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A normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508501.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508502.png" /> (cf. [[Normal space|Normal space]]) such that for no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508503.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508504.png" /> is satisfied, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508505.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508506.png" /> it is possible to find a finite open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508507.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508508.png" /> such that every finite covering refining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i0508509.png" /> has multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085010.png" />. Examples of infinite-dimensional spaces are the [[Hilbert cube|Hilbert cube]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085011.png" /> and the [[Tikhonov cube|Tikhonov cube]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085012.png" />. Most of the spaces encountered in functional analysis are also infinite-dimensional.
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A normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085013.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085014.png" /> is said to be infinite-dimensional in the sense of the large (small) inductive dimension if the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085016.png" />) is invalid for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085018.png" /> is an infinite-dimensional space, it is infinite-dimensional in the sense of the large inductive dimension. If in addition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085019.png" /> is compact, it is also infinite-dimensional in the sense of the small inductive dimension. The infinite dimensionality of a metric space is equivalent with its infinite dimensionality in the sense of the large inductive dimension. There exist finite-dimensional compacta that are infinite-dimensional in the sense of the small (and hence also in the sense of the large) inductive dimension. It is not known (1986) whether or not a compactum (or a metric space) that is finite-dimensional in the sense of the small inductive dimension and infinite-dimensional in the sense of the large inductive dimension exists.
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One of the most natural approaches to the study of infinite-dimensional spaces is to introduce the small transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085020.png" /> and the large transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085021.png" />. This approach consists in the extension of the definition of small and large inductive dimensions to infinite ordinal numbers. The transfinite dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085023.png" /> are not defined for all infinite-dimensional spaces. Thus, neither is defined for the Hilbert cube. The large transfinite dimension is not defined for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085024.png" />, which is the discrete sum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085025.png" />-dimensional cubes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085027.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085028.png" />.
+
A normal  $  T _ {1} $-
 +
space  $  X $(
 +
cf. [[Normal space|Normal space]]) such that for no  $  n = - 1, 0, 1 \dots $
 +
the inequality  $  \mathop{\rm dim}  X \leq  n $
 +
is satisfied, i.e. $  X \neq \emptyset $
 +
and for any  $  n = 0, 1 \dots $
 +
it is possible to find a finite open covering  $  \omega _ {n} $
 +
of  $  X $
 +
such that every finite covering refining  $  \omega _ {n} $
 +
has multiplicity  $  > n + 1 $.  
 +
Examples of infinite-dimensional spaces are the [[Hilbert cube|Hilbert cube]]  $  I  ^  \infty  $
 +
and the [[Tikhonov cube|Tikhonov cube]]  $  I  ^  \tau  $.  
 +
Most of the spaces encountered in functional analysis are also infinite-dimensional.
  
If the transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085030.png" />) is defined for a normal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085031.png" />, then it is equal to an ordinal number whose cardinality does not exceed the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085032.png" /> (respectively, the large weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085033.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085034.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085035.png" /> has a countable base, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085036.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085037.png" /> is compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085038.png" /> as well. For metric spaces, too, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085040.png" />, then there exist compacta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085042.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085044.png" />. For any ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085045.png" /> there exists a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085046.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085047.png" />.
+
A normal  $  T _ {1} $-
 +
space  $  X $
 +
is said to be infinite-dimensional in the sense of the large (small) inductive dimension if the inequality  $  \mathop{\rm Ind}  X \leq  n $(
 +
$  \mathop{\rm ind}  X \leq  n $)  
 +
is invalid for every  $  n = - 1, 0, 1 ,\dots $.  
 +
If  $  X $
 +
is an infinite-dimensional space, it is infinite-dimensional in the sense of the large inductive dimension. If in addition  $  X $
 +
is compact, it is also infinite-dimensional in the sense of the small inductive dimension. The infinite dimensionality of a metric space is equivalent with its infinite dimensionality in the sense of the large inductive dimension. There exist finite-dimensional compacta that are infinite-dimensional in the sense of the small (and hence also in the sense of the large) inductive dimension. It is not known (1986) whether or not a compactum (or a metric space) that is finite-dimensional in the sense of the small inductive dimension and infinite-dimensional in the sense of the large inductive dimension exists.
  
If the transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085048.png" /> is defined, the transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085049.png" /> is defined as well, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085050.png" />. Metric compacta for which the transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085051.png" /> is defined and for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085052.png" />, have also been constructed.
+
One of the most natural approaches to the study of infinite-dimensional spaces is to introduce the small transfinite dimension $  \mathop{\rm ind}  X $
 +
and the large transfinite dimension $  \mathop{\rm Ind}  X $.  
 +
This approach consists in the extension of the definition of small and large inductive dimensions to infinite ordinal numbers. The transfinite dimensions  $  \mathop{\rm ind}  X $
 +
and  $  \mathop{\rm Ind}  X $
 +
are not defined for all infinite-dimensional spaces. Thus, neither is defined for the Hilbert cube. The large transfinite dimension is not defined for the space  $  \cup I  ^ {n} $,
 +
which is the discrete sum of the  $  n $-
 +
dimensional cubes  $  I  ^ {n} $,
 +
$  n = 0, 1 \dots $
 +
but  $  \mathop{\rm ind}  \cup I  ^ {n} = \omega _ {0} $.
  
