GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 2, 1995, 201-210

TWO-DIMENSION-LIKE FUNCTIONS DEFINED ON THE CLASS OF ALL TYCHONOFF SPACES

I. TSERETELI

Abstract. Two-dimension-like functions are constructed on the class of all Tychonoff spaces. Several of their properties, analogous to those of the classical dimension functions, are established.

1. Introduction. All topological spaces discussed in this paper are assumed to be Tychonoff spaces.

As usual, Ind, ind, and dim denote the classical dimension functions (the large inductive, small inductive, and covering dimensions, respectively).

Let ℕ′ be the union of all integers ≥ −1 and of one element set consisting of a single formal symbol “+∞” provided with an essential order relation.

The set of all natural numbers is denoted by ℕ. Throughout the paper for any n ∈ ℕ, the symbol I^{n} denotes a standard

n-cube I^{n} ≡ [0; 1]^{n}, I^{0} ≡ {0}, and I^{−1} ≡ ∅ where ∅ stands for the empty set.

The family of all Tychonoff spaces is denoted by T. The class K of topological spaces is said to be permissible if K satisfies

the following conditions:

1) for any integer n ≥ −1 I^{n} ∈ K;

2) if X ∈ K and A ⊆ X, then A ∈ K;

3) if X_{1}, X_{2} ∈ K, then X_{1} × X_{2} ∈ K where X_{1} × X_{2} is the usual product of spaces.

The function d defined on a permissible class K of topological spaces with values in ℕ′ is called the generalized dimension-like function (GDF) if a) d∅ = −1 and b) dX = dY whenever X is homeomorphic to Y.

The GDF d defined on a permissible class K of topological spaces is said to be of the Tumarkin type if the following conditions T_{1}^{K}−T_{8}^{K} are satisfied:

T_{1}^{K}) for any integer n ≥ −1 dI^{n} = n; 201

1991 Mathematics Subject Classification. 54F45.

Key words and phrases. Dimension, dimension-like function.

1072-947X/95/0300-0201$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

T_{2}^{K}) if X ∈ K and A is a locally closed subspace of X (i.e., if A = F ∪ G, where F is closed and G is open in X), then dA ≤ dX;

T_{3}^{K}) if X ∈ K and X = ∪^{∞}_{i=1}A_{i} where for any i ∈ ℕ A_{i} is a closed subset

of the space X, then dX ≤ sup_{1≤i<+∞}{dA_{i}};

T_{4}^{K}) for any space X ∈ K there exists a Hausdorff compactification bX of the space X such that dbX ≤ dX;

T_{5}^{K}) for every pair of spaces X_{1}, X_{2} ∈ K at least one of which is nonempty we have d(X_{1} × X_{2}) ≤ dX_{1} + dX_{2};

T_{6}^{K}) if X ∈ K and X = A ∪ B, then dX ≤ dA + dB + 1;

T_{7}^{K}) if X ∈ K and there exists a nonnegative integer n such that dX ≤ n, then the space X can be represented as the union of n + 1 pairwise disjoint subsets X_{1}, X_{2}, …, X_{n+1} with dX_{i} ≤ 0 for any i = 1, 2, …, n + 1;

T_{8}^{K}) for any X ∈ K and an arbitrary subspace A of the space X there exists a G_{δ}-set H in X such that A ⊆ H ⊆ X and dH ≤ dA.

It is well-known fact that on the class of all separable metrizable spaces the classical dimension dim is a GDF of the Tumarkin type. On the other hand, as proved by L. Zambakhidze [1], there exists no GDF of the Tumarkin type on the class T. Moreover, there exists no GDF on T even satisfying the conditions T_{1}^{T}, T_{2}^{T}, T_{3}^{T}, T_{4}^{T}, T_{5}^{T} simultaneously [1]. Also, there is no GDF on T satisfying the conditions T_{1}^{T}, T_{2}^{T}, T_{8}^{T} simultaneously.

