# Dimension

of a topological space $X$

An integral invariant $\mathop{\rm dim} X$ defined as follows. $\mathop{\rm dim} X = - 1$ if and only if $X = \emptyset$. A non-empty topological space $X$ is said to be at most $n$- dimensional, written as $\mathop{\rm dim} X \leq n$, if in any finite open covering of $X$ one can inscribe a finite open covering of $X$ of multiplicity $\leq n + 1$, $n = 0, 1 , . . .$. If $\mathop{\rm dim} X \leq n$ for some $n = - 1, 0, 1 \dots$ then $X$ is said to be finite-dimensional, written as $\mathop{\rm dim} X < \infty$, and one defines

$$\mathop{\rm dim} X = \min \{ {n } : { \mathop{\rm dim} X \leq n } \} .$$

Here if $\mathop{\rm dim} X = n$, then the space is called $n$- dimensional. The concept of the dimension of a topological space generalizes the elementary geometrical concept of the number of coordinates of a Euclidean space (and a polyhedron), since the dimension of an $n$- dimensional Euclidean space (and any $n$- dimensional polyhedron) is equal to $n$( the Lebesgue–Brouwer theorem).

The importance of the concept of the dimension of a topological space is revealed by the Nöbeling–Pontryagin–Hurewicz–Kuratowski theorem: An $n$- dimensional metrizable space with a countable base can be imbedded in the $( 2n + 1)$- dimensional Euclidean space. Thus, the class of spaces that are topologically equivalent to subspaces of all possible $n$- dimensional Euclidean spaces, $n = 1, 2 \dots$ coincides with the class of finite-dimensional metrizable spaces with a countable base.

The dimension $\mathop{\rm dim} X$ is sometimes called the Lebesgue dimension, since its definition arises from Lebesgue's theorem on tilings: An $n$- dimensional cube has, for any $\epsilon > 0$, a finite closed covering of multiplicity $\leq n + 1$ such that all elements have diameter $< \epsilon$; there exists an $\epsilon _ {0} > 0$ for which the multiplicity of any finite closed covering of an $n$- dimensional cube is $\geq n + 1$ if the diameters of the elements of this covering are $< \epsilon _ {0}$.

Another, inductive, approach (see Inductive dimension) to the definition of the dimension of a topological space is possible, based on the separation of the space by subspaces of smaller dimension. This approach to the concept of dimension originates from H. Poincaré, L.E.J. Brouwer, P.S. Urysohn, and K. Menger. In the case of metrizable spaces it is equivalent to Lebesgue's definition.

The foundations of dimension theory were laid in the first half of the twenties of the 20th century in papers of Urysohn and Menger. In the later thirties, the dimension theory of metrizable spaces with a countable base was constructed, and by the start of the sixties the dimension theory of arbitrary metrizable spaces was finished.

Below, all topological spaces under consideration are supposed to be normal and Hausdorff (cf. Hausdorff space; Normal space). In this case, in the definition of dimension one can without harm replace the open coverings to be inscribed by closed ones.

Lebesgue's approach to the definition of dimension (in contrast to the inductive approach) makes it possible to geometrize the concept of dimension for any space by comparing the original topological space with most simple geometrical formations — polyhedra (cf. Polyhedron). Roughly speaking, a space is $n$- dimensional if and only if it differs arbitrarily little from an $n$- dimensional polyhedron. More precisely, there is Aleksandrov's theorem on $\omega$- mappings: $\mathop{\rm dim} X \leq n$ if and only if for any finite open covering $\omega$ of $X$ there is an $\omega$- mapping from $X$ onto an at most $n$- dimensional, $n = 0, 1 \dots$( compact) polyhedron. This theorem can be particularly visualized for compacta: A compactum $X$ has $\mathop{\rm dim} X \leq n$ if and only if for any $\epsilon > 0$ there is an $\epsilon$- mapping from $X$ onto an at most $n$- dimensional polyhedron. If $X$ also lies in a Euclidean or Hilbert space, then the $\epsilon$- mapping can be replaced by an $\epsilon$- shift (Aleksandrov's theorem on $\epsilon$- mappings and $\epsilon$- shifts).

