Georgian Mathematical Journal Volume 10 (2003), Number 2, 209–222

ON THE DIMENSION MODULO CLASSES OF TOPOLOGICAL SPACES AND FREE TOPOLOGICAL GROUPS

I. BAKHIA

Abstract. Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

2000 Mathematics Subject Classification: 54F45, 22A05.

Key words and phrases: Dimension, product of spaces, free topological groups, free topological semigroups.

The question when the product of two spaces is π-compact (i.e., has an open base, each element of which has a compact boundary) was studied in various particular cases in [17], [10] and in a general case in [5].

According to A. Lelek and J. de Groot (see, e.g., [1] and [14]), if a function of dimension type is considered modulo a class of compact spaces which is defined by the method of induction analogously to the function ind , then π-compact spaces turn out to be zero-dimensional (modulo a class of compact spaces).

In this paper, functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. These results can therefore be regarded as a generalization of the results of [17], [10] and [5]. Moreover, based on the same results of [17], the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo the class of compact spaces.

The paper consists of three parts. Part 1 contains the notation. In Part 2 the necessary definitions and statements are formulated. In Part 3 the conditions, under which the dimension modulo classes of a product of spaces is equal to zero, are established. Based on these results the structures of free topological semi-groups and free topological groups are determined.

1. Notation

All spaces considered in this paper are assumed to be Hausdorff and completely regular (Tychonoff). By 𝔐 (with or without an index) we denote topologically closed subclasses of the class of all completely regular spaces (that 𝔐

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is topologically closed means the following: if X ∈ 𝔐 and X is homeomorphic to Y, then Y ∈ 𝔐).

For each topological space X and for any of its subset U by [U]_{X} we denote the closure of U in X, and by Fr_{X}U the boundary of U in X.

Also, the following notation is used:

𝔐_{c} is a class of all compact spaces;

𝔐_{mc} is the class of all metrizable compact spaces;

𝔐_{dc} is the class of all Dugundji compacts [2];

𝔐_{cor} is the class of all Corson compacts [2];

𝔐_{ℵ_{0}} is the class of all countable spaces;

𝔐_{cn} is the class of spaces with countable networks in the sense of Arkhangelskii [2];

𝔐_{con} is the class of all connected spaces;

𝔐_{ps} is the class of all pseudocompact spaces;

𝔐_{cc} is the class of all countably compact spaces;

𝔐_{par} is the class of all paracompact spaces;

𝔐_{pp} is the class of all weakly paracompact spaces;

by N we denote the set of all integers ≥ −1, i.e., N = {−1, 0, 1, 2, … }.

2. Definitions and Necessary Auxiliary Propositions

Definition 2.1. Let 𝔐 be some topologically closed subclass of the class of completely regular spaces. Denote by loc 𝔐 the class of all spaces possessing the following property: X ∈ loc 𝔐 if and only if for each point x of the space X and each open set U_{x} such that x ∈ U_{x} there exists an open set V_{x} in X such that x ∈ V_{x} ⊆ [V_{x}]_{X} ⊆ U_{x} and [V_{x}]_{X} ∈ 𝔐.

Remark 2.1. It is obvious that for each topologically closed class 𝔐, loc 𝔐 is also topologically closed.

Definition 2.2. The class 𝔐 is called c-monotone if 𝔐 is topologically closed and any closed subspace of each space from the class 𝔐 also belongs to 𝔐.

Definition 2.3. Let 𝔐 be some class of spaces. For each space X (from the class of all completely regular spaces) the small inductive dimension in (X, 𝔐) modulo 𝔐 is defined as follows:

(i) in (X, 𝔐) = −1 if X ∈ 𝔐;

(ii) for each n ≥ 0, in (X, 𝔐) ≤ n if for each point x of the space X and for each neighborhood of this point U_{x} there exists an open neighborhood V_{x} in X such that x ∈ V_{x} ⊆ U_{x} and in (Fr_{X}V_{x}, 𝔐) ≤ n − 1; in (X, 𝔐) = n if in (X, 𝔐) ≤ n and in (X, 𝔐) ≰ n − 1 and finally, in (X, 𝔐) = +∞ if in (X, 𝔐) ≰ n for each n ∈ N.

Remark 2.2. If 𝔐 is the empty class, then for any space X we have: in (X, 𝔐) = ind X (where ind X is the small inductive dimension of X [14]).

Remark 2.3. in (X, 𝔐) ≤ 0 if and only if either 1) X ∈ 𝔐 or 2) there exists in X an open base {U_{a}}_{a∈M} such that Fr_{X}U_{a} ∈ 𝔐 for each a ∈ M. In particular, for each regular space X we have: in (X, 𝔐_{c}) ≤ 0 if and only if X is semicompact (≡ π-compact), see [8].

Definition 2.4. The class 𝔐 is called stable if the following conditions are fulfilled:

1) 𝔐 is topologically closed;

2) 𝔐 is c-monotone;

3) if X ∈ 𝔐 and Y is a continuous image of the space X, then Y ∈ 𝔐.

Definition 2.5. The class 𝔐 is called multiplicative if the following conditions are fulfilled:

1) 𝔐 is topologically closed;

2) if X ∈ 𝔐 and Y ∈ 𝔐, then X × Y ∈ 𝔐, where X × Y denotes the usual product of topological spaces X and Y.

The classes 𝔐_{c}, 𝔐_{mc}, 𝔐_{dc}, 𝔐_{cor}, 𝔐_{ℵ_{0}}, 𝔐_{cn} are examples of stable multiplicative classes.

Definition 2.6. The class 𝔐 is called weakly stable if it has the following property: If X ∈ 𝔐 and f : X → Y is a continuous closed mapping “onto”, then Y ∈ 𝔐.

Remark 2.4. One can easily show that if 𝔐 is a stable class and f : X → Y is an open continuous mapping of a space X onto a space Y and X ∈ loc 𝔐 then Y ∈ loc 𝔐.

Next, we formulate some needed definitions.

Definition 2.7 ([12]). Let X be a completely regular topological space. A pair (S(X), Θ) is called a free topological semigroup generated by the space X if the following conditions are fulfilled:

1) S(X) is a topological semigroup the space of which is completely regular;

2) Θ : X → S(X) is a topological embedding of the space X in S(X);

3) Θ(X) generates S(X) algebraically;

4) for each completely regular topological semigroup T and for each continuous mapping ω : X → T of the space X to T, there exists a unique continuous homomorphism Ω : S(X) → T such that the composition ΩΘ is equal to ω.

In [12] and [4] it is shown that for the completely regular topological space X, a free topological semigroup generated by X exists and is unique.