If the transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085053.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085054.png" />) of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085055.png" /> is defined, then also the transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085056.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085057.png" />) is defined for any (respectively, any closed) set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085058.png" />, and the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085059.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085060.png" />) is valid.
+
If the transfinite dimension $  { \mathop{\rm ind} }  X $(
 +
$  { \mathop{\rm Ind} }  X $)  
 +
is defined for a normal space $  X $,  
 +
then it is equal to an ordinal number whose cardinality does not exceed the weight  $  wX $(
 +
respectively, the large weight  $  WX $)  
 +
of  $  X $.
 +
In particular, if  $  X $
 +
has a countable base, then  $  { \mathop{\rm ind} }  X < \omega _ {1} $,  
 +
and if  $  X $
 +
is compact, then  $  { \mathop{\rm Ind} }  X < \omega _ {1} $
 +
as well. For metric spaces, too,  $  { \mathop{\rm Ind} }  X < \omega _ {1} $.  
 +
If  $  \alpha < \omega _ {1} $,
 +
then there exist compacta  $  S _  \alpha  $
 +
and  $  L _  \alpha  $
 +
for which  $  { \mathop{\rm Ind} }  S _  \alpha  = \alpha $,
 +
$  \mathop{\rm ind}  L _  \alpha  = \alpha $.
 +
For any ordinal number  $  \alpha $
 +
there exists a metric space  $  X _  \alpha  $
 +
with  $  \mathop{\rm ind}  X _  \alpha  = \alpha $.
  
For the maximal compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085061.png" /> of a normal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085062.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085063.png" /> is valid. A normal space of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085064.png" /> and of transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085065.png" /> has a compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085066.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085067.png" /> and dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085068.png" />. There exists a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085069.png" /> with a countable base having dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085070.png" /> for which no compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085071.png" /> with a countable base has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085072.png" />. A metrizable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085073.png" /> of transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085074.png" /> has a metric such that the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085075.png" /> with respect to it has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085076.png" />. A metrizable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085077.png" /> of transfinite dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085078.png" /> with a countable base has a metric such that the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085079.png" /> with respect to it has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085080.png" />.
+
If the transfinite dimension  $  { \mathop{\rm Ind} }  X $
 +
is defined, the transfinite dimension $  { \mathop{\rm ind} }  X $
 +
is defined as well, and $  \mathop{\rm ind}  X \leq  \mathop{\rm Ind}  X $.  
 +
Metric compacta for which the transfinite dimension $  { \mathop{\rm Ind} }  X $
 +
is defined and for which  $  \omega _ {0} < { \mathop{\rm ind} }  X < { \mathop{\rm Ind} }  X $,
 +
have also been constructed.
  
The class of spaces for which a large or a small transfinite dimension is defined is closely connected with the class of metric countable-dimensional spaces; if a complete metric space is countable-dimensional, then the small transfinite dimension is defined for it; if the small transfinite dimension is defined for a metric space with a countable base, the space is countable-dimensional; if for a metric space the large transfinite dimension is defined (in particular if the space is finite-dimensional), then the space is countable-dimensional; the large transfinite dimension is defined for a countable-dimensional metric compactum. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085081.png" /> is countable-dimensional and is infinite-dimensional. The Hilbert cube is not countable-dimensional.
+
If the transfinite dimension $  { \mathop{\rm ind} }  X $(
 +
$  { \mathop{\rm Ind} }  X $)
 +
of a space $  X $
 +
is defined, then also the transfinite dimension $  { \mathop{\rm Ind} }  A $(
 +
$  { \mathop{\rm Ind} }  A $)
 +
is defined for any (respectively, any closed) set  $  A \subseteq X $,  
 +
and the inequality  $  { \mathop{\rm ind} }  A \leq  { \mathop{\rm ind} }  X $(
 +
or  $  { \mathop{\rm Ind} }  A \leq  \mathop{\rm Ind}  X $)  
 +
is valid.
  