The following statement makes it possible to determine the dimension of a space by comparing it with all possible $n$- dimensional cubes: $\mathop{\rm dim} X \geq n$ if and only if the space has an essential mapping onto an $n$- dimensional cube, $n = 0, 1 , . . .$( Aleksandrov's theorem on essential mappings).

This theorem can be given the following form: $\mathop{\rm dim} X \leq n$ if and only if, for any set $A$ closed in $X$ and for any continuous mapping $f: A \rightarrow S ^ {n}$ into the $n$- dimensional sphere, there is a continuous extension $F: X \rightarrow S ^ {n}$, $n = 0, 1 \dots$ of $f$.

The following characterization of dimension indicates the role of this concept in problems of the existence of solutions to systems of equations: $\mathop{\rm dim} X \geq n$, $n = 1, 2 \dots$ if and only if $X$ has a system of disjoint pairs of closed sets $A _ {i}$, $B _ {i}$, $i = 1 \dots n$, such that for any functions $f _ {i}$ continuous on $X$ and satisfying the conditions $f _ {i} \mid _ {A _ {i} } > 0$, $f _ {i} \mid _ {B _ {i} } < 0$, $i = 1 \dots n$, there is a point $x \in X$ at which $f _ {i} ( x) = 0$, $i = 1 \dots n$( this is the Otto–Eilenberg–Hemmingsen theorem on partitions).

One of the most important properties of dimension is expressed by the Menger–Urysohn–Čech countable closed sum theorem: If the space $X$ is a finite or countable sum of closed subsets of dimension $\leq n$, then also $\mathop{\rm dim} X \leq n$, $n = 0, 1 , . . .$. In this theorem, the condition that the sum be finite or countable may be replaced by the condition of local finiteness. The statement for the large and small inductive dimensions analogous to this sum theorem already fails in the class of Hausdorff compacta. The following statements are among the fundamental general facts of dimension theory, and make it possible to reduce the consideration of arbitrary spaces to that of Hausdorff compacta. For any normal space

a) $\mathop{\rm dim} \beta X = \mathop{\rm dim} X$, $\mathop{\rm Ind} \beta X = \mathop{\rm Ind} X$, where $\beta X$ is the Stone–Čech compactification of $X$; at the same time, the inequality $\mathop{\rm ind} \beta X > \mathop{\rm ind} X$ is possible;

b) there exists a compactification $bX$ of $X$ with weight (cf. Weight of a topological space) $w ( bX)$ equal to the weight $wX$ and with dimension $\mathop{\rm dim} bX$ equal to the dimension $\mathop{\rm dim} X$; the analogous statement also holds for the large inductive dimension. The case of a countable weight of the space is especially interesting, since in this case the extension $bX$ is metrizable.

Statement b) can be strengthened: For any $n = 0, 1 \dots$ and any infinite cardinal number $\tau$ there is a Hausdorff compactum $\Pi _ \tau ^ {n}$ of weight $\tau$ and dimension $\mathop{\rm dim} \Pi _ \tau ^ {n} = n$ containing a homeomorphic image of every normal space $X$ of weight $\leq \tau$ and dimension $\mathop{\rm dim} X \leq n$( the theorem on the universal Hausdorff compactum of given weight and dimension). The analogous statement also holds for the large inductive dimension. Here for $\Pi _ {\aleph _ {0} } ^ {0}$ one can take the perfect Cantor set, and as $\Pi _ {\aleph _ {0} } ^ {1}$ the Menger universal curve.

It would seem that dimension should possess the monotonicity property: $\mathop{\rm dim} A \leq \mathop{\rm dim} X$ if $A \subset X$. This is so if a) the set $A$ is closed in $X$ or is strongly paracompact; or b) the space $X$ is metrizable (and even perfectly normal). However, already for a subset $A$ of a hereditarily normal space $X$ one may have $\mathop{\rm dim} A > \mathop{\rm dim} X$ and $\mathop{\rm Ind} A > \mathop{\rm Ind} X$. But always $\mathop{\rm ind} A \leq \mathop{\rm ind} X$ for $A \subset X$.

One of the main problems in dimension theory is the behaviour of dimension under continuous mappings. In the case of closed mappings (these also include all continuous mappings of Hausdorff compacta) the answer is given by the formulas of W. Hurewicz, which he originally obtained for the class of spaces with a countable base.

