UDC 513.831

MATHEMATICS

V. FEDORCHUK

ON BICOMPACTA WITH NONCOINCIDING DIMENSIONS

(Presented by Academician P. S. Aleksandrov on 29 January 1968)

In 1935 P. S. Aleksandrov raised the question whether, for bicompacta, the dimension defined by means of coverings ((\dim)) and the small inductive dimension ((\operatorname{ind})) coincide. A. G. Lund ((^2)), and then O. V. Lokutsievskii ((^1)), constructed bicompacta that are one-dimensional in the sense of (\dim) and two-dimensional in the sense of (\operatorname{ind}). At the same time, the bicompactum (S) of O. V. Lokutsievskii was considerably simpler than Lund’s bicompactum; moreover, the bicompactum (S) decomposed into the sum of two such bicompacta (S_1) and (S_2) that (\operatorname{ind} S_1 = \operatorname{ind} S_2 = 1). Thus, O. V. Lokutsievskii’s example also showed that the sum theorem for the small inductive dimension does not hold in bicompacta. With the aid of a sufficiently general construction, P. Vopenka ((^3)) constructed, for arbitrary natural numbers (m) and (n) ((1 \le m < n)), such bicompacta (X_m^n) that (\dim X_m^n = m), while (\operatorname{ind} X_m^n = n). All these bicompacta did not satisfy the first axiom of countability, and the dimensions there differed only on a nowhere dense set (namely, on the set of those points which do not have a countable fundamental system of neighborhoods).

Below we shall construct a separable bicompactum (B) with the first axiom of countability such that (\dim B = 2) and (3 \le \operatorname{ind}_b B \le 4) for every point (b \in B). One sufficiently general method for constructing examples of this kind will be indicated.

§ 1. Construction of the space (B). Denote by (J) the half-interval ((0, 1]) of the number line. Denote by (I_n) the segment ([1/(n+1), 1/n]) of the number line. Then

[
J = \bigcup_{n=1}^{\infty} I_n.
]

Let (g : J \to T^2) be such a continuous mapping of the half-interval (J) onto the torus (T^2) (instead of the torus one may take any two-dimensional connected and locally connected compactum) that each segment (I_n) is mapped by (g) onto the whole torus (T^2). Such a mapping (g) is easy to construct, starting, for example, from the standard Peano mapping of a segment onto a square.

Let now (S^2) be the two-dimensional sphere of diameter (1) (it may be regarded as a subset of three-dimensional Euclidean space), and let (x) be some point of the sphere (S^2). Define a mapping (h_x : S^2 \setminus {x} \to T^2) by the equality

[
h_x(x') = g(\rho(x, x')).
]

Here (\rho(x, x')) denotes the distance from the point (x) to the point (x'). The function (\rho) continuously maps the sphere with the point (x) removed, (S^2 \setminus {x}), onto the half-interval (J), and the mapping (h_x) is continuous as the superposition of the two continuous mappings (\rho) and (g).

The fundamental property of the mapping (h_x). For every nonempty open set (W \subset T^2) and every neighborhood (U) of the point (x), the set (U \cap h_x^{-1} W) contains a closed set (F) which separates the sphere (S^2) into such open sets (D) and (E) that (x \in D \subset U).

Let (B = S^2 \times T^2). We shall denote points of the set (B) by pairs ((x, y)), where (x \in S^2), (y \in T^2). Define on the set (B) a topology (\tau), relative to which (B) will be the desired space. A base of the topology (\tau) is formed by all sets of the form (O(U, x, W)), where (U) is an open subset of the sphere (S^2), (x \in U), (W) is an open subset of the torus (T^2), and

[
O(U,x,W)=({x}\times W)\cup{(U\cap h_x^{-1}W)\times T^2}=W^x\cup U(W^x).
]

It is easy to verify that the family ({O(U,x,W)}) forms a base for some topology.

Obviously, the topology (\tau) is different from the product topology of (S^2) on (T^2). At the same time the topology (\tau) induces on each layer ({x}\times T^2) the ordinary topology of the torus (T^2).

§ 2. Basic properties of the space (B).

Theorem 1. The space (B) is a separable bicompactum with the first axiom of countability such that (\dim B=2) and (3\le \operatorname{ind}_b B\le 4) for every point (b\in B).

Outline of the proof. 1. It is easy to verify that the space (B) is a separable bicompactum with the first axiom of countability. Moreover, the projection (\pi:B\to S^2) of the product (B=S^2\times T^2) onto the first factor is a continuous and irreducible mapping.

1. Let (b) be an arbitrary point of the bicompactum (B). The inequality (3\le \operatorname{ind}_b B\le 4) follows from the following assertion.

Lemma. Let (G) be an open subset of the bicompactum (B) such that (B\setminus [G]\ne\varnothing). Then there exists a point (x\in S^2) such that ({x}\times T^2=\pi^{-1}(x)\subset \operatorname{Fr}G).

Proof. Denote by (K) the small image of the set (G) under the mapping (\pi):

[
K=\pi^#G={x\in S^2\mid \pi^{-1}(x)\subset G},
]

[
L=\pi^#(B\setminus [G]), \qquad M=S^2\setminus K\cup L.
]

The sets (K) and (L), by virtue of the irreducibility of the mapping (\pi), are disjoint nonempty open subsets of the sphere (S^2). Hence the set (M) separates (S^2), and (\dim M=\operatorname{ind}M\ge 1). Let (x) be an arbitrary point of the set (M) for which (\operatorname{ind}_x M\ge 1). Using the basic property of the mapping (h_x), it is easy to show that (\pi^{-1}(x)\subset \operatorname{Fr}G).