Countable dimensionality of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085082.png" /> is equivalent to any one of the following properties: a) there exists a finite-to-one (but, in general, not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085083.png" />-to-one for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085084.png" />) continuous closed mapping of a zero-dimensional metric space onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085085.png" />; b) there exists a countable-to-one continuous closed mapping of a zero-dimensional metric space onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085086.png" />; and c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085087.png" /> is a [[Countably zero-dimensional space|countably zero-dimensional space]].
+
For the maximal compactification  $  \beta X $
 +
of a normal space $  X $
 +
the equality  $  { \mathop{\rm Ind} }  \beta X = { \mathop{\rm Ind} }  X $
 +
is valid. A normal space of weight  $  \tau $
 +
and of transfinite dimension  $  { \mathop{\rm Ind} }  X \leq  \alpha $
 +
has a compactification  $  bX $
 +
of weight  $  \tau $
 +
and dimension  $  { \mathop{\rm Ind} }  bX \leq  \alpha $.
 +
There exists a space  $  L $
 +
with a countable base having dimension  $  { \mathop{\rm ind} }  L = \omega _ {0} $
 +
for which no compactification  $  bX $
 +
with a countable base has dimension  $  { \mathop{\rm ind} }  bX = \omega _ {0} $.  
 +
A metrizable space  $  R $
 +
of transfinite dimension  $  { \mathop{\rm Ind} }  R = \alpha $
 +
has a metric such that the completion  $  cR $
 +
with respect to it has dimension  $  { \mathop{\rm Ind} }  cR = \alpha $.
 +
A metrizable space $  R $
 +
of transfinite dimension  $  { \mathop{\rm ind} }  R = \alpha $
 +
with a countable base has a metric such that the completion  $  cR $
 +
with respect to it has dimension  $  { \mathop{\rm ind} }  cR = \alpha $.
  
Theorems about the representability of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085088.png" />-dimensional metric space as a sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085089.png" /> zero-dimensional subsets or as the image of a zero-dimensional metric space under a continuous closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085090.png" />-to-one mapping indicate that it is natural to consider the class of countable-dimensional (metric) spaces and that it is close to the class of finite-dimensional spaces. As in the finite-dimensional case, there exists a countable-dimensional space which is universal in the sense of homeomorphic imbedding in the class of countable-dimensional metric spaces of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085091.png" />.
+
The class of spaces for which a large or a small transfinite dimension is defined is closely connected with the class of metric countable-dimensional spaces; if a complete metric space is countable-dimensional, then the small transfinite dimension is defined for it; if the small transfinite dimension is defined for a metric space with a countable base, the space is countable-dimensional; if for a metric space the large transfinite dimension is defined (in particular if the space is finite-dimensional), then the space is countable-dimensional; the large transfinite dimension is defined for a countable-dimensional metric compactum. The space  $  \cup I  ^ {n} $
 +
is countable-dimensional and is infinite-dimensional. The Hilbert cube is not countable-dimensional.
 +
 
 +
Countable dimensionality of a metric space  $  R $
 +
is equivalent to any one of the following properties: a) there exists a finite-to-one (but, in general, not a  $  k $-
 +
to-one for any  $  k = 1, 2 ,\  . . $)
 +
continuous closed mapping of a zero-dimensional metric space onto  $  R $;
 +
b) there exists a countable-to-one continuous closed mapping of a zero-dimensional metric space onto  $  R $;
 +
and c)  $  R $
 +
is a [[Countably zero-dimensional space|countably zero-dimensional space]].
 +
 
 +
Theorems about the representability of any  $  n $-
 +
dimensional metric space as a sum of $  n + 1 $
 +
zero-dimensional subsets or as the image of a zero-dimensional metric space under a continuous closed $  ( n + 1) $-
 +
to-one mapping indicate that it is natural to consider the class of countable-dimensional (metric) spaces and that it is close to the class of finite-dimensional spaces. As in the finite-dimensional case, there exists a countable-dimensional space which is universal in the sense of homeomorphic imbedding in the class of countable-dimensional metric spaces of weight $  \leq  \tau $.
  
 
If a normal space is represented as a finite or a countable sum of its countable-dimensional subspaces, then it is countable-dimensional. A subspace of a countable-dimensional perfectly-normal space is countable-dimensional.
 
If a normal space is represented as a finite or a countable sum of its countable-dimensional subspaces, then it is countable-dimensional. A subspace of a countable-dimensional perfectly-normal space is countable-dimensional.
  
The following theorem describes the relationships between countable and non-countable dimensional metric spaces: If a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085092.png" /> between metric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085094.png" /> is continuous and closed, if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085095.png" /> is countable-dimensional and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085096.png" /> is non-countable dimensional, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085097.png" /> is also non-countable dimensional.
+
The following theorem describes the relationships between countable and non-countable dimensional metric spaces: If a mapping $  f: R \rightarrow S $
 +
between metric spaces $  R $
 +
and $  S $
 +
is continuous and closed, if the space $  R $
 +
is countable-dimensional and the space $  S $
 +
is non-countable dimensional, then the set $  \{ {y \in S } : {| f ^ { - 1 } y | \geq  c } \} $
 +
is also non-countable dimensional.
  
