On Non-increasing of the Density and the Weak Density under Weakly Normal Functors of Finite Support

In the paper it is proved that if a covariant functor F : Comp → Comp is weakly normal, then for any infinite Tychonoff space X following inequalities hold: d(F β n (X)) ≤ d(X), d(F β ω (X)) ≤ d(X), wd(F β n (X)) ≤ wd(X), wd(F β ω (X)) ≤ wd(X).


Introduction
Let X be a T 1 -space. The set of all nonempty closed subsets of a space X denote by exp X. The family of all sets in the form , where U 1 , ..., U n is a sequence of open sets in X, generates a topology on the set exp X. This topology is called the Vietoris topology. The set exp X with the Vietoris topology is called the exponential space or the hyperspace of X [1]. Denote by exp n X the family of all nonempty finite subsets of a space X, consisting of at most n elements, i.e. exp n X = { F ∈ exp X : | F | ≤ n }. Denote by exp ω X the family of all finite subsets of X. It is easy to see that exp ω X = ∪ { exp n X : n = 1, 2, ...n, ...}. Denote by exp c X the family of all closed compact subsets of X. In [1] it is shown that the functor exp : Comp → Comp is normal. In the work [2] it is proved that for any hold.
It is known from [3] that for any T 1 -space X wd (X) = wd (exp n X) = wd (exp ω X) = wd (exp c X) = wd (exp X) hold. From above mentioned statements it is natural to put following 1.1. Question. For which normal functors F : Comp → Comp following equalities hold: where F β n , F β ω , F β are functors in the category of all Tychonoff spaces and their continuous mappings.

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On Non-increasing of the Density and the Weak Density under Weakly Normal Functors of Finite Support In the current work preserving of the density and the weak density by the functor of probability measures of finite support, is investigated.
Recall some necessary definitions. A covariant functor F : Comp → Comp is normal, if it is continuous, preserves the weight, intersections and pre-images, is monomorphic, is epimorphic, transforms a one-point set and empty set to a one-point and empty set, respectively [1].
A covariant functor F : Comp → Comp is weakly normal, if it satisfies all conditions of normality except of preserving pre-images [4].
A functor L : T ych → T ych is normal [5], if it is continuous, preserves the weight, embeddings, intersections, singletons, the empty set and translates k-covering mappings to surjections.
In the work [5] A.Ch.Chigogidze proved that if a functor F : Comp → Comp defined in the category of compacts and their continuous mappings, is normal, then it can be extended over the category of all Tychonoff spaces and their continuous mappings, preserving its normality. It is clear that for every normal functor L : T ych → T ych, every X ∈ Ob(T ych) and each point x ∈ L(X) there exists (by property of preserving intersections) smallest subspace A ⊆ X such that x ∈ L(A). The closure of this space in X, is called the support of a point x, the set itself is the kernel of the support. A functor L is a functor of finite support, if for any X ∈ Ob(T ych) and for any point x ∈ L(X) the support of x is compact.
A continuous mapping f : X → Y is a k-covering mapping, if for any compact B ⊂ Y there is a compact set A ⊂ X such that f (A) = B. Every perfect mapping is k-covering [6].
Let F : Comp → Comp be an arbitrary normal functor and let X ∈ Ob(T ych).
It is easy to see that F β is a covariant functor in the category T ych.
By P n denote the functor assigning to a compact space X the set of all elements a ∈ P (X), the support of which consists of at most n points.
Let X ∈ Comp. By C(X) denote the set of all continuous mappings φ : X → R of a space Xwith the usual sup-norm: For each c ∈ R by c X denote the constant function defined by the formula c For a compact X by O(X) denote the set of all weakly additive order-preserving normed functionals. Elements of the set O(X), are shortly called weakly additive functionals. Note that each functional ν ∈ O(X) is a continuous mapping of C(X) to R, i.e. the set O(X) is a subset of C p (C(X)). The set is equipped with the pointwise topology. Sets in the form In [4] T.Radul proved that the covariant functor O : Comp → Comp of weakly additive order-preserving normed functionals in the category of compacts, satisfies all conditions of normality except pre-image preserving.
A positive normed measure is called a probability measure. A space consisting of all probability measures, denote by P (X). A neighborhood base at a point µ ∈ P (X) consists of all the sets in the form A support supp(µ) of a measure µ ∈ P (X) is the smallest closed subset F ⊂ X such that µ(F ) = µ(X). For a compact X and a natural number n put P n (X) = {µ ∈ P (X) : |supp(µ)| ≤ n} [1].