We say that a subcollection {T_{i_{1}}^{T} , …, T_{i_{k}}^{T} } (1 ≤ i_{1} < ⋯ < i_{k} ≤ 8, k = 1, …, 8) of the collection {T_{1}^{T} , …, T_{8}^{T} } is realized if there exists a GDF on T which satisfies all conditions {T_{i_{1}}^{T} , …, T_{i_{k}}^{T} } simultaneously.

Clearly, if a subcollection {T_{i_{1}}^{T} , …, T_{i_{k}}^{T} } (1 ≤ i_{1} < ⋯ < i_{k} ≤ 8, k = 1, …, 8) of the collection {T_{1}^{T} , …, T_{8}^{T} } is realized, then any subcollection of {T_{i_{1}}^{T} , …, T_{i_{k}}^{T} } is realized, too. Also, if {T_{i_{1}}^{T} , …, T_{i_{k}}^{T} } is not realized, then no subcollection of the collection {T_{1}^{T} , …, T_{8}^{T} } containing the given one is realized.

L. Zambakhidze has shown [1] that the collections {T_{2}^{T} , T_{3}^{T} , T_{4}^{T} , T_{5}^{T} , T_{6}^{T} , T_{7}^{T} , T_{8}^{T} }, {T_{1}^{T} , T_{2}^{T} , T_{4}^{T} , T_{5}^{T} , T_{7}^{T} }, {T_{1}^{T} , T_{2}^{T} , T_{3}^{T} , T_{5}^{T} , T_{6}^{T} }, {T_{1}^{T} , T_{2}^{T} , T_{3}^{T} , T_{6}^{T} , T_{7}^{T} } and {T_{1}^{T} , T_{2}^{T} , T_{5}^{T} , T_{6}^{T} , T_{7}^{T} } are realized.

In this paper we prove that collections {T_{1}^{T} , T_{3}^{T} , T_{4}^{T} , T_{5}^{T} , T_{6}^{T} , T_{7}^{T} , T_{8}^{T} } and {T_{1}^{T} , T_{2}^{T} , T_{3}^{T} , T_{5}^{T} , T_{7}^{T} } are realized. To this end we construct two GDFs d_{1} and d_{2} on T such that d_{1} satisfies the conditions T_{1}^{T} , T_{3}^{T} , T_{4}^{T} , T_{5}^{T} , T_{6}^{T} , T_{7}^{T}, T_{8}^{T} and d_{2} satisfies the conditions T_{1}^{T} , T_{2}^{T} , T_{3}^{T} , T_{5}^{T} , T_{7}^{T}. Moreover, the functions d_{1} and d_{2} are the extensions of the classical dimension function dim from the class of all separable metrizable spaces over the class T.

2. GDF d_{1}. Let X ∈ T. It is assumed that d_{1}X = dim X if X has a countable base and d_{1}X = 0 otherwise.

Observe that d_{1} is a GDF on T and also is the extension of the function dim from the class of all separable metrizable spaces over the class T.

Theorem 1. The GDF d_{1} satisfies the conditions T_{1}^{T} , T_{3}^{T} , T_{4}^{T} , T_{5}^{T} , T_{6}^{T} , T_{7}^{T} , T_{8}^{T}, that is to say, the subcollection {T_{1}^{T} , T_{3}^{T} , T_{4}^{T} , T_{5}^{T} , T_{6}^{T} , T_{7}^{T} , T_{8}^{T} } of the collection {T_{1}^{T} , …, T_{8}^{T} } is realized.

Proof. The function d_{1} obviously satisfies the conditions T_{1}^{T} , T_{3}^{T} , T_{4}^{T} , T_{5}^{T}, T_{6}^{T}, and T_{7}^{T}. Therefore it remains for us to prove that d_{1} satisfies the condition T_{8}^{T}.

Let X ∈ T and A ⊆ X. Assume in the first place that ωX ≤ ℵ_{0} (here and below ωX denotes the weight of the space X and ℵ_{0} stands for a countable cardinal number). Then by Tumarkin’s theorem [2, Ch. 6, §3, Theorem 14] there exists a G_{δ}-set H in X such that A ⊆ H ⊆ X and dim H ≤ dim A. Hence, keeping in mind that ωA ≤ ℵ_{0} and ωH ≤ ℵ_{0}, we have d_{1}H ≤ d_{1}A.

