# 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