Definition 2.8 (A. A. Markov [11]). Let X be a completely regular space. A Hausdorff topological group F M(X) (a Hausdorff abelian topological group AM(X)) is called a free topological group in the sense of Markov (a free abelian topological group in the sense of Markov) of the space X if X is a subspace of F M(X) (resp. of AM(X)) and for each continuous mapping Φ : X → G of the space X into an arbitrary topological group (into an arbitrary abelian topological group) G there exists a unique continuous homomorphism Φ_{F} : F M(X) → G(resp. Φ_{A} : AM(X) → G) such that Φ_{F} |_{X} = Φ (resp. Φ_{A}|_{X} = Φ).

Definition 2.9 (M.N. Graev [9]). Let X be a completely regular space with a distinguished point e. A Hausdorff topological group F G(X) (a Hausdorff abelian topological group AG(X)) is called a free topological group in the sense of Graev (a free abelian topological group in the sense of Graev) of the space X if the following conditions are fulfilled:

1) X is a subspace of F G(X) (resp. of AG(X));

2) X algebraically generates F G(X) (resp. AG(X));

3) for each continuous mapping φ : X → Q of the space X into an arbitrary topological group (an arbitrary topological abelian group) Q, which sends the point e of X to the unit (to the zero element) of the group Q, there exists a continuous homomorphism Φ_{F} : F G(X) → Q (resp. Φ_{A} : AG(X) → Q) such that Φ_{F} |_{X} = φ (resp. Φ_{A}|_{X} = φ).

A. A. Markov and M. N. Graev showed ([11], [9]) the existence and uniqueness (in the corresponding categories) of the groups F M(X), AM(X), F G(X) and AG(X).

3. On the Zero-Dimensionality by Modulo Class of Topological

Spaces Topological Products, Free Topological Groups and Semigroups

The following theorem is valid:

Theorem 3.1. If Z = X × Y, where X and Y are topological spaces, Y is a nontrivial connected compact and in (X × Y, 𝔐) = 0, where 𝔐 is a weakly stable class, then X ∈ loc 𝔐.

Proof. Let us consider a point x ∈ X and an open neighborhood O_{x} of x. Let U be some open in Y set with U ≠ ∅ and U ≠ Y. Take some point y ∈ U. Let z = (x, y). Denote U_{z} = O_{x} × U ⊆ W_{x}. Since in (X × Y, 𝔐) ≤ 0, there exists open H_{z} such that z ∈ H_{z} ⊆ [H_{z}] ⊆ U_{z} and Fr_{Z}H_{z} ∈ 𝔐. Denote T_{x} = Pr_{X}(H_{z}), where Pr_{X} : X × Y → X is the projection. Obviously, x ∈ T_{x} ⊆ [T_{x}]_{X} ⊆ O_{x}. Show that T_{x} ⊆ Pr_{X}(Fr_{Z}H_{z}). Indeed, take any x′ ∈ T_{x} and consider the set {x′} × Y = A_{x′}. Denote A_{x′} ∩ H_{z} = B_{x′}. Show that B_{x′} ≠ ∅. Indeed, if B_{x′} = ∅, then Pr_{X}(B_{x′}) = ∅. But x′ ∈ Pr_{X}(B_{x′}), hence, it is not empty. On the other hand, since H_{z} is open in Z, the set B_{x′} is open in A_{x′}. Now let us show that 1) Fr_{A_{x′}}B_{x′} ⊆ Fr_{Z}H_{z} and 2) Fr_{A_{x′}}B_{x′} ≠ ∅.

First let us show 1). Fr_{A_{x′}}B_{x′} = [B_{x′}]_{A_{x′}}\B_{x′}, where [B_{x′}]_{A_{x′}} = [B_{x′}]_{Z} (because A_{x′} is closed in Z). But [B_{x′}]_{Z} ⊆ [A_{x′}]_{Z} ∩ [H_{z}]_{Z} = A_{x′} ∩ [H_{z}]_{Z} (here we use the inclusion [A ∩ B] ⊆ [A] ∩ [B] and the fact that A_{x′} is closed in Z).

Furthermore [B_{x′}]_{Z}\B_{x′} ⊆ (A_{x′} ∩ [H_{z}]_{Z})\B_{x′} (here we use the following: if A ⊆ B, then for any C we have A\C ⊆ B\C). Let us show that (A_{x′} ∩ [H_{z}]_{Z})\(A_{x′} ∩ H_{z}) ⊆ [H_{z}]_{Z}\H_{z} (i.e., (A_{x′} ∩ [H_{z}]_{Z})\B_{x′} ⊆ [H_{z}]_{Z}\H_{z}). Indeed if x′′ ∈ A_{x′} ∩ [H_{z}]_{Z}\A_{x′} ∩ H_{z}, then x′′ ∈ A_{x′} ∩ [H_{z}]_{Z} and x′′ ∉ A_{x′} ∩ H_{z}. Then x′′ ∈ A_{x′}, x′′ ∈ [H_{z}]_{Z} and x′′ ∉ H_{z}. Hence, in particular, x′′ ∈ [H_{z}]_{Z}\H_{z}. Thus [B_{x′}]_{Z}\B_{x′} ⊆ [H_{z}]_{Z}\H_{z}. So we have Fr_{Ax′}B_{x′} = [B_{x′}]_{Ax′}\B_{x′} = [B_{x′}]_{Z}\B_{x′} ⊆ [H_{z}]_{Z}\H_{z} = Fr_{Z}H_{z}. Thus 1) is proved.

Now let us prove 2). Let us consider the mapping π_{x′} : A_{x′} → Y determined as follows: π_{x′}(x′, y) = y. This mapping is one-to-one and surjective. The mapping π_{x′} is continuous because it is the restriction of the continuous mapping. Since Y is compact and {x′} × Y = A_{x′}, A_{x′} is compact and so the mapping π_{x′} is a homeomorphism (since continuous and one-to-one mapping from a compact space onto a Hausdorff space is a homeomorphism). If Fr_{A_{x}}B_{x′} = ∅, then B_{x′} is closed and open in A_{x′}. Let us note that π_{x′}(B_{x′}) ⊆ U. Since π_{x′} is a homeomorphism, the set π_{x′}(B_{x′}) is closed and open in Y. But this is impossible because π_{x′}(B_{x′}) ⊂ U ≠ Y, π_{x′}(B_{x′}) ≠ ∅ and Y is connected. Therefore our assumption is false. So Fr_{A_{x}}B_{x′} ≠ ∅.