Hurewicz' formula for mappings raising the dimension: If a mapping $f: X \rightarrow Y$ is continuous and closed, then

$$\mathop{\rm mult} f + \mathop{\rm dim} X - 1 \geq \mathop{\rm dim} Y,$$

where $\mathop{\rm mult} f = \sup \{ {| f ^ { - 1 } y | } : {y \in Y } \}$ is the multiplicity of $f$.

Hurewicz' formula for mappings lowering the dimension: For a continuous closed mapping $f: X \rightarrow Y$ onto a paracompactum $Y$, the inequality

$$\tag{1 } \mathop{\rm dim} X - \mathop{\rm dim} f \leq \mathop{\rm dim} Y$$

holds, where

$$\mathop{\rm dim} f = \ \sup \{ { \mathop{\rm dim} f ^ { - 1 } y } : {y \in Y } \} .$$

For an arbitrary normal space $Y$ this formula is, in general, false.

In the case of continuous mappings of finite-dimensional compacta, it has been established that a continuous mapping $f$ of dimension $\mathop{\rm dim} f = k$ is a superposition of $k$ continuous mappings of dimension 1 (this is a precization of formula (1), and an analogue of the fact that a $k$- dimensional cube is the product of $k$ intervals).

In the case of open mappings one can show that the image of a zero-dimensional Hausdorff compactum is zero-dimensional and, at the same time, that the Hilbert cube is the image of a one-dimensional compactum, even if the corresponding mapping $f$ has dimension $\mathop{\rm dim} f$ equal to zero. However, in the case of an open mapping $f: X \rightarrow Y$ of Hausdorff compacta $X$ and $Y$ with multiplicity $\leq \aleph _ {0}$, the equality $\mathop{\rm dim} X = \mathop{\rm dim} Y$ holds.

The behaviour of dimension under topological products is described by the following assertions:

a) there exist finite-dimensional compacta $X$ and $Y$ for which $\mathop{\rm dim} X \times Y < \mathop{\rm dim} X + \mathop{\rm dim} Y$;

b) if one of the factors of the product $X \times Y$ is a Hausdorff compactum or metrizable, then $\mathop{\rm dim} X \times Y \leq \mathop{\rm dim} X + \mathop{\rm dim} Y$;

c) there exist normal spaces $X$ and $Y$ for which $\mathop{\rm dim} X \times Y > \mathop{\rm dim} X + \mathop{\rm dim} Y$.

In the case of Hausdorff compacta $X$ and $Y$ one always has $\mathop{\rm Ind} X \times Y < \infty$ if $\mathop{\rm Ind} X < \infty$ and $\mathop{\rm Ind} Y < \infty$, but one may have $\mathop{\rm Ind} X \times Y > \mathop{\rm Ind} X + \mathop{\rm Ind} Y$. If, however, the Hausdorff compacta $X$ and $Y$ are perfectly normal or one-dimensional, then $\mathop{\rm Ind} X \times Y \leq \mathop{\rm Ind} X + \mathop{\rm Ind} Y$.

Dimension theory is most meaningful, first, for the class of metric spaces with a countable base, and, secondly, for the class of all metric spaces. In the class of metric spaces with a countable base one has the Urysohn equalities

$$\tag{2 } \mathop{\rm dim} X = \ \mathop{\rm ind} X = \ \mathop{\rm Ind} X.$$

In the class of arbitrary metric spaces one has the Katětov equality

$$\tag{3 } \mathop{\rm dim} X = \mathop{\rm Ind} X ,$$

and $\mathop{\rm ind} X = 0 < \mathop{\rm Ind} X = 1$ is possible.

In the case of metric spaces the concept of an $n$- dimensional space can be reduced to the concept of a zero-dimensional space by the following two methods. For a metric space $X$, $\mathop{\rm dim} X \leq n$, $n = 0, 1 \dots$ if and only if

a) $X$ can be represented by at most $n + 1$ zero-dimensional summands; or

b) there exists a continuous closed mapping of multiplicity $\leq n + 1$ from a zero-dimensional metric space onto $X$.

For any subset $A$ of a metric space $X$ there is a subset $B \supset A$ of type $G _ \delta$ in $X$ for which $\mathop{\rm dim} B = \mathop{\rm dim} A$.