1. The proof of the inequality (\operatorname{ind}_t B\le 4) reduces to the sufficiently cumbersome verification that the small inductive dimension of the boundary of any basic set (O(U,x,W)) is at most 3. Whether in fact the small inductive dimension of the bicompactum (B) is equal to 3 or to 4 has not been established.

2. (\dim B=2). Since the space (B) contains, as a closed subset, a space homeomorphic to the two-dimensional torus, (\dim B\ge 2). Let (\omega) be an arbitrary cover of the bicompactum (B). Since the mapping (\pi:B\to S^2) is closed, there exists a finite set of points (x_1,\ldots,x_l) of the sphere (S^2) such that the preimage (\pi^{-1}(x)) of every point (x\ne x_j), (j=1,\ldots,l), is contained in some element of the cover (\omega) together with the indicated neighborhood (\pi^{-1}(U_x)). For each layer ({x_j}\times T^2=\pi^{-1}(x_j)), (j=1,\ldots,l), one can choose a system (\beta_j), consisting of basic sets of the space (B) and inscribed in the cover (\omega), such that the multiplicity of the system (\beta_j\le 3) and the body of the system (\beta_j) coincides with the set (U_j\times T^2=\pi^{-1}U_j), where (U_j) is a neighborhood of the point (x_j). This can be done so that the sets (U_j) are pairwise disjoint and the multiplicity of the system

[
\beta=\bigcup_{j=1}^{l}\beta_j
]

in some neighborhood (G) of the set (B\setminus \bigcup_{j=1}^{l}\pi^{-1}U_j) does not exceed one. There exists a system (\alpha={A_1,\ldots,A_m}) of multiplicity (\le 3), consisting of open subsets of the sphere (S^2) and covering the compactum (S^2\setminus \bigcup_{j=1}^{l}U_j), such that the system

[
\pi^{-1}(\alpha)={\pi^{-1}A_s\mid s=1,\ldots,m}
]

is inscribed in the cover (\omega) and has multiplicity (\le 2) on the complement of the set (B\setminus \bigcup_{j=1}^{l}\pi^{-1}U_j). Moreover, one may assume that the body of the system (\pi^{-1}(\alpha)) is contained in the set (G), where the multiplicity of the system (\beta) is not greater than one. Then the system (\beta\cup \pi^{-1}(\alpha)) will be a cover of the bicompactum (B), inscribed in the cover (\omega) and having multiplicity (\le 3).

The bicompactum (B) is not perfectly normal. It contains a subset of cardinality continuum which is discrete in the induced topology. The question remains open as to whether the dimensions (\dim) and (\operatorname{ind}) coincide for perfectly normal bicompacta.

§ 3. General construction. Let (X) be an arbitrary bicompactum, and let to each point (x \in X) there be assigned a certain bicompactum (Y_x). Suppose that for each point (x \in X) a continuous mapping
(h_x: X \setminus {x} \to Y_x) is defined. Then the system ({X, Y_x, h_x \mid x \in X}) defines in the following way a bicompactum (B), continuously mapped onto the bicompactum (X) by means of a projection (\pi: B \to X) such that (\pi^{-1}(x) = Y_x) for all (x \in X): the bicompactum (B) is defined on the set (\bigcup_{x \in X} Y_x); a base for the topology is formed by all possible sets of the form

[
O(U, x, W) = W \cup \pi^{-1}(U \cap h_x^{-1}W),
]

where (W) is an open subset of the bicompactum (Y_x), and (U) is a neighborhood of the point (x) in the bicompactum (X). We shall denote the bicompactum (B) by (B{X, Y_x, h_x}). In what follows we shall always assume that the functions (h_x) satisfy the following property: for every nonempty open set (W \subset Y_x) and every neighborhood (U) of the point (x), the set (U \cap h_x^{-1}W) contains a closed set (F) which partitions the space (X) into such open sets (D) and (E) that (x \in D \subset U).

Theorem 2. If (\operatorname{Ind} X \le n) and (\dim Y_x \le n) for all (x \in X), then
[
\dim B{X, Y_x, h_x} \le n .
]

Theorem 3. If (\dim X \ge n) and (\operatorname{ind} Y_x \ge n) for all (x \in X), then
[
\operatorname{ind} B{X, Y_x, h_x} \ge 2n - 1 .
]

Theorem 4. If the bicompactum (X) is hereditarily normal, (\operatorname{Ind} X \le n), and (\operatorname{ind} Y_x \le n) for all (x \in X), then
[
\operatorname{ind} B{X, Y_x, h_x} \le 2n .
]

Corollary. If the bicompactum (X) is hereditarily normal, (\dim X = \operatorname{Ind} X = n), and (\operatorname{ind} Y_x = n) for all (x \in X), then
[
\dim B \le n
\quad\text{and}\quad
2n - 1 \le \operatorname{ind} B \le 2n .
]

Remark. There exist families ({X, Y_x, h_x}) satisfying all the conditions of this paragraph. Indeed, one such family is the following: the space (X) coincides with the (n)-dimensional sphere (S^n), the spaces (Y_x) for all (x \in X) coincide with the (n)-dimensional torus (T^n) (instead of (T^n) one may take an arbitrary (n)-dimensional connected and locally connected compactum), and the mappings (h_x: S^n \setminus {x} \to T^n) are obtained from a mapping (g: J \to T^n) in the same way as, in the first paragraph, the mappings (h_x: S^2 \setminus {x} \to T^2) are obtained from the mapping (g: J \to T^2).

Mechanics and Mathematics Faculty
Moscow State University
named after M. V. Lomonosov

Received
25 I 1968

REFERENCES

1. O. V. Lokutsievskii, DAN, 67, No. 2, 217 (1949).
2. A. Lunts, DAN, 66, No. 5, 801 (1949).
3. P. Vopěnka, Czechoslov. Math. J., 8, No. 3, 319 (1958).