Thus we see that for every x′ ∈ T_{x} there exists a point z′ ∈ Fr_{z}H_{z} such that x′ = Pr_{x}(z′). This means that T_{x} ⊆ Pr_{X}(Fr_{Z}H_{z}). H_{z} is dense in [H_{z}]. Therefore T_{x} = Pr_{X}(H_{z}) is dense in Pr_{X}([H_{z}]_{Z}). Hence, [T_{x}]_{X} = Pr_{X}([H_{z}]_{Z}). But since Y is compact, Pr_{X} : X × Y → X is the closed mapping, i.e., Pr_{X}([H_{z}]_{Z}) is closed in X. On the other hand, Fr_{Z}H_{z} ⊆ [H_{z}]_{Z}. Thus we have T_{x} ⊆ Pr_{X}(Fr_{Z}H_{z}) ⊆ Pr_{X}([H_{z}]_{Z}). But [T_{x}]_{X} = Pr_{X}([H_{z}]_{Z}), so [T_{x}]_{X} ⊆ [Pr_{X}(Fr_{Z}H_{z})]_{X} ⊆ [Pr_{X}[H_{z}]_{Z}]_{X}. Thus [T_{x}]_{X} = [Pr_{X}(Fr_{Z}H_{z})]_{X} = Pr_{X}(Fr_{Z}H_{z}). Since Pr_{X} : X × Y → X is the closed mapping and Fr_{Z}H_{z} ∈ 𝔐, by weak stability, Pr_{X}(Fr_{Z}H_{z}) ∈ 𝔐. So we have found a neighborhood T_{x} of the point x such that T_{x} ⊂ [T_{x}]_{X} ⊂ O_{x} and [T_{x}]_{X} ∈ 𝔐. Thus X ∈ loc 𝔐. □

Corollary 3.1. If Z = X × Y where X and Y are topological spaces, Y is a connected compact and in (X × Y, 𝔐_{con}) = 0, then X ∈ loc 𝔐_{con}.

Corollary 3.2. If Z = X × Y where X and Y are topological spaces, Y is a connected compact and in (X × Y, 𝔐_{ps}) = 0, then X ∈ loc 𝔐_{ps}.

Corollary 3.3. If Z = X × Y where X and Y are topological spaces, Y is a connected compact and in (X × Y, 𝔐_{cor}) = 0, then X ∈ loc 𝔐_{cor}.

Corollary 3.4. If Z = X × Y, where X and Y are topological spaces, Y is a connected compact and in (X × Y, 𝔐_{par}) = 0, then X ∈ loc 𝔐_{par}.

Corollary 3.5. If Z = X × Y, where X and Y are topological spaces, Y is a connected compact and in (X × Y, 𝔐_{pp}) = 0, then X ∈ loc 𝔐_{pp}.

Corollary 3.6. If Z = X × Y, where X and Y are topological spaces, Y is a connected compact and in (X × Y, 𝔐_{c}) = 0, then X ∈ loc 𝔐_{c}.

As it is shown in [1], the following theorem holds.

Theorem 3.2. If X × Y is a topological product of the spaces X and Y such that in (X × Y, 𝔐) ≤ 0, where 𝔐 is a stable class and ind X > 0, then Y ∈ loc 𝔐.

In the sequel we shall need the following propositions which are easy to prove.

Proposition 3.1. If 𝔐 is a c-monotone class and X ∈ loc 𝔐, then in (X, 𝔐) ≤ 0.

Proof. Let x ∈ X and O_{x} be some open neighborhood of x in X. Since X ∈ loc 𝔐, there is an open neighborhood V_{x} such that x ∈ V_{x} ⊆ [V_{x}]_{X} ⊆ O_{x} and [V_{x}]_{X} ∈ 𝔐. But Fr_{X}V_{x} ⊆ [V_{x}]_{X} and since [V_{x}]_{X} ∈ 𝔐 and Fr_{X}V_{x} is closed in X (and therefore in [V_{x}]_{X} too), then, by c-monotonity, Fr_{X}V_{x} ∈ 𝔐, which means that in (X, 𝔐) ≤ 0. □

Proposition 3.2. If 𝔐 is a multiplicative class, then the class loc 𝔐 is also multiplicative.

Proof. X ∈ loc 𝔐 means that for every point x of X and for every its open neighborhood O_{x} there exists an open neighborhood V_{x} of x such that x ∈ V_{x} ⊆ [V_{x}]_{X} ⊆ O_{x} and [V_{x}]_{X} ∈ 𝔐. Also, Y ∈ loc 𝔐 means that for every point y ∈ Y and any its open neighborhood O_{y} there is an open neighborhood V_{y} of y such that y ∈ V_{y} ⊆ [V_{y}]_{Y} ⊆ O_{y} and [V_{y}]_{Y} ∈ 𝔐. Denote Z = X × Y. Consider the point z = (x, y) ∈ Z and its open neighborhood O_{z} = O_{x} × O_{y}. We have z = (x, y) ∈ V_{x} × V_{y} ⊆ [V_{x} × V_{y}]_{Z} = [V_{x}]_{X} × [V_{y}]_{Y} ⊆ O_{x} × O_{y}. Since 𝔐 is multiplicative, [V_{x}]_{X} × [V_{y}]_{Y} ∈ 𝔐, i.e., [V_{x} × V_{y}]_{Z} ∈ 𝔐. Since the point z = (x, y) was selected arbitrarily in (X × Y) and since the family of neighborhoods of the form O_{z} = O_{x} × O_{y} is the base of the space X × Y at the point z = (x, y), then X × Y ∈ loc 𝔐, i.e., loc 𝔐 is multiplicative. □

Proposition 3.3. If 𝔐 is a c-monotone class, A is closed in X and in (X, 𝔐) ≤ 0, then in (A, 𝔐) ≤ 0.

Proof. Let us consider a point x ∈ A and an open in A neighborhood O_{x} of x. Then there exists an open neighborhood O′_{x} of x in X such that O′_{x} ∩ A = O_{x}. Then there is V′_{x}, open in X, such that x ∈ V′_{x} ⊆ O′_{x} and Fr_{X}V′_{x} ∈ 𝔐. Let us consider V′_{x} ∩ A = V_{x}.

Clearly, V_{x} is open in A, V_{x} ⊆ O′_{x} and Fr_{A}V_{x} ⊆ Fr_{X}V′_{x}. Furthermore Fr_{A}V_{x} is closed in Fr_{X}V′_{x}. Since Fr_{X}V′_{x} ∈ 𝔐, by c-monotonity, Fr_{A}V_{x} ∈ 𝔐. Thus we obtain that for every x ∈ A and any its neighborhood O_{x} there is an open neighborhood V_{x} of x such that x ∈ V_{x} ⊆ O_{x} and Fr_{A}V_{x} ∈ 𝔐. This means that in (A, 𝔐) ≤ 0. □

The following is true.