In addition to countable-dimensional spaces, a natural extension of the class of finite-dimensional spaces is the class of weakly countable-dimensional spaces. If one considers metrizable spaces only, weakly countable-dimensional spaces occupy a place which is intermediate between finite-dimensional and countable-dimensional spaces. There exist countable-dimensional metric compacta that are not weakly countable-dimensional, while the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085098.png" /> is both weakly countable-dimensional and infinite-dimensional. A closed subspace of a weakly countable-dimensional space is weakly countable-dimensional. A normal space is weakly countable-dimensional if it is representable as a finite or a countable sum of its weakly countable-dimensional closed subsets.
+
In addition to countable-dimensional spaces, a natural extension of the class of finite-dimensional spaces is the class of weakly countable-dimensional spaces. If one considers metrizable spaces only, weakly countable-dimensional spaces occupy a place which is intermediate between finite-dimensional and countable-dimensional spaces. There exist countable-dimensional metric compacta that are not weakly countable-dimensional, while the space $  \cup I  ^ {n} $
 +
is both weakly countable-dimensional and infinite-dimensional. A closed subspace of a weakly countable-dimensional space is weakly countable-dimensional. A normal space is weakly countable-dimensional if it is representable as a finite or a countable sum of its weakly countable-dimensional closed subsets.
  
In the classes of normal weakly countable-dimensional and metric weakly countable-dimensional spaces there exist universal (in the sense of homeomorphic imbedding) spaces. In the case of spaces with a countable base, an example is the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i05085099.png" /> of the Hilbert cube which consists of all points with only a finite number of non-zero coordinates. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850100.png" /> has no weakly countable-dimensional compactifications.
+
In the classes of normal weakly countable-dimensional and metric weakly countable-dimensional spaces there exist universal (in the sense of homeomorphic imbedding) spaces. In the case of spaces with a countable base, an example is the subspace $  I  ^  \omega  $
 +
of the Hilbert cube which consists of all points with only a finite number of non-zero coordinates. The space $  I  ^  \omega  $
 +
has no weakly countable-dimensional compactifications.
  
All classes of infinite-dimensional spaces considered so far are  "not very infinite-dimensional"  as compared with, for example, the Hilbert cube. The problem of distinguishing  "not very infinite-dimensional"  from  "very infinite-dimensional"  spaces was solved by P.S. Aleksandrov and Yu.M. Smirnov, who introduced the classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850102.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850104.png" />-weakly infinite-dimensional and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850106.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850108.png" />-strongly infinite-dimensional normal spaces (cf. [[Weakly infinite-dimensional space]]). Any finite-dimensional space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850109.png" />-weakly infinite-dimensional, while any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850110.png" />-weakly infinite-dimensional space is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850111.png" />-weakly infinite-dimensional. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850112.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850113.png" />-weakly infinite-dimensional, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850114.png" />-strongly infinite-dimensional.
+
All classes of infinite-dimensional spaces considered so far are  "not very infinite-dimensional"  as compared with, for example, the Hilbert cube. The problem of distinguishing  "not very infinite-dimensional"  from  "very infinite-dimensional"  spaces was solved by P.S. Aleksandrov and Yu.M. Smirnov, who introduced the classes of $  A $-  
 +
and $  S $-
 +
weakly infinite-dimensional and of $  A $-  
 +
and $  S $-
 +
strongly infinite-dimensional normal spaces (cf. [[Weakly infinite-dimensional space]]). Any finite-dimensional space is $  S $-
 +
weakly infinite-dimensional, while any $  S $-
 +
weakly infinite-dimensional space is also $  A $-
 +
weakly infinite-dimensional. The space $  \cup I  ^ {n} $
 +
is $  A $-
 +
weakly infinite-dimensional, but $  S $-
 +
strongly infinite-dimensional.
  
In the case of compacta the definitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850115.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850116.png" />-weak (strong) infinite dimensionality are equivalent, and for this reason <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850117.png" />-weakly (strongly) infinite-dimensional compacta are simply called strongly (weakly) infinite-dimensional. The Hilbert cube is strongly infinite-dimensional. There exist infinite-dimensional and weakly infinite-dimensional compacta.
+
In the case of compacta the definitions of $  A $-  
 +
and $  S $-
 +
weak (strong) infinite dimensionality are equivalent, and for this reason $  A $-
 +
weakly (strongly) infinite-dimensional compacta are simply called strongly (weakly) infinite-dimensional. The Hilbert cube is strongly infinite-dimensional. There exist infinite-dimensional and weakly infinite-dimensional compacta.
  
A closed subspace of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850118.png" />- (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850119.png" />-) weakly infinite-dimensional space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850120.png" />- (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850121.png" />-) weakly infinite-dimensional. A normal space which is the sum of a finite number of its closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850122.png" />-weakly infinite-dimensional sets, is itself <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850123.png" />-weakly infinite-dimensional. A paracompactum which is the sum of a finite or countable system of its closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850124.png" />-weakly infinite-dimensional sets is itself <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850125.png" />-weakly infinite-dimensional. A hereditarily-normal space which is the sum of a finite or countable system of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850126.png" />-weakly infinite-dimensional sets is itself <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850127.png" />-weakly infinite-dimensional.
+
A closed subspace of an $  A $-  
 +
( $  S $-)  
 +
weakly infinite-dimensional space is $  A $-  
 +
( $  S $-)  
 +
weakly infinite-dimensional. A normal space which is the sum of a finite number of its closed $  S $-
 +
weakly infinite-dimensional sets, is itself $  S $-
 +
weakly infinite-dimensional. A paracompactum which is the sum of a finite or countable system of its closed $  A $-
 +
weakly infinite-dimensional sets is itself $  A $-
 +
weakly infinite-dimensional. A hereditarily-normal space which is the sum of a finite or countable system of its $  A $-
 +
weakly infinite-dimensional sets is itself $  A $-
 +
weakly infinite-dimensional.
  
