Cantor manifold

An $ n $- dimensional compact space $ X $, $ \mathop{\rm dim} X = n $, in which any partition $ B $ between non-empty sets has dimension $ \mathop{\rm dim} B \geq n - 1 $. An equivalent definition is: An $ n $- dimensional Cantor manifold is an $ n $- dimensional compact space $ X $ such that for each representation of $ X $ as the union of two non-empty closed proper subsets $ X _ {1} $ and $ X _ {2} $, $ \mathop{\rm dim} ( X _ {1} \cap X _ {2} ) \geq n - 1 $. One-dimensional metrizable Cantor manifolds are one-dimensional continua or Cantor curves (cf. Cantor curve).

The concept of a Cantor manifold was introduced by P.S. Urysohn (see [1]). An $ n $- dimensional closed ball, and therefore an $ n $- dimensional closed manifold, are Cantor manifolds; $ n $- dimensional Euclidean space cannot be partitioned by a set of dimension $ \leq n - 2 $( for $ n = 3 $, this is Urysohn's theorem, for $ n > 3 $, Aleksandrov's theorem). An $ ( n - 1 ) $- dimensional Cantor manifold is the common boundary of two regions of $ n $- dimensional Euclidean space, one of which is bounded (Aleksandrov's theorem). The main fact in the theory of Cantor manifolds is that every $ n $- dimensional compact space contains an $ n $- dimensional Cantor manifold (Aleksandrov's theorem).

A maximal $ n $- dimensional Cantor manifold in an $ n $- dimensional compact space $ X $ is called a dimensional component of $ X $. An $ n $- dimensional Cantor submanifold of a compact Hausdorff space $ X $ is contained in a unique dimensional component of $ X $. The intersection of two distinct dimensional components of an $ n $- dimensional compact Hausdorff space $ X $ has dimension $ \leq n - 2 $. In particular, dimensional components of a one-dimensional compact Hausdorff space are components of it. The set of dimensional components of a finite-dimensional compact metric space is finite, countable or has the cardinality of the continuum. If $ A $ is an arbitrary dimensional component of a perfectly-normal compact space $ X $ and $ B $ is the union of all remaining dimensional components, then $ \mathop{\rm dim} ( A \cap B ) \leq m - 2 $( Aleksandrov's theorem). In a hereditarily-normal first-countable compact Hausdorff space, a dimensional component may be contained in the union of the remaining dimensional components.

The union $ K _ {X} $ of all dimensional components of an $ n $- dimensional compact space $ X $ is called the interior dimensional kernel of the space. In view of the monotonicity of dimension, it is always true that $ \mathop{\rm dim} K _ {X} = \mathop{\rm dim} X $ and $ \mathop{\rm dim} ( X \setminus K _ {X} ) \leq \mathop{\rm dim} X $ when $ X $ is a perfectly-normal compact space. The set $ X \setminus K _ {X} $ contains no $ n $- dimensional compact set. But even for Hausdorff compacta it is not known (1978) whether $ \mathop{\rm dim} ( X \setminus K _ {X} ) = \mathop{\rm dim} X $. With regard to hereditarily-normal compact spaces, the interior dimensional kernel and its complement can have all permissible dimensions; that is to say, assuming the validity of the continuum hypothesis, for any triple of integers $ n $, $ n _ {1} $ and $ n _ {2} $ with $ n \geq 1 $, $ n _ {1} \geq n $ and $ n _ {2} \geq 0 $, there exists a hereditarily-normal compact space $ X $ of dimension $ n $ such that $ \mathop{\rm dim} K _ {X} = n _ {1} $ and $ \mathop{\rm dim} ( X \setminus K _ {X} ) = n _ {2} $.