Theorem 3.3. Let 𝔐 be a stable multiplicative class. Then the inequality, in (X × Y, 𝔐) ≤ 0 takes place if and only if one of the following four conditions is fulfilled:

1) ind X ≤ 0, ind Y ≤ 0;

2) X ∈ loc 𝔐, Y ∈ loc 𝔐;

3) X ∈ loc 𝔐, ind X = 0 and in (Y, 𝔐) ≤ 0;

4) Y ∈ loc 𝔐, ind Y = 0 and in (X, 𝔐) ≤ 0.

Proof. Necessity. Let us show that if in (X × Y, 𝔐) ≤ 0, then one of the conditions given above is fulfilled. Let us consider four possible cases:

1. ind X = 0, ind Y = 0;

2. ind X > 0, ind Y > 0;

3. ind X > 0, ind Y = 0; 4. ind X = 0, ind Y > 0.

1. If ind X = 0 and ind Y = 0, then there is nothing to prove.

2. If ind X > 0 and ind Y > 0, then, by Theorem 3.2, Y ∈ loc 𝔐 and X ∈ loc 𝔐.

3. If ind X > 0 and ind Y = 0, then, by Theorem 3.1, Y ∈ loc 𝔐 and ind Y = 0.

4. The proof is similar to the proof in case 3.

Sufficiency. 1) If ind X ≤ 0, ind Y ≤ 0, then ind (X × Y) ≤ 0 (see [7]). Then since (by c-monotonity) ∅ ∈ 𝔐, we have in (X × Y, 𝔐) ≤ 0.

2) If X ∈ loc 𝔐 and Y ∈ loc 𝔐, then, by Proposition 3.2, X × Y ∈ loc 𝔐. Then by Proposition 3.1, we have: in (X × Y, 𝔐) ≤ 0. Suppose X ∈ loc 𝔐, ind X = 0 and in (Y, 𝔐) ≤ 0. Let us show that in (X × Y, 𝔐) ≤ 0.

3) Suppose z = (x, y) is any point of the space X × Y and U_{z} is any neighborhood of this point. Then there is a neighborhood O_{x} of x and there is a neighborhood O_{y} of y such that z ∈ O_{x} × O_{y} ⊆ U_{z}. Since X ∈ loc 𝔐, there exists V_{x} such that x ∈ V_{x} ⊆ [V_{x}] ⊆ O_{x} and [V_{x}]_{X} ∈ 𝔐. Since ind X = 0, there is W_{X} such that x ∈ W_{x} ⊆ V_{x} ⊆ [V_{x}] ⊆ O_{x} and W_{X} is closed and open in X (i.e., Fr_{X}W_{x} = ∅). Thus W_{x} is closed and open in [V_{x}]_{X} too. Since 𝔐 is c-monotone, W_{x} ∈ 𝔐. So W_{x} is closed and open and W_{x} ∈ 𝔐. Since in (Y, 𝔐) ≤ 0, there exists an open set W_{y} such that y ∈ W_{y} ⊆ [W_{y}]_{Y} ⊆ O_{y} and Fr_{Y} W_{y} ∈ 𝔐.

Let us consider W_{z} = W_{x} × W_{y}. Then z ∈ W_{z} = W_{x} × W_{y} ⊆ O_{x} × O_{y} ⊆ U_{z} and Fr_{Z}W_{z} = ([W_{x}]_{X} × Fr_{Y} W_{y}) ∩ (Fr_{X}W_{x} × [W_{y}]_{Y}). Since Fr_{X}W_{x} = ∅, (Fr_{X}W_{x}×[W_{y}]_{Y}) = ∅, i.e., Fr_{Z}W_{z} = [W_{x}]_{X} ×Fr_{Y} W_{y} = W_{x}×Fr_{Y} W_{y} because [W_{x}]_{X} = W_{x} (since it is closed and open). But W_{x} ∈ 𝔐 and Fr_{Y} W_{y} ∈ 𝔐. Since 𝔐 is multiplicative, we have W_{x} × Fr_{Y} W_{y}. So Fr_{Z}W_{z} ∈ 𝔐. Since the point x and its open neighborhood O_{z} were chosen arbitrarily, in (X × Y, 𝔐) ≤ 0.

The fourth case is similar to the third one. □

Corollary 3.7. in (X × Y, 𝔐_{c}) ≤ 0 if and only if one of the following four conditions is fulfilled:

1) ind X = 0, ind Y = 0;

2) X ∈ loc 𝔐_{c}X ∈ loc 𝔐_{c};

3) X ∈ loc 𝔐_{c}, ind X = 0 and in (Y, 𝔐_{c}) ≤ 0;

4) Y ∈ loc 𝔐_{c}, ind Y = 0 and in (X, 𝔐_{c}) ≤ 0.

Corollary 3.8. in (X × Y, 𝔐_{mc}) ≤ 0 if and only if one of the following four conditions is fulfilled:

1) ind X = 0, ind Y = 0;

2) X ∈ loc 𝔐_{mc}, X ∈ loc 𝔐_{mc};

3) X ∈ loc 𝔐_{mc}, ind X = 0 and in (Y, 𝔐_{mc}) ≤ 0;

4) Y ∈ loc 𝔐_{mc}, ind Y = 0 and in (X, 𝔐_{mc}) ≤ 0.

Corollary 3.9. in (X × Y, 𝔐_{dc}) ≤ 0 if and only if one of the following four conditions is fulfilled:

1) ind X = 0, ind Y = 0;

2) X ∈ loc 𝔐_{dc}, X ∈ loc 𝔐_{dc}; 3) X ∈ loc 𝔐_{dc}, ind X = 0 and in (Y, 𝔐_{dc}) ≤ 0;

4) Y ∈ loc 𝔐_{dc}, ind Y = 0 and in (X, 𝔐_{dc}) ≤ 0.

Corollary 3.10. in (X × Y, 𝔐_{cor}) ≤ 0 if and only if one of the following four conditions is fulfilled:

1) ind X = 0, ind Y = 0;

2) X ∈ loc 𝔐_{cor}, X ∈ loc 𝔐_{cor};

3) X ∈ loc 𝔐_{cor}, ind X = 0 and in (Y, 𝔐_{cor}) ≤ 0;

4) Y ∈ loc 𝔐_{cor}, ind Y = 0 and in (X, 𝔐_{cor}) ≤ 0.