Now let ωX > ℵ_{0}. If A = ∅, it can be assumed that H = ∅. If A ≠ ∅, then assume that H = X.

3. GDF d_{2}. We begin by defining the function H constructed by Hayashi [3].

Definition 1 ([3]). A subset X′ of the space X ∈ T is called quasiclosed in X if there exists a finite family {F_{1}, …, F_{k}} of closed subsets of the space X such that X′ = F_{1} ± F_{2} ± ⋯ ± F_{k}, where + and − denote respectively the union and the difference of sets, and whenever ± is written one should take either + or −.

Clearly, every closed subset as well as every open subset of the space X is quasiclosed in X.

The function H is defined on the class T as follows:

Let X ∈ T. H(X) = −1 iff X = ∅; H(X) = 0 iff X ≠ ∅ and X = ∪^{∞}_{i=1}X_{i}

where for any i ∈ ℕ X_{i} is quasiclosed in X and ind X_{i} ≤ 0; H(X) ≤ n (n ∈ ℕ) iff X = X_{1} ∪ X_{2}, where H(X_{1}) ≤ n − 1 and H(X_{2}) ≤ 0; H(X) = n (n = 0, 1, 2, …) iff H(X) ≤ n and H(X) ≰ n − 1.

Finally, H(X) = ∞ iff the inequality H(X) ≤ n does not hold for any n = −1, 0, 1, 2, . . ..

Now we shall define the function d_{2}.

Let X ∈ T. d_{2}(X) = −1 iff X = ∅; d_{2}(X) ≤ n (n = 0, 1, 2, …) if

X = ∪^{∞}_{t=1}X_{t} where H(X_{t}) ≤ 0 for any t ∈ ℕ and ∪^{n+1}_{k=1}X_{t_{k}} = X for any

n + 1 pairwise disjoint natural numbers t_{1}, t_{2}, …, t_{n+1}; d_{2}(X) = n (n = 0, 1, 2, …) iff d_{2}(X) ≤ n and d_{2}(X) ≰ n − 1; d_{2}(X) = ∞ if d_{2}(X) ≰ n for any n = −1, 0, 1, 2, . . ..

Clearly, d_{2} is a GDF on the class T.

Lemma 1. We have d_{2}(X) = dim X for any X ∈ T with a countable base.

Proof. Let X ∈ T and ωX ≤ ℵ_{0}. Suppose that d_{2}(X) ≤ n. Then X =

∪^{∞}_{t=1}X_{t}, where H(X_{t}) ≤ 0 for any t ≥ 1 and ∪^{n+1}_{i=1}X_{t_{i}} = X for any pairwise

disjoint t_{1}, t_{2}, …, t_{n+1} ∈ ℕ. Since ω(X_{t}) ≤ ℵ_{0} for any t ∈ ℕ, it follows from [3, Theorem 4.3, Corollary 2] that dim X_{t} = ind X_{t} = H(X_{t}) ≤ 0. Therefore

we have X = ∪^{∞}_{t=1}X_{t} where dim X_{t} ≤ 0 for any t ∈ ℕ and ∪^{n+1}_{i=1}X_{t_{i}} = X for

any pairwise disjoint t_{1}, …, t_{n+1} ∈ ℕ. Hence [4, Theorem 1.5.8] dim X ≤ n. Conversely, let ωX ≤ ℵ_{0} and dim X ≤ n, (n ≥ 0). Then by Ostrand’s

theorem [5] X = ∪^{∞}_{t=1}X_{t}, where dim X_{t} ≤ 0 for any t ≥ 1 and ∪^{n+1}_{i=1}X_{t_{i}} = X for any n + 1 pairwise disjoint natural numbers t_{1}, t_{2}, …, t_{n+1}. Applying again [4, Theorem 4.3, Corollary 2], we obtain H(X_{t}) = dim X_{t} ≤ 0. Hence, by the definition of the function d_{2}, we have d_{2}(X) ≤ n.

Corollary 1. The GDF d_{2} is the extension of the function dim from the class of all separable metrizable spaces over the class T.

Corollary 2. The equalities d_{2}(I^{n}) = dim I^{n} = n hold for any integer n ≥ −1.

