An
-dimensional
compact space
,
,
in which any
partition
between non-empty sets has
dimension
.
An equivalent definition is: An
-dimensional
Cantor manifold is an
-dimensional
compact space
such that for each representation of
as the union of two non-empty closed proper subsets
and
,
.
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
-dimensional
closed ball, and therefore an
-dimensional
closed manifold, are Cantor manifolds;
-dimensional
Euclidean space cannot be partitioned by a set of dimension
(for
,
this is
Urysohn's theorem,
for
,
Aleksandrov's theorem).
An
-dimensional
Cantor manifold is the common boundary of two regions of
-dimensional
Euclidean space, one of which is bounded
(Aleksandrov's theorem).
The main fact in the theory of Cantor manifolds is that every
-dimensional
compact space contains an
-dimensional
Cantor manifold
(Aleksandrov's theorem).
A maximal
-dimensional
Cantor manifold in an
-dimensional
compact space
is called a
dimensional component
of
.
An
-dimensional
Cantor submanifold of a compact Hausdorff space
is contained in a unique dimensional component of
.
The intersection of two distinct dimensional components of an
-dimensional
compact Hausdorff space
has dimension
.
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
is an arbitrary dimensional component of a perfectly-normal compact space
and
is the union of all remaining dimensional components, then
(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
of all dimensional components of an
-dimensional
compact space
is called the
interior dimensional kernel
of the space. In view of the monotonicity of dimension, it is always true that
and
when
is a perfectly-normal compact space. The set
contains no
-dimensional
compact set. But even for Hausdorff compacta it is not known
(1978)
whether
.
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
,
and
with
,
and
,
there exists a hereditarily-normal compact space
of dimension
such that
and
.
If
,
then
(as defined by Urysohn) is the
inductive dimensional kernel,
that is, the set of all
for which
.
The inductive dimensional kernel
of a compact metric set
is always an
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
set. At each point
,
if
is compact metric
(Menger's theorem).
Therefore for an arbitrary compact metric space
,
is everywhere dense in
.
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
whose interior dimensional kernel
is everywhere dense in
is called a
generalized Cantor manifold.
The common boundary of two open subsets of
-dimensional
Euclidean space is an
-dimensional
generalized Cantor manifold. In a metrizable
-dimensional
generalized Cantor manifold
there may be an everywhere-dense set of points
for which
.
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
is called an
infinite-dimensional Cantor manifold
if there is no method of partitioning it by a weakly infinite-dimensional closed subset.