Corollary 3.11. in (X × Y, 𝔐_{ℵ_{0}}) ≤ 0 if and only if one of the following four conditions is fulfilled:

1) ind X = 0, ind Y = 0;

2) X ∈ loc 𝔐_{ℵ_{0}}, X ∈ loc 𝔐_{ℵ_{0}};

3) X ∈ loc 𝔐_{ℵ_{0}}, ind X = 0 and in (Y, 𝔐_{ℵ_{0}}) ≤ 0;

4) Y ∈ loc 𝔐_{ℵ_{0}}, ind Y = 0 and in (X, 𝔐_{ℵ_{0}}) ≤ 0.

Corollary 3.12. in (X × Y, 𝔐_{cn}) ≤ 0 if and only if one of the following four conditions is fulfilled:

1) ind X = 0, ind Y = 0;

2) X ∈ loc 𝔐_{cn}, X ∈ loc 𝔐_{cn};

3) X ∈ loc 𝔐_{cn}, ind X = 0 and in (Y, 𝔐_{cn}) ≤ 0;

4) Y ∈ loc 𝔐_{cn}, ind Y = 0 and in (X, 𝔐_{cn}) ≤ 0.

Based on the obtained results, we determine the structure of free topological semigroups (in the sense of Marxen) and free topological groups (in the sense of Markov [11] and Graev [9]) in the case where these objects are zero-dimensional modulo a certain class.

Theorem 3.4. Let 𝔐 be a stable multiplicative class. Then the space S(X) of free topological semigroups of the space X has in (S(X), 𝔐) ≤ 0 if and only if either ind X = 0 or X ∈ loc 𝔐.

Proof. Marxen showed in [12] that the space of the topological semi-group S(X)

of any space X is of the form S(X) = ∞ ; ⋁_{n=1}X^{n}, where ⋁ is a symbol for a

topological sum and X^{n} is the n^{th} power of X.

As is known, if ind X = 0, then for every n ≥ 1 we have ind X^{n} = 0, but the topological sum of zero-dimensional (in sense of ind ), spaces is still zerodimensional (in sense of ind ). Thus ind S(X) = 0. Since 𝔐 is a stable class, we have in (S(X), 𝔐) ≤ 0. If X ∈ loc 𝔐, then for every n ≥ 1 (by multiplicity of 𝔐) we have X^{n} ∈ loc 𝔐. Now let us show that S(X) ∈ loc 𝔐. Suppose x ∈ S(X) is a point chosen arbitrarily and O_{x} is any its neighborhood in S(X). Then there is unique n such that x ∈ X^{n}. Let us consider O′_{x} = O_{x} ∩ X^{n}. Since X^{n} ∈ loc 𝔐, there is a neighborhood V′_{x} such that x ∈ V′_{x} ⊆ [V′_{x}]_{X^{n}} ⊆ O′_{x} and [V′_{x}]_{X} ∈ 𝔐. V′_{x} is open in X^{n} and X^{n} is open in S(X). So V′_{x} is open in S(X). Thus we have found an open neighborhood V_{x} in S(X) such that x ∈ V′_{x} ⊆ [V′_{x}]_{X^{n}} = [V′_{x}]_{S(X)} ⊆ O′_{x} ⊆ O_{x} and [V′_{x}]_{S(X)} ∈ 𝔐. Consequently, S(X) ∈ loc 𝔐. Therefore, by Proposition 3.1, in (S(X), 𝔐) ≤ 0.

And vice versa suppose in (S(X), 𝔐) ≤ 0. We have to show that either

ind X = 0 or X ∈ loc 𝔐. Since S(X) = ∞ ; ⋁_{n=1}X^{n}, X × X is the closed subset

of S(X). So, by Proposition 3.3, in (X × X, 𝔐) ≤ in (S(X), 𝔐) ≤ 0. Thus in (X × X, 𝔐) ≤ 0. Since 𝔐 is a stable multiplicative class, by Theorem 3.2, either ind X = 0 or X ∈ loc 𝔐, which was to be proved. □

Definition 3.1. Let 𝔐 be a topologically closed class. The uncountable space X is called 𝔐-dispersed if for every uncountable and closed subset B we have AM(B) ∉ loc 𝔐, where AM(B) is a free abelian topological group of B in the sense of Markov.

Remark 3.1. One can show that every uncountable compact space is 𝔐_{c}, 𝔐_{dc}-, 𝔐_{mc}-, 𝔐_{ℵ_{0}}-dispersed (see, e.g., [6]).

Theorem 3.5. Let X be an uncountable compact and 𝔐-dispersed space, where 𝔐 is a stable and multiplicative class. Then the following conditions are equivalent: 1) ind X ≤ 0, 2) ind AG(X) ≤ 0, 3) ind AM(X) ≤ 0,

4) in (AG(X), 𝔐) ≤ 0, 5) in (AM(X), 𝔐) ≤ 0, 6) Ind AM(X) ≤ 0,

7) Ind AG(X) ≤ 0.

Proof. The proof will follow the scheme: 1)⇒2)⇒3)⇒4)⇒ 5)⇒6)⇒7)⇒1). First let us show that 1)⇒2). Since X is a compact space, AG(X) is a

K_{ω}-space (see [9]), i.e., AG(X) = ∞ ; ⋃_{n=1}A_{i}(X), where A_{i}(X) symbolizes a sub-

space of the group AG(X), which consists of words the length of which is ≤ i. It is known (see [9]) that for every n = 1, 2, …, A_{n}(X) is a compact subspace of AG(X). Let us show by induction that dim A_{n}(X) = 0. Indeed, when n = 1 we have A_{1}(X) = X ⋁(−X) ⋁{0}. Since ind X = 0, we have dim X = 0. Since (−X) is homeomorphic to X, we have dim (−X) = 0, i.e., dim A_{1}(X) = dim X = ind X = 0. Suppose for every p ≤ n − 1 it is known that dim A_{p}(X) = 0. Let us discuss A_{n}(X). It is obvious that A_{n}(X) = A_{n−1}(X)∪(A_{n}(X)\A_{n−1}(X)). Since A_{n−1}(X) is closed in AG(X) (see [9]), it is closed in A_{n}(X). Therefore A_{n}(X)\A_{n−1}(X) is open in A_{n}(X). Now let us show that ind (A_{n}(X)\A_{n−1}(X)) ≤ 0. Suppose α ∈ A_{n}(X)\A_{n−1}(X). This means that α is a non-cancellable word of length n, i.e., α = ε_{1}x_{1} + ⋯ + ε_{n}x_{n}, where ε_{i} is equal to 1 or −1 for every i = 1, 2, …, n.