Lemma 2. We have d_{2}(X′) ≤ d_{2}(X) for each X ∈ T and an arbitrary subspace X′ of the space X.

Proof. Assume that X ∈ T and X′ is an arbitrary subspace of X. Let d_{2}(X) ≤ n (n ≥ −1). It will be shown that d_{2}(X′) ≤ n holds too. Indeed, since d_{2}(X) ≤ n, we have X = ∪^{∞}_{t=1}X_{t} where H(X_{t}) ≤ 0 for each t ≥ 1 and

X = ∪^{n+1}_{i=1}X_{t_{i}} for any pairwise disjoint numbers t_{1}, …, t_{n+1}. Introduce the

notation X′_{t} ≡ X_{t} ∩ X′. Obviously, X′ = ∪^{∞}_{t=1}X′_{t}.

Further, since H(X_{t}) ≤ 0 for any t ≥ 1, by the definition of the function H we have X_{t} = ∪^{∞}_{i=1}X_{ti} ≤ 0, where each X_{ti} is quasiclosed in X_{t} and

ind X_{ti} ≤ 0 (i = 1, 2, …). Observe that X′_{t} = X_{t} ∩ X′ = ( ∪^{∞}_{i=1} X_{ti}) ∩ X′ =

∪^{∞}_{i=1}(X_{ti} ∩X′). Since each X_{ti} is quasiclosed in X_{t}, X_{ti} ∩(X_{t} ∩X′) = X_{ti} ∩X′

will be quasiclosed in X_{t}∩X′ = X′_{t} [3, Theorem 1.4]. Introduce the notation X′_{ti} = X_{ti} ∩ X′. Then X′_{t} = ∪^{∞}_{i=1}X′_{ti}, where each X′_{ti} is quasiclosed in X′_{t}.

Moreover, since X′_{ti} = X_{ti} ∩ X′ ⊆ X_{ti}, we have ind X′_{ti} ≤ ind X_{ti} ≤ 0.

Now we shall show that X′_{t} = ∪^{n+1}_{i=1}X′_{t_{i}} for any pairwise disjoint natural

numbers t_{1}, …, t_{n+1}. Indeed, ∪^{n+1}_{i=1}X′_{t_{i}} = ∪^{n+1}_{i=1}(X_{t_{i}} ∩ X′) = ( ∪^{n+1}_{i=1} X_{t_{i}}) ∩ X′ =

X ∩ X′ = X′. The inequality d_{2}(X′) ≤ n is proved and so is the inequality d_{2}(X′) ≤ d_{2}(X).

Lemma 3. Let X ∈ T and X = ∪^{∞}_{i=1}X_{i} where each X_{i} is quasiclosed in

X. Also assume there exists a natural number n such that d_{2}(X_{i}) ≤ n for any i ≥ 1. Then d_{2}(X) ≤ n.

Proof. It is obvious that if n = −1, the assertion is true.

Let us consider the case n = 0. By definition, d_{2}(X) = 0 iff H(X) = 0. Applying Theorem 3.2 from [3], we conclude that the assertion of the lemma is true in this case too.

Now consider the case n ≥ 1. It can be assumed without loss of generality that X_{i} ∩ X_{j} = ∅ whenever i ≠ j. (Indeed, otherwise we have to consider

a new covering {X′_{i}}_{i=1}^{∞} of the space X, where X′_{1} = X_{1}, X′_{k} = X_{k}\ ∪^{k−1}_{i=1} X′_{i}

for k > 1. Then [3, Theorems 1.1 and 1.3] each X′_{i} is quasiclosed in X and X′_{i} ∩ X′_{j} = ∅ whenever i ≠ j. Since X′_{k} ⊆ X_{k} for any k ≥ 1, by Lemma 2 we have d_{2}(X′_{k}) ≤ d_{2}(X_{k}) ≤ n). By the definition of the function d_{2} and since d_{2}X_{i} ≤ n, we have X_{i} = ∪^{∞}_{t=1}X_{it}, where H(X_{it}) ≤ 0 for each t ≥ 1 and

∪^{n+1}_{j=1}X_{it_{j}} = X_{i} for any pairwise disjoint natural numbers t_{1}, …, t_{n+1}.