Main Results
2. The density and the weak density of the space of probability measures of finite support in the category of Tychonoff spaces.
In this chapter we shall prove that the functor of probability measures of finite support preserves the density and the weak density in the category of compacts.
Proof. Let's prove first d(P n (X)) ≤ d(X). Suppose X is an infinite compact and d(X) = τ ≥ ℵ 0 . It is clear that d(X n ) = τ for every n ∈ N . By Basmanov's theorem ( [7]) we see that the space P n (X) can be represented as the continuous image of the product X n × σ n−1 , where σ n−1 is (n − 1)-dimensional simplex. The mapping π : X n × σ n−1 → P n (X) is defined by the formula: ..., x n , m 1 , ..., m n where (m 1 , ..., m n ) ∈ σ n−1 , ∑ n i=1 m i = 1 and m i ≥ 0 for every i ∈ N , δ xi are Dirac measures at points x i , respectively. Hence, d(X n ×σ n−1 ) ≤ τ and from the fact that the density of the space is preserved under continuous mappings, we obtain d(P n (X)) ≤ τ .
Let us now prove that d(X) ≤ d(P n (X)). Let d(P n (X)) = τ ≥ ℵ 0 . Then there is a dense subset Ω = {µ α : a ∈ A} of P n (X) such that |A| = τ . We construct the subset M of X with the following way: Suppose it doesn't hold, i.e. there is a point x 0 and its neighborhood Ox 0 such that Consider the neighborhood O(δ x0 ; φ; 0, 5) of the point δ x0 in P n (X).
Since Ω is dense in P n (X), there exists 5. This contradiction to density of the set Ω in P n (X) shows that the set M is dense in X, i.e. d(X) ≤ τ . Theorem 2.1 is proved.
Let us now prove that d(X) ≤ d(P ω (X)).Suppose d(P ω (X)) = τ ≥ ℵ 0 . Then there exists a dense subset Ω = {µ α : a ∈ A} of P ω (X), where |A| = τ . Let us construct the subset M of X with the following way: It is obvious that |M | = τ . We shall show that M is dense in X. Suppose that M is not dense in X. Then there is a point x 0 and its neighborhood Ox 0 such that M Consider the neighborhood O(δ x0 ; ϕ; 1 4 ) of the point δ x0 in P ω (X).
Since Ω is dense in P ω (X), there is the intersection is not empty, i.e. there is an element ν ∈ Ω ∩ O(δ x0 ; ϕ; 1 4 ). Let supp(ν) = {x 1 , x 2 , ..., x n }. Then This contradicts density of the set Ω in P ω (X). Hence the set M is dense in X. This implies d(X) ≤ τ . Theorem 2.2 is proved. From theorems 2.1 and 2.2 we obtain the following Corollary 2.1. For any infinite compact X and any n ∈ N we have d(X) = d(P n (X)) = d(P ω (X)).
3. Density and weak density of the space of probability spaces in the category of Tychonoff spaces.
In this chapter preserving of the density and the weak density by weakly normal functors of finite support, is investigated.
We need following definition and theorems.

