View source for Cantor manifold

An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202301.png" />-dimensional compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202303.png" />, in which any [[Partition|partition]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202304.png" /> between non-empty sets has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202305.png" />. An equivalent definition is: An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202306.png" />-dimensional Cantor manifold is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202307.png" />-dimensional compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202308.png" /> such that for each representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c0202309.png" /> as the union of two non-empty closed proper subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023012.png" />. One-dimensional metrizable Cantor manifolds are one-dimensional continua or Cantor curves (cf. [[Cantor curve|Cantor curve]]).

The concept of a Cantor manifold was introduced by P.S. Urysohn (see [[#References|[1]]]). An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023013.png" />-dimensional closed ball, and therefore an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023014.png" />-dimensional closed manifold, are Cantor manifolds; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023015.png" />-dimensional Euclidean space cannot be partitioned by a set of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023016.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023017.png" />, this is Urysohn's theorem, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023018.png" />, Aleksandrov's theorem). An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023019.png" />-dimensional Cantor manifold is the common boundary of two regions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023020.png" />-dimensional Euclidean space, one of which is bounded (Aleksandrov's theorem). The main fact in the theory of Cantor manifolds is that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023021.png" />-dimensional compact space contains an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023022.png" />-dimensional Cantor manifold (Aleksandrov's theorem).

A maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023023.png" />-dimensional Cantor manifold in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023024.png" />-dimensional compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023025.png" /> is called a dimensional component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023026.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023027.png" />-dimensional Cantor submanifold of a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023028.png" /> is contained in a unique dimensional component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023029.png" />. The intersection of two distinct dimensional components of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023030.png" />-dimensional compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023031.png" /> has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023032.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023033.png" /> is an arbitrary dimensional component of a perfectly-normal compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023035.png" /> is the union of all remaining dimensional components, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023036.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023037.png" /> of all dimensional components of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023038.png" />-dimensional compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023039.png" /> is called the interior dimensional kernel of the space. In view of the monotonicity of dimension, it is always true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023041.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023042.png" /> is a perfectly-normal compact space. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023043.png" /> contains no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023044.png" />-dimensional compact set. But even for Hausdorff compacta it is not known (1978) whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023045.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023048.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023051.png" />, there exists a hereditarily-normal compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023052.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023053.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023055.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023057.png" /> (as defined by Urysohn) is the inductive dimensional kernel, that is, the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023058.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023059.png" />. The inductive dimensional kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023060.png" /> of a compact metric set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023061.png" /> is always an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023062.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023063.png" /> set. At each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023064.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023065.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023066.png" /> is compact metric (Menger's theorem). Therefore for an arbitrary compact metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023068.png" /> is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023069.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023070.png" /> whose interior dimensional kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023071.png" /> is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023072.png" /> is called a generalized Cantor manifold. The common boundary of two open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023073.png" />-dimensional Euclidean space is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023074.png" />-dimensional generalized Cantor manifold. In a metrizable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023075.png" />-dimensional generalized Cantor manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023076.png" /> there may be an everywhere-dense set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023077.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023078.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023079.png" /> is called an infinite-dimensional Cantor manifold if there is no method of partitioning it by a weakly infinite-dimensional closed subset.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Urysohn,   "Works on topology and other areas of mathematics" , '''1''' , Moscow-Leningrad  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P [P.S. Aleksandrov] Alexandroff,   "Untersuchungen über Gestalt und lage abqeschlossener Menge beliebiqer Dimension"  ''Ann. of Math.'' , '''30'''  (1929)  pp. 101–187</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,   "On the dimension of normal spaces"  ''Proc. Royal. Soc. London Ser. A'' , '''189'''  (1947)  pp. 11–39</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.S. Aleksandrov,   B.A. Pasynkov,   "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Menger,   "Dimensiontheorie" , Teubner  (1928)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.G. Sklyarenko,   "Dimensionality properties of infinite-dimensional spaces"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''23''' :  2  (1959)  pp. 197–212  (In Russian)</TD></TR></table>



====Comments====
The theorems attributed to Aleksandrov are not only due to him: The theorem on partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023080.png" />-dimensional Euclidean space is attributed to K. Menger [[#References|[a5]]] and Urysohn [[#References|[a1]]] and [[#References|[a2]]].

The Cantor manifold theorem for compact metric spaces is due to W. Hurewicz and Menger [[#References|[a3]]] and L.A. Tumarkin [[#References|[a6]]]. Aleksandrov generalized it to arbitrary compact Hausdorff spaces in [[#References|[3]]]. Finally, the theorem on intersections of dimensional components was proved by S. Mazurkiewicz in [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020230/c02023081.png" /> of cubes of increasing dimension.

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