# Cantor manifold

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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)

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].
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.