A weakly countable-dimensional paracompactum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850128.png" />-weakly infinite-dimensional. A hereditarily-normal countable-dimensional space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850129.png" />-weakly infinite-dimensional. A weakly infinite-dimensional, not countable-dimensional metric compactum has been constructed by R. Pol [[#References|[3]]].
+
A weakly countable-dimensional paracompactum is $  A $-
 +
weakly infinite-dimensional. A hereditarily-normal countable-dimensional space is $  A $-
 +
weakly infinite-dimensional. A weakly infinite-dimensional, not countable-dimensional metric compactum has been constructed by R. Pol [[#References|[3]]].
  
The study of arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850130.png" />-weakly infinite-dimensional metrizable spaces is reduced to the compact case by the following: A metrizable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850131.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850132.png" />-weakly infinite-dimensional if and only if it can be represented as a sum of a weakly infinite-dimensional compactum and finite-dimensional open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850134.png" /> such that for any discrete sequence of points
+
The study of arbitrary $  S $-
 +
weakly infinite-dimensional metrizable spaces is reduced to the compact case by the following: A metrizable space $  R $
 +
is $  S $-
 +
weakly infinite-dimensional if and only if it can be represented as a sum of a weakly infinite-dimensional compactum and finite-dimensional open sets $  O _ {n} $,
 +
$  n = 1, 2 \dots $
 +
such that for any discrete sequence of points
  
<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/i/i050/i050850/i050850135.png" /></td> </tr></table>
+
$$
 +
x _ {i}  \in  R,\ \
 +
i = 1, 2 \dots
 +
$$
  
there exists a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850136.png" /> (depending on the sequence) containing all the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850137.png" />, beginning with some such point.
+
there exists a set $  O _ {n} $(
 +
depending on the sequence) containing all the points $  x _ {i} $,  
 +
beginning with some such point.
  
The following theorems provide another way of studying infinite-dimensional compacta instead of arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850138.png" />-weakly infinite-dimensional spaces: The maximal compactification of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850139.png" />-weakly infinite-dimensional space is weakly infinite-dimensional; any normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850140.png" />-weakly infinite-dimensional space of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850141.png" /> has a weakly infinite-dimensional compactification of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850142.png" />. All compactifications of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850143.png" />-weakly infinite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850144.png" /> are strongly infinite-dimensional.
+
The following theorems provide another way of studying infinite-dimensional compacta instead of arbitrary $  S $-
 +
weakly infinite-dimensional spaces: The maximal compactification of an $  S $-
 +
weakly infinite-dimensional space is weakly infinite-dimensional; any normal $  S $-
 +
weakly infinite-dimensional space of weight $  \tau $
 +
has a weakly infinite-dimensional compactification of weight $  \tau $.  
 +
All compactifications of the $  A $-
 +
weakly infinite-dimensional space $  I  ^  \omega  $
 +
are strongly infinite-dimensional.
  
A compactum is strongly infinite-dimensional if and only if there exists a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850145.png" /> such that for any set
+
A compactum is strongly infinite-dimensional if and only if there exists a continuous mapping $  f: X \rightarrow I  ^  \infty  $
 +
such that for any set
  
<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/i/i050/i050850/i050850146.png" /></td> </tr></table>
+
$$
 +
I  ^ {n}  = \{ {y = ( y _ {i} ) \in I  ^  \infty  } : {
 +
y _ {i} = 0, i > n } \}
 +
$$
  
(which is homeomorphic to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850147.png" />-dimensional cube) the restriction of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850148.png" /> to the inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850149.png" /> is an [[Essential mapping|essential mapping]].
+
(which is homeomorphic to an $  n $-
 +
dimensional cube) the restriction of the mapping $  f $
 +
to the inverse image $  f ^ { - 1 } I  ^ {n} $
 +
is an [[Essential mapping|essential mapping]].
  
 
There exists an infinite-dimensional metric compactum any non-empty subspace of which is either zero-dimensional or infinite-dimensional. Moreover, any strongly infinite-dimensional metric compactum contains a subcompactum any non-empty subspace of which is either zero-dimensional or infinite-dimensional. Any strongly infinite-dimensional compactum contains an infinite-dimensional Cantor manifold.
 
There exists an infinite-dimensional metric compactum any non-empty subspace of which is either zero-dimensional or infinite-dimensional. Moreover, any strongly infinite-dimensional metric compactum contains a subcompactum any non-empty subspace of which is either zero-dimensional or infinite-dimensional. Any strongly infinite-dimensional compactum contains an infinite-dimensional Cantor manifold.
  