Let us consider any open neighborhood O_{α} of the point α in A_{n}(X). According to Arkhangelskii–Joiner’s lemma (see, e.g., [3]) there exist closed and open subsets U_{1}, U_{2}, …, U_{n} which satisfy the following:

1. x_{i} ∈ U_{i} for every i = 1, 2, …, n;

2. U_{i} ∩ U_{j} = ∅ as soon as i ≠ j (i, j = 1, 2, …, n);

3. the set ε_{1}U_{1} + ε_{2}U_{2} + ⋯ + ε_{n}U_{n} is an open neighborhood of the point α in A_{n}(X);

4. ε_{1}U_{1}+ε_{2}U_{2}+⋯+ε_{n}U_{n} ⊆ O_{α} because for every i, U_{i} is compact. Therefore ε_{1}U_{1} + ε_{2}U_{2} + ⋯ + ε_{n}U_{n} is an open compact neighborhood of the point α ∈ A_{n}(X)\A_{n−1}(X).

Since the point α and its neighborhood O_{α} were chosen arbitrarily, then ind (A_{n}(X)\A_{n−1}(X)) ≤ 0. Furthermore, for any closed subset F of A_{n}, where A_{n}(X) ⊆ A_{n}(X)\A_{n−1}(X), we have dim F ≤ ind F ≤ ind (A_{n}(X)\A_{n−1}(X)) ≤ 0. Then, by the inductive assumption and according to Dowker’s theorem [7],

we obtain dim A_{n}(X) ≤ 0. Since AG(X) = ∞ ; ⋃_{n=1}A_{i}(X), we have dim AG(X) ≤ 0.

Thus ind AG(X) ≤ 0.

1)⇒2) is proved.

2)⇒3). According to F. Ward’s theorem AM(X) ≈ AG(X) × Z, where Z is

the set of integers with discrete topology. Therefore AM(X) = ∞ ; ⋁_{n=1}AG(X)_{i} and

thus 2)⇒3).

3)⇒4). Since 𝔐 is a stable class, ∅ ∈ 𝔐 and according to Nishiura’s Theorem 1.8 from [14] we have in (AM(X), 𝔐) ≤ ind AM(X) = ind AG(X) = 0. 3)⇒4) is proved.

4)⇒5) follows also from [16]. Indeed, in (AG(X), 𝔐)=0 ⇒ in (AM(X), 𝔐)=

in (AG(X) × Z, 𝔐)=in ( ∞ ; ⋁_{n=1}AG(X)_{i}, 𝔐) = 0.

5)⇒6). We have to prove that in (AM(X), 𝔐) ≤ 0 ⇒ Ind AM(X) ≤ 0. Suppose, on the contrary, that Ind AM(X) > 0. Then ind X > 0 because if ind X ≤ 0, then ind AG(X) ≤ 0 and ind AM(X) ≤ 0. Since AM(X) is σcompact and thus, finally compact, we have dim AM(X) ≤ Ind AM(X) = 0. This contradicts our assumption. Therefore ind X > 0. Since X is compact and ind X > 0, we have dim X > 0. By Proposition 2.6.3 from [17], there are compact subspaces F_{1} and F_{2} of the compact X such that F_{1} ∩ F_{2} = ∅, dim F_{1} > 0, dim F_{2} > 0. Let us consider the compact space F = F_{1} ∪ F_{2}. Since F_{1} ∩ F_{2} = ∅, the compact F is homeomorphic to the topological sum of F_{1} and F_{2}, i.e., F = F_{1} ⋁ F_{2}.

According to Ward’s theorem from [16], the free topological group AM(X) is topologically isomorphic to AG(X) × Z. Since in (AM(X), 𝔐) ≤ 0 we have in (AG(X), 𝔐) ≤ 0. Denote by G(F) the subgroup of AG(X) which is generated algebraically by the compact subspace F ⊆ X. It is well-known that [9] G(F) is closed in AG(X) and G(F) is topologically isomorphic to AG(F).

Thus in (G(F), 𝔐) = in (AG(F), 𝔐) ≤ 0.

By Ward’s theorem, AM(F) = AG(F) × Z, i.e., AM(F) = ∞ ; ⋁_{n=1}AG(F)_{i},

where for every i = 1, 2, …, AG(F)_{i} is homeorphic to AG(F). It is easy to show that we have in (AM(F), 𝔐) ≤ 0. But F = F_{1} ⋁ F_{2}. It is known that AM(F_{1} ⋁ F_{2}) ≅ AM(F_{1}) × AM(F_{2}). Then in (AM(F_{1}) × AM(F_{2}), 𝔐) ≤ 0. Consequently, by Theorem 3.3, one of the following four cases will take place:

1. ind AM(F_{1}) ≤ 0, ind AM(F_{2}) ≤ 0,

2. AM(F_{1}) ∈ loc 𝔐, AM(F_{2}) ∈ loc 𝔐,

3. AM(F_{1}) ∈ loc 𝔐, ind AM(F_{1}) = 0 and in AM(F_{2}), 𝔐) ≤ 0,

4. AM(F_{2}) ∈ loc 𝔐, ind AM(F_{2}) = 0 and in AM(F_{1}), 𝔐) ≤ 0.

But in accordance with the condition it is impossible for AM(F_{1}) or AM(F_{2}) to belong to loc 𝔐. Therefore only case 1) remains.

Since F_{1} ⊆ AM(F_{1}) and F_{2} ⊆ AM(F_{2}), we have ind F_{1} ≤ 0 and ind F_{2} ≤ 0. Then dim F_{1} ≤ 0 and dim F_{2} ≤ 0. But this contradicts dim F_{1} > 0, dim F_{2} > 0.

Thus our assumption that Ind AM(X) > 0 is false. So 5)⇒6) is proved. 6)⇒7) follows from Ward’s theorem. 7)⇒1). Since Ind AG(X) ≤ 0, we have Ind X ≤ 0 because X is the closed

subspace of AG(X). Hence ind X ≤ 0. □

Theorem 3.6. Let X be an uncountable K_{ω}-space [13] and an 𝔐-dispersed space, where 𝔐 is a stable multiplicative class. Then the following conditions are equivalent: 1) ind X ≤ 0, 2) ind F G(X) ≤ 0, 3) ind F M(X) ≤ 0,

4) in (F G(X), 𝔐) ≤ 0, 5) in (F M(X), 𝔐) ≤ 0.

Proof. The proof will follow the scheme 1)⇒2)⇒3)⇒4)⇒5) ⇒1).