We introduce the notation X_{(t)} ≡ ∪^{∞}_{i=1}X_{it}. It is obvious that ∪^{∞}_{t=1}X_{(t)} = X.

We shall prove that H(X_{(t)}) ≤ 0 for any t ≥ 1.

Since X_{i} ∩ X_{j} = ∅ for i ≠ j, it is obvious that X_{it} = X_{i} ∩ X_{(t)}. Hence due to the quasiclosedness of X_{i} in X this implies [3, Theorem 1.4] that X_{it} is quasiclosed in X_{(t)}. On the other hand, since H(X_{it}) ≤ 0, we have

X_{it} = ∪^{∞}_{k=1}X_{itk}, where each X_{itk} is quasiclosed in X_{it} (and, accordingly,

in X_{i} and X_{(t)} as well [3, Theorem 1.5]) and for any i, t, k ≥ 1 we have

ind X_{itk} ≤ 0. But it is clear that X_{(t)} = ∪^{∞}_{i,k=1}X_{itk} for any t ≥ 1. Hence, by

the definition of the function H, H(X_{(t)}) ≤ 0 for any t ≥ 1.

Now let us consider natural numbers t_{1}, …, t_{n+1} such that t_{i} ≠ t_{j} whenever i ≠ j (i, j = 1, …, n + 1). We have

∪^{n+1}_{m=1}X_{(t_{m})} = ∪^{n+1}_{m=1}( ∪^{∞}_{i=1} X_{it_{m}}) = ∪^{n+1}_{i=1}( ∪^{n+1}_{m=1} X_{it_{m}}) = ∪^{∞}_{i=1}X_{i} = X.

Hence d_{2}(X) ≤ n.

Corollary 3. let X ∈ T and X = ∪^{∞}_{i=1}X_{i}, where each X_{i} is closed in X.

Also assume that there exists a natural number n such that d_{2}(X_{i}) ≤ n for any i ≥ 1. Then d_{2}(X) ≤ n.

Lemma 4. If X_{1} is a quasiclosed subset of X ∈ T and Y_{1} is a quasiclosed subset of Y ∈ T, then X_{1}×Y_{1} is a quasiclosed subset of the space X×Y ∈ T.

Proof. By the assumption

X_{1} = F_{0} ± F_{1} ± ⋯ ± F_{k−1} ± F_{k}

and

Y_{1} = Φ_{0} ± Φ_{1} ± ⋯ ± Φ_{s−1} ± Φ_{s},

where each F_{i} (o ≤ i ≤ k) is a closed subset of X and each Φ_{j} (0 ≤ j ≤ s) is a closed subset of Y. The sign “+” denotes the usual union of sets, the sign “−” the usual difference of sets, and so whenever ± is written one should take either + or −.

The lemma will be proved by double induction (with respect to k and s). If k = s = 0, then X_{1} = F_{0} and Y_{1} = Φ_{0}, where F_{0} is a closed subset of

the space X and Φ_{0} is a closed subset of the space Y. Therefore X_{1} × Y_{1} will be a closed subset and thus it will also be a quasiclosed subset of X ×Y.

Assume that Lemma 4 has already been proved in two cases: 1) 0 ≤ k ≤ m − 1 and 0 ≤ s ≤ n; 2) 0 ≤ k ≤ m and 0 ≤ s ≤ n − 1, and prove it for k = m and s = n. For this note that the following (easily verifiable) point-set equations hold for any sets A, B, C, D:

(a) (A ∪ B) × (C ∪ D) = (A × C) ∪ (A × D) ∪ (B × C) ∪ (B × D);

(b) (A ∪ B) × (C\D) = {[(A × C) ∪ (B × C)]\(A × D)}\(B × D);

(c) (A\B) × (C ∪ D) = {[(A × C) ∪ (A × D)]\(B × C)}\(B × D);

(d) (A\B) × (C\D) = [(A × C)\(A × D)]\(B × C).