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On Non-increasing of the Density and the Weak Density under Weakly Normal Functors of Finite Support Definition 3.1 [9]. The weak density of a topological space X is the smallest cardinal number τ ≥ ℵ 0 such that there is a π-base in X coinciding with τ centered systems of open sets, i.e. there is a π-base B = ∪ {B α : α ∈ A}, where B α is a centered system of open sets for each α ∈ A and |A| = τ .
The weak density of a topological space X is denoted by wd(X). If wd (X) = ℵ 0 then we say that a topological space X is weakly separable [10].
Theorem 3.2 [8]. If Y is dense in X, then wd(Y ) = wd(X). Theorem 3.3 [11]. Let P be a functor of probability measures. Then P ω (X) is dense in the space P (X) for any compact X.
The following proposition can be easily obtained from [12]. Proposition 3.1. Let X be an infinite Tychonoff space, bX its arbitrary compact extension. Then for any natural number n ∈ N the space P n (X) is dense in the space P n (bX).
Proposition 3.2. Suppose that X and Y are Tychonoff spaces such that X is a dense subspace of Y . Then P β ω (X) is dense subspace of P β ω (Y ). Proposition 3.3. Let X be an infinite Tychonoff space, bX its arbitrary compact extension. Then P β ω (X) is a dense subspace of P (bX).
Proposition 3.4. Let X be an infinite Tychonoff space. If X is a dense subspace of a Tychonoff space Y , then Proof. Assume that X is a dense subspace of a Tychonoff space Y . Then X is dense in βY . By proposition 3.3 the set P β ω (X) is dense in the compact P (βY ).
Proof. Let X be an infinite Tychonoff space and let wd (X) = τ ≥ ℵ 0 . Then by theorem 3.2 [8] for an arbitrary compact extension bX of X we have wd (X) = d (bX) = τ . By corollary 2.1 we obtain Since spaces bX and P n (bX) are compact, by propositions 3.1 and 3.2 we see that the space P n (X) is dense in P n (bX), the space P β ω (X) is dense in P β ω (bX). Therefore, P β ω (X) is dense in P β (X). It is known that the weak density is hereditary with respect to open subspaces and compact extensions. Therefore, from theorems 4 [9] and 2.3 [10] it follows wd (X) = wd (P n (X)) = wd(P β ω (X)) = wd(P β (X)) = τ. Theorem 3.4 is proved.
Theorem 3.5. If a covariant functor F : Comp → Comp is weakly normal, then for an arbitrary infinite Tychonoff space X we have Proof. 1) Let Y be a dense subspace of a Tychonoff space X such that |Y | = τ = d (X). We have |exp n Y | = τ , because Y is infinite. Since F preserves weight, for any finite compact F (Z) = F β (Z) is metrizable, and therefore, is separable. For each Z ∈ exp n Y fix a countable dense subset D Z of F (Z). Since all sets D Z are countable, . Let bX be an arbitrary compact extension of the space X. By lemma 2.1 [8] the set F β n (Y ) is dense in the set F β n (X). Consequently, the set D is dense in F β n (X). Therefore d(F β n (X)) ≤ d(X). 2) Let Y be a dense subset of a Tychonoff space X such that |Y | = τ = d (X). Since Y is infinite, we have |exp ω Y | = τ . By the fact that F preserves weight, for any finite Z the compact F (Z) = F β (Z) is metrizable, and therefore, is separable. For each Z ∈ exp ω Y fix a countable dense subset D Z of F (Z). Since all sets D Z are countable, we have . Let bX be an arbitrary compact extension of the space X. By proposition 2.1 [8] the set F β ω (Y ) is dense in the set F β ω (X). Consequently, the set D is dense in F β ω (X). Hence d(F β ω (X)) ≤ d(X). Theorem 3.5 is proved. Corollary 3.1. If a covariant functor F : Comp → Comp is weakly normal, then for any infinite Tychonoff space X we have Corollary 3.2. If a covariant functor F : Comp → Comp is weakly normal, then for any infinite separable space X spaces F β n (X), F β ω (X), F β (X) are separable too.

Conclusion
In the future works, we are planning to investigate cardinal invariants and functors preserving them. Now, we are trying to give answers to the following open questions: Does the functor F preserve the tightness, the density, the weak density, the local density and the local weak density, where F can be the exponential functor exp, the functor of superextension λ, the functor of probability measures P , the functor of weakly additive functionals O and the functor of semi-additive functionals OS?