1)⇒2). Since X is a K_{ω}-space, there exists a K_{ω}-decomposition {X_{i}} of X, i.e., there exists a system {X_{i}}, i = 1, 2, …, of subspaces of the space X such that:

1. X = ⋃_{i=1}^{∞} X_{i};

2. X_{i} is compact for every i = 1, 2, …;

3. X_{i} ⊆ X_{j}, for every i < j (i, j = 1, 2, …);

4. the set A is closed in X if and only if A∩X_{i} is compact for every i = 1, 2, …

(If {X_{i}} is a K_{ω}-decomposition of X, we write X =^{ω} ⋃_{i=1}^{∞} X_{i}).

As we know (see [13]), if X =^{ω} ⋃_{i=1}^{∞} X_{i}, then F G(X) =^{ω} ∞ ; ⋃_{j=1} ⋃_{i=1}^{∞} F_{j}(X_{i}), where

F_{j}(X_{i}) denotes the set of words generated by elements of X_{i}, the length of which is ≤ j. It is clear that F_{j}(X_{i}) ⊆ F G(X_{i}) for every j = 1, 2, . . .. Let us show that ind F G(X_{i}) = 0 for every i = 1, 2, . . .. We know that F_{n}(X_{i}) is compact for every n. Let us show by induction that dim F_{n}(X_{i}) = 0 for every i . For n = 1 we have dim X_{i} = 0. Then dim F_{1}(X_{i}) = 0, where F_{1}(X_{i}) = X_{i} ⋁ X_{i}^{−1} ⋁{1}. Suppose for every p ≤ n−1 the assertion is already proved and that dim F_{p}(X_{i}) = 0. Let us consider F_{n}(X_{i}) = F_{n−1}(X_{i}) ∪ (F_{n}(X_{i})\F_{n−1}(X_{i})). Since F_{n−1}(X_{i}) is closed in F G(X_{i}), then it is also closed in F_{n}(X_{i}). Thus F_{n}(X_{i})\F_{n−1}(X_{i}) is open in F_{n}(X_{i}). Let us show that ind (F_{n}(X_{i})\F_{n−1}(X_{i})) ≤ 0. Indeed, suppose α ∈ F_{n}(X_{i})\F_{n−1}(X_{i}). Thus α = x_{1}^{ε_{1}}x_{2}^{ε_{2}} … x_{n}^{ε_{n}}, where ε_{i} is equal to 1 or −1. Let us consider any open neighborhood O_{α} of the point α in F_{n}(X_{i}). Then according to Arkhangelskii–Joiner’s lemma (which is also true for the group F G(X_{i})), there exist open compact sets U_{1}, U_{2}, …, U_{n} which satisfy the following conditions:

1. x_{i} ∈ U_{i}, i = 1, 2, …;

2. U_{i} ∩ U_{j} = ∅, i ≠ j (i, j = 1, 2, …);

3. the set U_{1}^{ε_{1}}U_{2}^{ε_{2}} … U_{n}^{ε_{n}} is an open neighborhood of the point α in F_{n}(X_{i});

4. U_{1}^{ε_{1}}U_{2}^{ε_{2}} … U_{n}^{ε_{n}} ⊆ O_{a}.

Since U_{i} is compact for every i = 1, 2, …, n, U_{1}^{ε_{1}}U_{2}^{ε_{2}} … U_{n}^{ε_{n}} is an open compact neighborhood of the point α ∈ F_{n}(X_{i})\F_{n−1}(X_{i}). Since the point α and its neighborhood O_{α} where chosen arbitrarily, we have ind (F_{n}(X_{i})\F_{n−1}(X_{i})) ≤ 0.

For every compact F ⊆ F_{n}(X_{i})\F_{n−1}(X_{i}) we have: dim F ≤ 0. Since according to the inductive assumption, dim F_{n−1}(X_{i}) ≤ 0, by Dowker’s theorem (see[7]) we conclude that dim F_{n}(X_{i}) ≤ 0, i.e., for every i we have shown that dim F n(X_{i}) ≤ 0. Hence F G(X_{i}) ≤ 0. Since F G(X) is compact and

F G(X) = ∞ ; ⋃_{n=1} ⋃_{i=1}^{∞} F_{n}(X_{i}), we have dim F G(X) ≤ 0. Thus ind F G(X) ≤ 0.

2)⇒3). Suppose ind F G(X) = 0. Then ind X = 0 because X ⊆ F G(X). Let us consider the space Y = X ⋁{p}, where p ∉ X, which is a topological sum of the space X and the singleton {p}. It is clear that since X is the K_{ω}-space, we have ind Y = 0. So by means of 1)⇒2) we get ind F G(Y) = 0. But it is known (see [9]) that F M(X) is topologically isomorphic to F G(Y). Thus ind F M(X) = 0.

So ind F M(X) = 0. According to Morris’ theorem (see [13]), the group F M(X) is topologically isomorphic to the coproduct of the group F G(X) and the discrete group Z of integers i.e., F M(X) ≅ F G(X) ∗ Z. Hence F G(X) is a closed subspace of F M(X). Thus ind F G(X) = 0 is proved.

3)⇒4). Since 𝔐 is the stable class and ∅ ∈ 𝔐, according to Nishiura’s theorem (see [14]), in (F M(X), 𝔐) ≤ in (F M(X), ∅) = ind F M(X) ≤ 0.

4)⇒5). Suppose in (F M(X), 𝔐)≤0. We have to show that in (F G(X), 𝔐)≤ 0. According to Morris’ theorem, F M(X) ≅ F G(X) ∗ Z. Since F G(X) is a topological group the space of which is a K_{ω}-space, it is complete in the sense of A. Weil (see [9]). Moreover, F G(X) is a closed subgroup of F M(X). Therefore, by c-monotonity of the function in (X, P), we get in (F G(X), 𝔐) ≤ 0.

Now let us prove 5)⇒1). Suppose in (F G(X), 𝔐) ≤ 0. We have to show that ind X = 0.

Let us assume that the opposite is true, i.e., ind X > 0. Then dim X > 0.

Since X is the K_{ω}-space, there exists its ω decomposition X =^{ω} ⋃_{i=1}^{∞} X_{i}, where

each X_{i} is compact.

Suppose i_{0} is a natural number such that dim X_{i_{0}} > 0 (if there is no such i_{0} and for every i dim X_{i} is equal to 0, then dim X is also equal to 0).

Since X_{i_{0}} is compact, then, based on the well-known result that dim X_{i_{0}} is equal to the supremum of dim dimensions of the components of X_{i_{0}}, there exists a connected compact ϕ^{i_{0}} ⊆ X_{i_{0}} such that dim ϕ^{i_{0}} > 0.

