $ to $ S \cap V $ is a proper mapping into $ \lambda ( V ) $, while the restriction of $ \lambda $ to $ ( S \cap V ) \setminus {\lambda ^ {-1} } ( \sigma ) $ is a finite-to-one locally biholomorphic cover over $ \lambda ( V ) \setminus \sigma $. For irreducible one-dimensional analytic sets there exists thus (after a suitable linear change in coordinates) a local parametric representation of the form

$$ z _ {1} = \zeta ^ {m} ,\ z _ {2} = f _ {2} ( \zeta ) \dots z _ {n} = f _ {n} ( \zeta ) , $$

where $ \zeta \in \mathbf C , | \zeta | < r $, $ m $ is a positive integer and the functions $ f _ {i} $ are holomorphic in the disc $ | \zeta | < r $. Thus, in a neighbourhood of each irreducibility point, a one-dimensional analytic set is a topological manifold. For an analytic set of higher dimension this is usually not true.

The union of a finite number and the intersection of any family of analytic sets in $ U $ are analytic sets in $ U $. Any analytic set in $ U $ is closed in $ U $. Any compact analytic set in $ U \subset \mathbf C ^ {n} $ consists of a finite number of points. If $ U $ is connected and the analytic set $ S \neq U $, then $ U \setminus S $ is open, everywhere-dense in $ U $ and is also connected. The set of all isolated points of an analytic set in $ U $ has no limit points in $ U $. Moreover, all analytic sets are locally connected. A connected analytic set is pathwise connected.

Any analytic set in $ U $ of dimension $ d $ has locally in $ U $ finite $ 2d $- dimensional Hausdorff measure $ \mathop{\rm mes} _ {2d} $. If $ { \mathop{\rm dim} _ {a} } S = k $, there exist positive constants $ c $ and $ C $( which depend on $ a $ and $ S $) such that

$$ cr ^ {2k} \leq \mathop{\rm mes} _ {2k} ( S \cap \{ | z - a | < r \} ) \leq Cr ^ {2k} , $$

for all sufficiently small $ r < 0 $.

The family of analytic sets is invariant under biholomorphic mappings (cf. Biholomorphic mapping). Moreover, if $ S $ is an analytic set in $ U $ and if $ f: U \rightarrow U _ {1} $ is a proper holomorphic mapping, then $ f(S) $ is an analytic set in $ U _ {1} $.

The definition of analytic sets on complex manifolds is similar to the definition for $ \mathbf C ^ {n} $, and all the properties listed above are preserved except for one: in the general case there exist compact non-discrete analytic sets. In special manifolds analytic sets can have certain additional properties. For example, in the complex $ n $- dimensional projective space all analytic sets are algebraic, i.e. coincide with the set of common zeros of some finite tuple of homogeneous polynomials (Chow's theorem).

Real-analytic sets in open subsets of $ \mathbf R ^ {n} $ are defined in the same manner, except that real-analytic functions must be taken instead of holomorphic functions. Any real-analytic set is the intersection of some analytic set (in some open subset of $ \mathbf C ^ {n} $) with the real subspace $ \mathbf R ^ {n} \subset \mathbf C ^ {n} $.

References

[1]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)
[2]	M. Hervé, "Several complex variables: local theory" , Oxford Univ. Press (1963)

E.M. Chirka

Comments

Analytic sets are also called analytic varieties or complex-analytic sets