All separable Banach spaces are mutually homeomorphic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050850/i050850150.png" />-strongly infinite-dimensional and homeomorphic to the product of a countable system of straight lines.
+
All separable Banach spaces are mutually homeomorphic, $  A $-
 +
strongly infinite-dimensional and homeomorphic to the product of a countable system of straight lines.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  B.A. Pasynkov,  "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Engelking,  "Transfinite dimension"  G.M. Reed (ed.) , ''Surveys in general topology'' , Acad. Press  (1980)  pp. 131–161</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Pol,  "A weakly infinite-dimensional compactum which is not countable dimensional"  ''Proc. Amer. Math. Soc.'' , '''82'''  (1981)  pp. 634–636</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  B.A. Pasynkov,  "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Engelking,  "Transfinite dimension"  G.M. Reed (ed.) , ''Surveys in general topology'' , Acad. Press  (1980)  pp. 131–161</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Pol,  "A weakly infinite-dimensional compactum which is not countable dimensional"  ''Proc. Amer. Math. Soc.'' , '''82'''  (1981)  pp. 634–636</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 22:12, 5 June 2020


A normal $ T _ {1} $- space $ X $( cf. Normal space) such that for no $ n = - 1, 0, 1 \dots $ the inequality $ \mathop{\rm dim} X \leq n $ is satisfied, i.e. $ X \neq \emptyset $ and for any $ n = 0, 1 \dots $ it is possible to find a finite open covering $ \omega _ {n} $ of $ X $ such that every finite covering refining $ \omega _ {n} $ has multiplicity $ > n + 1 $. Examples of infinite-dimensional spaces are the Hilbert cube $ I ^ \infty $ and the Tikhonov cube $ I ^ \tau $. Most of the spaces encountered in functional analysis are also infinite-dimensional.

A normal $ T _ {1} $- space $ X $ is said to be infinite-dimensional in the sense of the large (small) inductive dimension if the inequality $ \mathop{\rm Ind} X \leq n $( $ \mathop{\rm ind} X \leq n $) is invalid for every $ n = - 1, 0, 1 ,\dots $. If $ X $ is an infinite-dimensional space, it is infinite-dimensional in the sense of the large inductive dimension. If in addition $ X $ is compact, it is also infinite-dimensional in the sense of the small inductive dimension. The infinite dimensionality of a metric space is equivalent with its infinite dimensionality in the sense of the large inductive dimension. There exist finite-dimensional compacta that are infinite-dimensional in the sense of the small (and hence also in the sense of the large) inductive dimension. It is not known (1986) whether or not a compactum (or a metric space) that is finite-dimensional in the sense of the small inductive dimension and infinite-dimensional in the sense of the large inductive dimension exists.

One of the most natural approaches to the study of infinite-dimensional spaces is to introduce the small transfinite dimension $ \mathop{\rm ind} X $ and the large transfinite dimension $ \mathop{\rm Ind} X $. This approach consists in the extension of the definition of small and large inductive dimensions to infinite ordinal numbers. The transfinite dimensions $ \mathop{\rm ind} X $ and $ \mathop{\rm Ind} X $ are not defined for all infinite-dimensional spaces. Thus, neither is defined for the Hilbert cube. The large transfinite dimension is not defined for the space $ \cup I ^ {n} $, which is the discrete sum of the $ n $- dimensional cubes $ I ^ {n} $, $ n = 0, 1 \dots $ but $ \mathop{\rm ind} \cup I ^ {n} = \omega _ {0} $.

If the transfinite dimension $ { \mathop{\rm ind} } X $( $ { \mathop{\rm Ind} } X $) is defined for a normal space $ X $, then it is equal to an ordinal number whose cardinality does not exceed the weight $ wX $( respectively, the large weight $ WX $) of $ X $. In particular, if $ X $ has a countable base, then $ { \mathop{\rm ind} } X < \omega _ {1} $, and if $ X $ is compact, then $ { \mathop{\rm Ind} } X < \omega _ {1} $ as well. For metric spaces, too, $ { \mathop{\rm Ind} } X < \omega _ {1} $. If $ \alpha < \omega _ {1} $, then there exist compacta $ S _ \alpha $ and $ L _ \alpha $ for which $ { \mathop{\rm Ind} } S _ \alpha = \alpha $, $ \mathop{\rm ind} L _ \alpha = \alpha $. For any ordinal number $ \alpha $ there exists a metric space $ X _ \alpha $ with $ \mathop{\rm ind} X _ \alpha = \alpha $.

If the transfinite dimension $ { \mathop{\rm Ind} } X $ is defined, the transfinite dimension $ { \mathop{\rm ind} } X $ is defined as well, and $ \mathop{\rm ind} X \leq \mathop{\rm Ind} X $. Metric compacta for which the transfinite dimension $ { \mathop{\rm Ind} } X $ is defined and for which $ \omega _ {0} < { \mathop{\rm ind} } X < { \mathop{\rm Ind} } X $, have also been constructed.

