The Openness of Certain Subfunctors of the Probability Measure Functor and the Topological Properties of Spaces of the Form F(X)\ŋF(X)

It is known that the functor P probability measures is an open functor of compacts and continuous maps into itself acting in the category comp [1]. In this note we show some subfunctors of the functor P of probability measures also being open functors. This means that these functors translate open mappings between compacts into open mappings. On the other hand, it is known that for any infinite compactum X space P(X) homeomorphism to a Hubert cube Q. The question naturally arises in what cases from the homeomorphism of the spaces F(X) and F(Y) implies the homeomorphism of compact sets X and Y, for normal functors F: Comp→Comp. And also in this note it is shown that for a functor Pf: Comp→Comp of homeomorphism Pf(X)\δ(X) and Pf(Y)\δ(Y)implies homeomorphism X and Y. It is further shown that for a compactum hereditary normality of space is equivalent to metrizability.


Introduction
It is known that the functor P probability measures is an open functor of compacts and continuous maps into itself acting in the category comp [1]. In this note we show some subfunctors of the functor P of probability measures also being open functors. This means that these functors translate open mappings between compacts into open mappings. On the other hand, it is known that for any infinite compactum X space P(X) homeomorphism to a Hubert cube Q. The question naturally arises in what cases from the homeomorphism of the spaces F(X) and F(Y) implies the homeomorphism of compact sets X and Y, for normal functors F: Comp→Comp. And also in this note it is shown that for a functor P f : Comp→Comp of homeomorphism P f (X)\δ(X) and P f (Y)\δ(Y)implies homeomorphism X and Y. It is further shown that for a compactum hereditary normality of space is equivalent to metrizability.

Preliminaries
We recall the definition and some properties of the normality of the covariant functor F: Comp→Comp acting in the category of compacta. We say that the functor F: , defined by the equality F(i) (α)=F(i α )(α), is an embedding for every ; Saves the preimages if for every map f: X→Y and every closed set Preserves the weight if ω(F(X))=ω(X) for an infinite bicompactum X; 7. It is continuous if for any inverse spectrum S={X α ;π α β :α€A} of bicompacta, the homeomorphism is the map f:F(lim S)→ limF(S), which has the limit of the maps F(π α ), if π α :limS→X α end-to-end projections of the spectrum.
In what follows we assume that all the functors under consideration are monomorphic and preserve intersections. We also assume that all functors preserve non-empty spaces. This restriction is irrelevant, since by this we exclude from consideration only the empty functor, i.e. the functor F, which takes every space into an empty set. In fact, let F(X)=Ø for some nonempty bicompactum X.
Then F(X)=F(1)=Ø by the monomorphism of F. Now let Y-be an arbitrary non-empty bicompactum. Consider the constant mapping f:Y→1. Then ( ) ( ) Consequently, the space F(Y) is empty, since it is mapped to an empty set. Thus, we have proved that there exists a unique monomorphic functor preserving non-empty sets.
Let F:Comp→Comp be a functor. We denote by C(X,Y) the space of continuous mappings from X and Y in a bicompact-open topology. In particular({K},Y) is naturally homeomorphism to the k-power of For the functor F, the bicompactum X of the natural number K, we define the map π F,X,k :C({k},X)×F({k})→F(X) by the equality When it is clear which functor and which bicompactum Y we are talking about, we denote the map π F,X,k by π F,k or π k .
By the Shchepin theorem [2], the map F:C(Z,Y)→F(F(Z),F(Y) is continuous for every continuous functor F and bicompacts Z and Y. Therefore takes place.

Proposition 1
For a continuous functor F, a bicompactum X, and a natural number k, the mapping F πF,X,k is continuous [3].
We define the subfunctor F k (X) of the functor F in the following way: for the compact space X, the space F k (X) is the image of the map π F,X,k , and for the mapping f:X→Y the map F k (f) is the restriction of the map F(f) to F k (X). From the easily verifiable commutatively *Corresponding author: Jumayev EE, Professor, Department of Mathematics, National Academy of Sciences of Ukraine, Ukraine, Tel: 215841282; E-mail: and, therefore, the functoriality of the construction F k . A functor F is called a functor of degree n if F n (X)=F(X) for every bicompact X, but F n-1 (X)≠F(X) for some X.