Four cases are possible:

(++) \begin{cases}X_{1} = F_{0} ± F_{1} ± ⋯ ± F_{m−1} + F_{m} \\ Y_{1} = Φ_{0} ± Φ_{1} ± ⋯ ± Φ_{n−1} + Φ_{n} \end{cases} ,

(+−) \begin{cases}X_{1} = F_{0} ± F_{1} ± ⋯ ± F_{m−1} + F_{m} \\ Y_{1} = Φ_{0} ± Φ_{1} ± ⋯ ± Φ_{n−1} − Φ_{n} \end{cases} ,

(−+) \begin{cases}X_{1} = F_{0} ± F_{1} ± ⋯ ± F_{m−1} − F_{m} \\ Y_{1} = Φ_{0} ± Φ_{1} ± ⋯ ± Φ_{n−1} + Φ_{n} \end{cases} ,

(−−) \begin{cases}X_{1} = F_{0} ± F_{1} ± ⋯ ± F_{m−1} − F_{m} \\ Y_{1} = Φ_{0} ± Φ_{1} ± ⋯ ± Φ_{n−1} − Φ_{n} \end{cases} .

Let us consider each of these cases separately.

Introduce the notation

F_{0} ± F_{1} ± ⋯ ± F_{m−1} ≡ F̃_{m−1};
Φ_{0} ± Φ_{1} ± ⋯ ± Φ_{n−1} ≡ Φ̃_{n−1}.

Case (++). Due to (a) we have

X_{1} × Y_{1} = (F̃_{m−1} ∪ F_{m}) × (Φ̃_{n−1} ∪ Φ_{n}) = (F̃_{m−1} × Φ̃_{n−1}) ∪
∪ (F̃_{m−1} × Φ̃_{n}) ∪ (F_{m} × Φ̃_{n−1}) ∪ (F_{m}) × Φ_{n}).

By the assumption of induction F̃_{m−1} ×Φ̃_{n−1}, F̃_{m−1} ×Φ_{n} and F_{m} ×Φ̃_{n−1} are quasiclosed subsets of X × Y. Since the sets F_{m} and Φ_{n} are closed in X and Y, respectively, F_{m} × Φ_{n} is closed (and thus is also quasiclosed) in X ×Y. This means that the union of these sets will be quasiclosed in X ×Y as well [3, Theorem 1.1].

Case (+−). Applying (b), we have

X_{1} × Y_{1} = {[(F̃_{m−1} × Φ_{n}) ∪ (F_{m} × Φ̃_{n−1})]\
\(F̃_{m−1} × Φ̃_{n−1})}\(F_{m} × Φ_{n}).

By the assumption the sets F̃_{m−1} × Φ_{n}, F_{m} × Φ̃_{n−1} and F̃_{m−1} × Φ̃_{n−1} are quasiclosed in X × Y. The set F_{m} × Φ_{n} is obviously closed in X × Y. Hence X_{1} × Y_{1} is quasiclosed in X × Y [3, Theorems 1.1 and 1.3].

Case (−+). By (c) we have

X_{1} × Y_{1} = {[(F̃_{m−1} × Φ̃_{n−1}) ∪ (F̃_{m−1} × Φ_{n})]\
\(F_{m} × Φ̃_{n−1})}\(F_{m} × Φ_{n}).

By the assumption of induction and Theorem 1.3 from [3] one can prove that X_{1} × Y_{1} is quasiclosed in X × Y.

Case (−−). From (d) it follows that

X_{1} × Y_{1} = [(F̃_{m−1} × Φ̃_{n−1})\(F̃_{m−1} × Φ_{n})]\(F_{m} × Φ̃_{n−1}).

By the assumption the sets F̃_{m−1} ×Φ̃_{n−1}, F̃_{m−1} ×Φ_{n} and F_{m} ×Φ̃_{n−1} are quasiclosed in X ×Y. Hence by Theorem 1.3 from [3] X_{1} ×Y_{1} is quasiclosed in X × Y as well.

Proposition 1. For any pair of spaces X, Y ∈ T, if ind X ≤ 0 and ind Y ≤ 0, then ind X × Y ≤ 0.

The proof is trivial.