If the transfinite dimension $ { \mathop{\rm ind} } X $( $ { \mathop{\rm Ind} } X $) of a space $ X $ is defined, then also the transfinite dimension $ { \mathop{\rm Ind} } A $( $ { \mathop{\rm Ind} } A $) is defined for any (respectively, any closed) set $ A \subseteq X $, and the inequality $ { \mathop{\rm ind} } A \leq { \mathop{\rm ind} } X $( or $ { \mathop{\rm Ind} } A \leq \mathop{\rm Ind} X $) is valid.

For the maximal compactification $ \beta X $ of a normal space $ X $ the equality $ { \mathop{\rm Ind} } \beta X = { \mathop{\rm Ind} } X $ is valid. A normal space of weight $ \tau $ and of transfinite dimension $ { \mathop{\rm Ind} } X \leq \alpha $ has a compactification $ bX $ of weight $ \tau $ and dimension $ { \mathop{\rm Ind} } bX \leq \alpha $. There exists a space $ L $ with a countable base having dimension $ { \mathop{\rm ind} } L = \omega _ {0} $ for which no compactification $ bX $ with a countable base has dimension $ { \mathop{\rm ind} } bX = \omega _ {0} $. A metrizable space $ R $ of transfinite dimension $ { \mathop{\rm Ind} } R = \alpha $ has a metric such that the completion $ cR $ with respect to it has dimension $ { \mathop{\rm Ind} } cR = \alpha $. A metrizable space $ R $ of transfinite dimension $ { \mathop{\rm ind} } R = \alpha $ with a countable base has a metric such that the completion $ cR $ with respect to it has dimension $ { \mathop{\rm ind} } cR = \alpha $.

The class of spaces for which a large or a small transfinite dimension is defined is closely connected with the class of metric countable-dimensional spaces; if a complete metric space is countable-dimensional, then the small transfinite dimension is defined for it; if the small transfinite dimension is defined for a metric space with a countable base, the space is countable-dimensional; if for a metric space the large transfinite dimension is defined (in particular if the space is finite-dimensional), then the space is countable-dimensional; the large transfinite dimension is defined for a countable-dimensional metric compactum. The space $ \cup I ^ {n} $ is countable-dimensional and is infinite-dimensional. The Hilbert cube is not countable-dimensional.

Countable dimensionality of a metric space $ R $ is equivalent to any one of the following properties: a) there exists a finite-to-one (but, in general, not a $ k $- to-one for any $ k = 1, 2 ,\ . . $) continuous closed mapping of a zero-dimensional metric space onto $ R $; b) there exists a countable-to-one continuous closed mapping of a zero-dimensional metric space onto $ R $; and c) $ R $ is a countably zero-dimensional space.

Theorems about the representability of any $ n $- dimensional metric space as a sum of $ n + 1 $ zero-dimensional subsets or as the image of a zero-dimensional metric space under a continuous closed $ ( n + 1) $- to-one mapping indicate that it is natural to consider the class of countable-dimensional (metric) spaces and that it is close to the class of finite-dimensional spaces. As in the finite-dimensional case, there exists a countable-dimensional space which is universal in the sense of homeomorphic imbedding in the class of countable-dimensional metric spaces of weight $ \leq \tau $.

If a normal space is represented as a finite or a countable sum of its countable-dimensional subspaces, then it is countable-dimensional. A subspace of a countable-dimensional perfectly-normal space is countable-dimensional.

The following theorem describes the relationships between countable and non-countable dimensional metric spaces: If a mapping $ f: R \rightarrow S $ between metric spaces $ R $ and $ S $ is continuous and closed, if the space $ R $ is countable-dimensional and the space $ S $ is non-countable dimensional, then the set $ \{ {y \in S } : {| f ^ { - 1 } y | \geq c } \} $ is also non-countable dimensional.

In addition to countable-dimensional spaces, a natural extension of the class of finite-dimensional spaces is the class of weakly countable-dimensional spaces. If one considers metrizable spaces only, weakly countable-dimensional spaces occupy a place which is intermediate between finite-dimensional and countable-dimensional spaces. There exist countable-dimensional metric compacta that are not weakly countable-dimensional, while the space $ \cup I ^ {n} $ is both weakly countable-dimensional and infinite-dimensional. A closed subspace of a weakly countable-dimensional space is weakly countable-dimensional. A normal space is weakly countable-dimensional if it is representable as a finite or a countable sum of its weakly countable-dimensional closed subsets.

In the classes of normal weakly countable-dimensional and metric weakly countable-dimensional spaces there exist universal (in the sense of homeomorphic imbedding) spaces. In the case of spaces with a countable base, an example is the subspace $ I ^ \omega $ of the Hilbert cube which consists of all points with only a finite number of non-zero coordinates. The space $ I ^ \omega $ has no weakly countable-dimensional compactifications.