The Main Part
For compacta X,P(X) denotes spaces of probability measures. It is known that for an infinite compactum X, this space P(X) is homeomorphism to a Hilbert cube Q [4]. For a natural number n, denotes the set of all probability measures with at most n supports.  for some i [4,5].
For a natural number n, we put ,

Definition 1
A seminormal functor is called retroactively stable if for any compactum the subspace is a retract for the compactum. those. there exists a continuous retraction

Proposition 1
The mapping f:X→Y is a coretraction, if and only if there exists a multiplicative extension operator for f.

Proposition 1
A semi-normal functor F:Comp→Comp retroactively ŋ F is stable if and only if the embedding is a correction for any X Comp ∈ .
Obviously, for convex compact sets the functor P of probability measures is retractively stable [5]. hence, AR-compacta are retractively stable for any seminormal functors F. It was shown [4,6] that the subfunctors P f ,P f,n ,P C f,n and P C f of the functor P of probability measures are retractively stable. It follows from the definition of retractively stable functors that the retraction ( ) ( ) is closed and perfect.

Proposition 3
If X is contained in Y, then the Banach space C(X) admits a linear and multiplicative extension operator in C(Y) if and only if X is a retract of the space Y [7].

Corollary 1
For any retractively ŋ F of a stable functor F:Comp→Comp space ŋ F (X) C-is embedded in the space F(X).
If X is a metrizable compactum, then X n ×F(n) is also a metrizable compactum, and the map π F,X,n :X n ×F(n)→F(X) is perfect. Hence F(X) is metrizable, where F is a retractive ŋ F stable functor of degree ≤n. Using the reduced properties of retractively ŋ F stable functors of finite degree ŋ F and properties of perfect mappings [8]. we can assert.

Theorem 1
For the compactum X and retractively ŋ F of stable functors F of degree ≤n the following conditions are equivalent: 1) X is metrizable;

Corollary 2
For the functors F=Pf,Pf,n,PCf,n and P C f,n the following conditions are equivalent: 1) X is metrizable;

2) F(X) is metrizable
Let Q-be a topological property. We say that the space X has the property outside Q the set A, in the space X if the space X\A has the property Q, where , A X A ⊂ ≠ ∅ . It is known that the normal Δ of the compactum X normal outside the diagonal satisfies the first axiom of countability [9].

Theorem 2
For compact subsets of X and Y, the spaces P f (X) and P f (Y) are homeomorphism, respectively, outside the sets δ(X) and δ(Y) if and only if δ(X) and δ(Y) are homeomorphism.
Evidence. Let X and Y be compact sets such that h:P f (X)\δ(X)→Pf(Y)\δ(Y) we denote by h this homeomorphism h:P f (X)\δ(X)→Pf(Y)\δ(Y). Now we establish a homeomorphism It is known that for any ( ) x X δ δ ∈ the preimage of (r X f ) -1 (δ x ) contains the point δ x . The homeomorphism h maps the set (r X f ) -1 (δ x )\δ x to some set Similarly, as Theorem 2, we prove the following for the functors F=P f,n ,P C f,n and P C f .

Theorem 3
For compacta X and Y, the spaces F(X) and F(Y) are homeomorphism, respectively, outside the sets ŋ F (X) and ŋ F (Y) if and only if ŋ F (X) and ŋ F (Y) are homeomorphism.
The following is given [5]. Let X and Y be openly generated compacta without points of countable character, and h:P n (X)→P n (Y) homeomorphism. Then h(P k (X))=P k (Y) for any natural k<n, and a quotient, X homeomorphism of Y.
Theorem 2 implies the following, which is a generalization of Theorem [5].

Corollary 3
Let X and Y be infinite compacta, and let h:P f,n (X)→P f,n (Y) homeomorphism. Then h(P f,k (X))=P f,k (Y) for any natural k<n, in particular, the X homeomorphism Y.

Recall that Y X
⊂ is a C-embedded in X if every continuous real function defined on Y extends to a continuous function on X [7].

Theorem 5
Let F:Comp→Comp be the normal functor of the ( ) AR M space in ( ) AR M space. Then ŋ F C-is embedded in F(X) for any X Comp