Infinite-dimensional space

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)