All classes of infinite-dimensional spaces considered so far are "not very infinite-dimensional" as compared with, for example, the Hilbert cube. The problem of distinguishing "not very infinite-dimensional" from "very infinite-dimensional" spaces was solved by P.S. Aleksandrov and Yu.M. Smirnov, who introduced the classes of $ A $- and $ S $- weakly infinite-dimensional and of $ A $- and $ S $- strongly infinite-dimensional normal spaces (cf. Weakly infinite-dimensional space). Any finite-dimensional space is $ S $- weakly infinite-dimensional, while any $ S $- weakly infinite-dimensional space is also $ A $- weakly infinite-dimensional. The space $ \cup I ^ {n} $ is $ A $- weakly infinite-dimensional, but $ S $- strongly infinite-dimensional.

In the case of compacta the definitions of $ A $- and $ S $- weak (strong) infinite dimensionality are equivalent, and for this reason $ A $- weakly (strongly) infinite-dimensional compacta are simply called strongly (weakly) infinite-dimensional. The Hilbert cube is strongly infinite-dimensional. There exist infinite-dimensional and weakly infinite-dimensional compacta.

A closed subspace of an $ A $- ( $ S $-) weakly infinite-dimensional space is $ A $- ( $ S $-) weakly infinite-dimensional. A normal space which is the sum of a finite number of its closed $ S $- weakly infinite-dimensional sets, is itself $ S $- weakly infinite-dimensional. A paracompactum which is the sum of a finite or countable system of its closed $ A $- weakly infinite-dimensional sets is itself $ A $- weakly infinite-dimensional. A hereditarily-normal space which is the sum of a finite or countable system of its $ A $- weakly infinite-dimensional sets is itself $ A $- weakly infinite-dimensional.

A weakly countable-dimensional paracompactum is $ A $- weakly infinite-dimensional. A hereditarily-normal countable-dimensional space is $ A $- weakly infinite-dimensional. A weakly infinite-dimensional, not countable-dimensional metric compactum has been constructed by R. Pol [3].

The study of arbitrary $ S $- weakly infinite-dimensional metrizable spaces is reduced to the compact case by the following: A metrizable space $ R $ is $ S $- weakly infinite-dimensional if and only if it can be represented as a sum of a weakly infinite-dimensional compactum and finite-dimensional open sets $ O _ {n} $, $ n = 1, 2 \dots $ such that for any discrete sequence of points

$$ x _ {i} \in R,\ \ i = 1, 2 \dots $$

there exists a set $ O _ {n} $( depending on the sequence) containing all the points $ x _ {i} $, beginning with some such point.

The following theorems provide another way of studying infinite-dimensional compacta instead of arbitrary $ S $- weakly infinite-dimensional spaces: The maximal compactification of an $ S $- weakly infinite-dimensional space is weakly infinite-dimensional; any normal $ S $- weakly infinite-dimensional space of weight $ \tau $ has a weakly infinite-dimensional compactification of weight $ \tau $. All compactifications of the $ A $- weakly infinite-dimensional space $ I ^ \omega $ are strongly infinite-dimensional.

A compactum is strongly infinite-dimensional if and only if there exists a continuous mapping $ f: X \rightarrow I ^ \infty $ such that for any set

$$ I ^ {n} = \{ {y = ( y _ {i} ) \in I ^ \infty } : { y _ {i} = 0, i > n } \} $$

(which is homeomorphic to an $ n $- dimensional cube) the restriction of the mapping $ f $ to the inverse image $ f ^ { - 1 } I ^ {n} $ is an essential mapping.

There exists an infinite-dimensional metric compactum any non-empty subspace of which is either zero-dimensional or infinite-dimensional. Moreover, any strongly infinite-dimensional metric compactum contains a subcompactum any non-empty subspace of which is either zero-dimensional or infinite-dimensional. Any strongly infinite-dimensional compactum contains an infinite-dimensional Cantor manifold.

All separable Banach spaces are mutually homeomorphic, $ A $- strongly infinite-dimensional and homeomorphic to the product of a countable system of straight lines.

References

[1] P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)
[2] R. Engelking, "Transfinite dimension" G.M. Reed (ed.) , Surveys in general topology , Acad. Press (1980) pp. 131–161
[3] R. Pol, "A weakly infinite-dimensional compactum which is not countable dimensional" Proc. Amer. Math. Soc. , 82 (1981) pp. 634–636

Comments

A space is called a countable-dimensional space if it can be written as the union of a countable family of finite-dimensional subsets, see also Countably zero-dimensional space.

References

[a1] R. Engelking, "Dimension theory" , North-Holland & PWN (1978)
[a2] R. Engelking, E. Pol, "Countable dimensional spaces: a survey" Diss. Math. , 216 (1983) pp. 1–41
[a3] L.A. Ljuksemburg, "On compact metric spaces with non-coinciding transfinite dimensions" Pac. J. Math. , 93 (1981) pp. 339–386
[a4] C. Bessaga, A. Pelczyński, "Selected topics in infinite-dimensional topology" , PWN (1975)
How to Cite This Entry:
Infinite-dimensional space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinite-dimensional_space&oldid=47341
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article