In the class of metric spaces of weight $\leq \tau$ and dimension $\leq n$ there exists a universal space (in the sense of imbedding). Dowker's theorem has played an important role in the dimension theory of metric (and more general) spaces: $\mathop{\rm dim} X \leq n$ if and only if in any locally finite open covering of $X$ one can inscribe an open covering of multiplicity $\leq n + 1$.

One of the most important problems in dimension theory is the problem of the relations between the Lebesgue dimension and the inductive dimensions. Although for an arbitrary space $X$ the values of the dimensions $\mathop{\rm dim} X$, $\mathop{\rm ind} X$, $\mathop{\rm Ind} X$ are, in general, pairwise distinct, for some classes of spaces that are in some sense close to metric spaces one has, e.g., the following:

a) if the space $X$ admits a continuous closed mapping $f$ of dimension $\mathop{\rm dim} f = 0$ onto a metric space, then (3) holds, whence follow the equalities (2) for locally compact Hausdorff groups and their quotient spaces;

b) if there exists a continuous closed mapping from a metric space onto $X$, then (2) holds.

One more general condition for equality (3) to hold for a paracompactum $X$ is as follows: $\mathop{\rm dim} X = n$ and $X$ is the image of a zero-dimensional space under a closed mapping of multiplicity $\leq n + 1$, $n = 0, 1 , . . .$.

In the case of an arbitrary space $X$ one always has the inequalities $\mathop{\rm dim} X \leq \mathop{\rm Ind} X$ and $\mathop{\rm ind} X \leq \mathop{\rm Ind} X$, while the equalities $\mathop{\rm dim} X = 0$ and $\mathop{\rm Ind} X = 0$ are equivalent. For a strongly paracompact (in particular, for a Hausdorff compact or Lindelöf compact) space $X$ one has the inequality $\mathop{\rm dim} X \leq \mathop{\rm ind} X$. For Hausdorff compacta the equalities $\mathop{\rm ind} X = 1$ and $\mathop{\rm Ind} X = 1$ are equivalent. There exist Hausdorff compacta satisfying the first axiom of countability (and even perfectly-normal Hausdorff compacta, if one assumes the continuum hypothesis), for which $\mathop{\rm dim} X = 1$ and $\mathop{\rm ind} X = n$, $n = 2, 3 , . . .$. An example of a topologically homogeneous Hausdorff compactum with $\mathop{\rm dim} X < \mathop{\rm ind} X$ has been constructed. For perfectly-normal Hausdorff compacta one always has $\mathop{\rm ind} X = \mathop{\rm Ind} X$. There exist Hausdorff compacta, satisfying even the first axiom of countability, for which $\mathop{\rm ind} X < \mathop{\rm Ind} X$. It is not known (1983) whether there exists an $m$ such that for every $n > m$ there is a Hausdorff compactum (a metric space) $X$ with $\mathop{\rm ind} X = m$, $\mathop{\rm Ind} X = n$.

In the case of non-metrizable spaces, the dimension may not only fail to be monotone, but it also has other pathological properties. For any $n = 2, 3 \dots$ an example of a Hausdorff compactum $\theta _ {n}$ in which any closed set has dimension either 0 or $n = \mathop{\rm dim} \theta _ {n}$ has been constructed. An analogous example for the inductive dimension is impossible. Also, for each $n = 1, 2 \dots$ an example of a Hausdorff compactum $\Phi _ {n}$ for which any closed set separating $\Phi _ {n}$ has dimension $n = \mathop{\rm dim} \Phi _ {n}$ has been constructed. Thus, the approach to the definition of dimension in the case of a non-metrizable space differs in principle from the inductive approach of Poincaré based on the separation of the space by spaces of a smaller number of coordinates. The Hausdorff compacta $\Phi _ {n}$ are directly related to the following statement: Any $n$- dimensional Hausdorff compactum contains an $n$- dimensional Cantor manifold.

A subset of an $n$- dimensional Euclidean space $E ^ {n}$ is $n$- dimensional if and only if it contains interior points with respect to $E ^ {n}$. A compactum has dimension $\leq n$ if and only if it has a mapping of dimension zero into $E ^ {n}$, hence, up to zero-dimensional mappings, $n$- dimensional compacta are indistinguishable from the bounded closed subsets of $E ^ {n}$ containing interior points (with respect to $E ^ {n}$).