According to Zambakhidze’s Theorem 2.6.3 (see [17]) there exist subsets F^{i_{0}(1)} and F^{i_{0}(2)} closed in ϕ^{i_{0}} (and therefore in X too) such that F^{i_{0}(1)} ∩ F^{i_{0}(2)} = ∅, dim F^{i_{0}(1)} > 0, dim F^{i_{0}(2)} > 0. Since ϕ^{i_{0}} is connected and compact, there exists a point z = ϕ^{i_{0}}\F^{i_{0}(1)} ⋃ F^{i_{0}(2)} (otherwise ϕ^{i_{0}} would be decomposed into the sum of two separate closed subsets, which contradicts the connectedness of ϕ^{i_{0}}). Suppose Y_{i_{0}} = F^{i_{0}(1)} ⋃ F^{i_{0}(2)} and Y′_{i_{0}} = F^{i_{0}(1)} ⋃ F^{i_{0}(2)} ⋃{z}. Since F^{i_{0}(1)} ⋂ F^{i_{0}(2)} = ∅ and z ∉ F^{i_{0}(1)} ⋃ F^{i_{0}(2)}, we have Y_{i_{0}} = F^{(i_{0}(1)} ⋁ F^{i_{0}(2)}, Y′_{i_{0}} = F^{i_{0}(1)} ⋁ F^{i_{0}(2)} ⋁{z}. Obviously, Y_{i_{0}} and Y′_{i_{0}} are compact subspaces of the space X. Then according to Graev’s theorem (see [9]), F M(Y_{i_{0}}) is topologically isomorphic to the group F G(Y′_{i_{0}}). Since Y′_{i_{0}} is a compact subspace of the K_{ω}-space, the subgroup G(Y′_{i_{0}}) of the group F G(X) (which is generated algebraically from the compact Y′_{i_{0}}) is closed in F G(X) and is topologically isomorphic to the free topological group F G(Y′_{i_{0}}). By the condition, in (F G(X), 𝔐) ≤ 0. So in (F G(Y′_{i_{0}}), 𝔐) ≤ 0 and thus in (F G(Y_{i_{0}}), 𝔐) ≤ 0.

Furthermore, Y_{i_{0}} is homeomorphic to the topological sum of the spaces F^{i_{0}(1)} and F^{i_{0}(2)}. According to Morris’ theorem (see [13]) the group F M(Y_{i_{0}}) is topologically isomorphic to the free product F M(F^{i_{0}(1)}) ∗ F M(F^{i_{0}(2)}) of topological groups F M(F^{i_{0}(1)}) and F M(F^{i_{0}(2)}). Thus in (F M(F^{i_{0}(1)}) ∗ F M(F^{i_{0}(2)}), 𝔐) ≤ 0. According to Ordman’s theorem (see [15]), in the sequence F M(F^{i_{0}(1)}) ×

F M(F^{i_{0}(2)}) →^{i} F M(F^{i_{0}(1)}) ∗ F M(F^{i_{0}(2)}) →^{ρ} F M(F^{i_{0}(1)}) × F M(F^{i_{0}(2)}), where

i is a homeomorphic inclusion of F M(F^{i_{0}(1)}) × F M(F^{i_{0}(2)}) into F M(F^{i_{0}(1)}) ∗ F M(F^{i_{0}(2)}) and ρ is a continuous mapping of F M(F^{i_{0}(1)}) ∗ F M(F^{i_{0}(2)}) onto F M(F^{i_{0}(1)}) × F M(F^{i_{0}(2)}), the composition iρ is the identical mapping. Hence i(F M(F^{i_{0}(1)}) × F M(F^{i_{0}(2)})) is closed in F M(F^{i_{0}(1)}) ∗ F M(F^{i_{0}(2)}). Therefore in (F M(F^{i_{0}(1)} × F M(F^{i_{0}(2)}), 𝔐) ≤ 0.

According to Theorem 3.3 in (F M(F^{i_{0}(1)}) × F M(F^{i_{0}(2)}), 𝔐) ≤ 0 is true if and only if one of the following conditions is fulfilled:

1. ind F M(F^{i_{0}(1)}) = 0, ind F M(F^{i_{0}(2)}) = 0;

2. F M(F^{i_{0}(1)}) ∈ loc 𝔐, F M(F^{i_{0}(2)}) ∈ loc 𝔐;

3. F M(F^{i_{0}(1)}) ∈ loc 𝔐, ind F M(F^{i_{0}(1)}) = 0 and in (F M(F^{i_{0}(2)}), 𝔐) ≤ 0;

4. F M(F^{i_{0}(2)}) ∈ loc 𝔐, ind F M(F^{i_{0}(2)}) = 0 and in (F M(F^{i_{0}(1)}), 𝔐) ≤ 0. As we know (see [11]), abelian free topological groups AM(F^{i_{0}(1)}) and

AM(F^{i_{0}(2)}) are topologically isomorphic to the factor groups F M(F^{i_{0}(1)})/K_{1} and F M(F^{i_{0}(2)})/K_{2}, where K_{1} and K_{2} are commutants. The mappings

π_{1} : F M(F^{i_{0}(1)}) → F M(F^{i_{0}(1)})/K_{1} = AM(F^{i_{0}(1)}),
π_{2} : F M(F^{i_{0}(2)}) → F M(F^{i_{0}(2)})/K_{2} = AM(F^{i_{0}(2)})

are open. If F M(F^{α_{0}(1)}) ∈ loc 𝔐 and F M(F^{α_{0}(2)}) ∈ loc 𝔐, then AM(F^{i_{0}(1)}) ∈ loc 𝔐 and AM(F^{i_{0}(2)}) ∈ loc 𝔐, respectively. But this is impossible because the space X is 𝔐-dispersed. Thus we exclude cases 2), 3) and 4). So we have ind F M(F^{i_{0}(1)}) = 0 and ind F M(F^{i_{0}(2)}) = 0. On the other hand, we have dim F M(F^{i_{0}(1)}) > 0 and dim F M(F^{i_{0}(2)}) > 0 (since dim F^{i_{0}(1)} > 0 and dim F^{i_{0}(2)} > 0). So, F M(F^{i_{0}(1)}) and F M(F^{i_{0}(2)}) are σ-compacts and ind F M(F^{i_{0}(1)}) > 0, ind F M(F^{i_{0}(2)}) > 0. This is the contradiction. Thus our assumption that ind X > 0 is not true. Therefore ind X ≤ 0. □

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(Received 20.11.2002; revised 14.04.2003)

Author’s address:

Department of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 0143 Georgia