Lemma 5. Let X, Y ∈ T and let either X ≠ ∅ or Y ≠ ∅. Then d_{2}(X × Y) ≤ d_{2}(X) + d_{2}(Y).

Proof. If either d_{2}(X) = 0 and d_{2}(Y) = −1 or d_{2}(X) = −1 and d_{2}(Y) = 0, then the inequality d_{2}(X × Y) ≤ d_{2}(X) + d_{2}(Y) is obvious.

Assume that d_{2}(X) = n ≥ 0 and d_{2}(Y) = m ≥ 0. Then X = ∪^{∞}_{t=1}X_{t},

Y = ∪^{∞}_{l=1}Y_{l}, where H(X_{t}) ≤ 0 and H(Y_{l}) ≤ 0 for any t, l ≥ 1, and, moreover, the equalities

X = ∪^{n+1}_{i=1}X_{t_{i}}, Y = ∪^{m+1}_{j=1}Y_{l_{j}}

hold for any sequences t_{1}, …, t_{n+1} and l_{1}, …, l_{m+1} of natural numbers with pairwise disjoint numbers.

Introduce the notation Z_{p} ≡ X_{p} × Y_{p} for any p ≥ 1. Since H(X_{p}) ≤ 0

and H(Y_{p}) ≤ 0 for each p ∈ ℕ, we have X_{p} = ∪^{∞}_{i=1}X_{pi} and Y_{p} = ∪^{∞}_{j=1}Y_{pj},

where each X_{pi} is quasiclosed in X_{p} and each Y_{pj} is quasiclosed in Y_{p}, and for any i, j ∈ ℕ we have ind X_{pi} ≤ 0, ind Y_{pj} ≤ 0.

Lemma 4 implies that X_{pi} × Y_{pj} is quasiclosed in X_{p} × Y_{p} = Z_{p} and by Proposition 1 we have ind(X_{pi} × Y_{pj}) ≤ 0. Moreover, it is obvious that

Z_{p} = X_{p} × Y_{p} = ∪^{∞}_{i,j=1}(X_{pi} × Y_{pj}).

Let us now prove that if we are given n + m + 1 natural numbers p_{1}, p_{2}, …, p_{n+m+1} such that p_{i} ≠ p_{j} for any i ≠ j (1 ≤ i, j ≤ n + m + 1),

then ∪^{n+m+1}_{i=1}Z_{p_{i}} = X ×Y. (This, in particular, implies that ∪^{∞}_{p=1}Z_{p} = X ×Y.)

The inclusion X × Y ⊇ Z_{p_{1}} ∪ ⋯ ∪ Z_{p_{n+m+1}} is obvious. Let us prove the inverse inclusion. Assume that (x, y) ∈ X × Y. It remains for us to show that if (x, y) does not belong to some m + n members of the system {Z_{p_{1}}, …, Z_{p_{m+n+1}}}, then (x, y) necessarily belongs to the remaining member of this system.

Consider the case where (x, y) ∈ X × Y and (x, y) ∉ Z_{p_{1}} ∪ ⋯ ∪ Z_{p_{m+n}}. It will be shown that (x, y) ∈ Z_{p_{m+n+1}}. (All other cases are considered analogously.) Let x not belong to exactly k (0 ≤ k ≤ m + n) members of the system {X_{p_{i}}}_{i=1}^{m+n} and belong to the remaining m + n − k members of this system. Then, since each subsystem of the system {X_{t}}_{t=1}^{∞} consisting of n + 1 elements covers the space X, we have k ≤ n.

By the assumption (x, y) ∉ Z_{p_{1}} ∪ ⋯ ∪ Z_{p_{m+n}}. Now if x ∈ X_{p_{i}}, we shall necessarily have y ∉ Y_{p_{i}} (1 ≤ i ≤ m + n). Hence y does not belong to at least m + n − k members of the system {Y_{p_{i}}}_{i=1}^{m+n}. Since each subsystem of the system {Y_{l}}_{l=1}^{∞} consisting of m + 1 elements covers the space Y, we have m + n − k ≤ m. Therefore n ≤ k.