If $ \mathop{\rm dim} X = \mathop{\rm ind} X $, then $ K _ {X} \subset N _ {X} $( as defined by Urysohn) is the inductive dimensional kernel, that is, the set of all $ x \in X $ for which $ \mathop{\rm ind} _ {x} X = n $. The inductive dimensional kernel $ N _ {X} $ of a compact metric set $ X $ is always an $ F _ \sigma $ set. It is not known whether the same holds for the interior dimensional kernel. For compact Hausdorff spaces however, neither the inductive dimensional kernel nor the interior dimensional kernel need be an $ F _ \sigma $ set. At each point $ x \in N _ {X} $,

$$ \mathop{\rm ind} _ {x} N _ {X} = \mathop{\rm ind} _ {x} X , $$

if $ X $ is compact metric (Menger's theorem). Therefore for an arbitrary compact metric space $ X $, $ K _ {X} $ is everywhere dense in $ N _ {X} $. This does not carry over to arbitrary compact Hausdorff spaces. It remains an open question (1978) whether a point is contained in the inductive dimensional kernel along with some non-degenerate continuum.

A finite-dimensional continuum $ X $ whose interior dimensional kernel $ K _ {X} $ is everywhere dense in $ X $ is called a generalized Cantor manifold. The common boundary of two open subsets of $ n $- dimensional Euclidean space is an $ ( n - 1 ) $- dimensional generalized Cantor manifold. In a metrizable $ n $- dimensional generalized Cantor manifold $ X $ there may be an everywhere-dense set of points $ x $ for which $ \mathop{\rm ind} _ {x} X < n $. Neither products nor continuous mappings preserve the property of being a generalized Cantor manifold. The same is true concerning the property of being a Cantor manifold.

A compact space $ X $ is called an infinite-dimensional Cantor manifold if there is no method of partitioning it by a weakly infinite-dimensional closed subset.

References

[1]	P.S. Urysohn, "Works on topology and other areas of mathematics" , 1 , Moscow-Leningrad (1951) (In Russian)
[2]	P [P.S. Aleksandrov] Alexandroff, "Untersuchungen über Gestalt und lage abqeschlossener Menge beliebiqer Dimension" Ann. of Math. , 30 (1929) pp. 101–187
[3]	P.S. Aleksandrov, "On the dimension of normal spaces" Proc. Royal. Soc. London Ser. A , 189 (1947) pp. 11–39
[4]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)
[5]	V.V. Fedorchuk, "On dimensional components of compact spaces" Soviet Math. Dokl. , 15 : 2 (1974) pp. 505–509 Dokl. Akad. Nauk SSSR , 215 : 2 (1974) pp. 289–292
[6]	K. Menger, "Dimensiontheorie" , Teubner (1928)
[7]	E.G. Sklyarenko, "Dimensionality properties of infinite-dimensional spaces" Izv. Akad. Nauk SSSR Ser. Mat. , 23 : 2 (1959) pp. 197–212 (In Russian)

Comments

The theorems attributed to Aleksandrov are not only due to him: The theorem on partitions of $ n $- dimensional Euclidean space is attributed to K. Menger [a5] and Urysohn [a1] and [a2].

The Cantor manifold theorem for compact metric spaces is due to W. Hurewicz and Menger [a3] and L.A. Tumarkin [a6]. Aleksandrov generalized it to arbitrary compact Hausdorff spaces in [3]. Finally, the theorem on intersections of dimensional components was proved by S. Mazurkiewicz in [a4] for compact metric spaces, Aleksandrov generalized it to perfectly-normal compact spaces.

It is not true that every infinite-dimensional compact space contains an infinite-dimensional Cantor manifold, as there are many compact metric weakly infinite-dimensional spaces, e.g. the one-point compactification of the topological sum $ \oplus _ {n=1} ^ \infty I ^ { n } $ of cubes of increasing dimension.

References

[a1]	P.S. Urysohn, "Mémoire sur les multiplicités cantoriennes" Fund. Math. , 7 (1925) pp. 30–137
[a2]	R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50
[a3]	W. Hurewicz, K. Menger, "Dimension and Zusammenhangsstuffe" Math. Ann. , 100 (1928) pp. 618–633
[a4]	S. Mazurkiewicz, "Ein Satz über dimensionelle Komponenten" Fund. Math. , 20 (1933) pp. 98–99
[a5]	K. Menger, "Über die dimension von Punktmengen II" Monatsh. für Math. and Phys. , 34 (1926) pp. 137–161
[a6]	L.A. Tumarkin, "Sur la structure dimensionelle des ensembles fermés" C.R. Acad. Paris , 186 (1928) pp. 420–422