From the inequalities k ≤ n and n ≤ k we obtain the equality n = k. Therefore x does not belong to exactly n elements of the system {X_{p_{i}}}_{i=1}^{m+n}. Assume that they are sets X_{p_{i_{1}}} , …, X_{p_{in}} and consider the system {X_{p_{i_{1}}} , …, X_{p_{in}} , X_{p_{m+n+1}}}. Since the latter system consists of n + 1 elements, we have ( ∪^{n}_{j=1}X_{p_{i_{j}}}) ∪ X_{p_{m+n+1}} = X and, consequently, since x ∉

∪^{n}_{j=1}X_{p_{i_{j}}}, we have x ∈ X_{p_{m+n+1}}.

Analogously, y does not belong to exactly m + n − k = m + n − n = m elements of the system {Y_{p_{i}}}_{i=1}^{m+n}. Assume that they are sets Y_{p_{j_{1}}} , …, Y_{p_{jm}}. (It is obvious that {p_{i_{1}}, …, p_{i_{n}}} ∪ {p_{j_{1}}, …, p_{j_{m}}} = {p_{1}, …, p_{m+n}} and {p_{i_{1}}, …, p_{i_{n}}} ∩ {p_{j_{1}}, …, p_{j_{m}}} = ∅.) Consider the system {Y_{p_{j_{1}}} , …, Y_{p_{jm}} ,

Y_{p_{m+n+1}}}. Since this system consists of m + 1 members, we have ( ∪^{m}_{i=1}Y_{p_{j_{i}}}) ∪

Y_{p_{m+n+1}} = Y. But y ∉ ∪^{m}_{i=1}Y_{p_{j_{i}}} and thus y ∈ Y_{p_{m+n+1}}. Therefore (x, y) ∈

X_{p_{m+n+1}} × Y_{p_{m+n+1}} ⊆ Z_{p_{1}} ∪ ⋯ ∪ Z_{p_{m+n+1}}.

Lemma 6. Let X ∈ T and d_{2}(X) ≤ n (where 0 ≤ n < +∞). Then there

exist n + 1 subspaces X_{1}, …, X_{n+1} of the space X such that X = ∪^{n+1}_{i=1}X_{i}

and d_{2}(X_{i}) ≤ 0 holds for any i = 1, …, n + 1.

Proof. d_{2}(X) ≤ n implies X = ∪^{∞}_{t=1}X_{t} where for each t ≥ 1 H(X_{t}) ≤ 0

(which in turn implies d_{2}(X_{t}) ≤ 0) and for any pairwise disjoint natural

numbers t_{1}, …, t_{n+1} we have X = ∪^{n+1}_{i=1}X_{t_{i}}, in particular, X = ∪^{n+1}_{k=1}X_{k} where

d_{2}(X_{k}) ≤ 0 for any k = 1, …, n + 1.

Applying Lemmas 2, 5, 6 and Corollaries 2, 3, we arrive at

Theorem 2. The GDF d_{2} satisfies the conditions T_{1}^{T} , T_{2}^{T} , T_{3}^{T} , T_{5}^{T} , T_{7}^{T} simultaneously. In other words, the subsystem {T_{1}^{T} , T_{2}^{T} , T_{3}^{T} , T_{5}^{T} , T_{7}^{T} } of the system {T_{1}^{T} , …, T_{8}^{T} } is realized.

References

1. L. G. Zambakhidze and I. G. Tsereteli, On the realizability of dimension-like functions in the class of Tychonoff spaces. (Russian) Soobshch. Akad. Nauk Gruz. SSR 126(1987), No. 2, 265-268.

2. P. S. Alexandrov and B. A. Pasynkov, Introduction to the dimension theory. Introduction to the theory of topological spaces and the general dimension theory. (Russian) Nauka, Moscow, 1973.

3. Y. Hayashi, On the dimension of topological spaces. Math. Japon. 3(1954), No. 2, 71-843.

4. R. Engelking, General topolgy. PWN–Polish Scientific Publishers, Warszawa, 1977.

5. P. A. Ostrand, Covering dimension in general spaces. Gen. Topol. and Appl. 1(1971), No. 3, 209-221.

(Received 21.09.1993)

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia