SOME PROPERTIES OF TOPOLOGICAL SPACES RELATED TO THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY

In the paper the local density and the local weak density of topological spaces are investigated. It is proved that for stratifiable spaces the local density and the local weak density coincide, these cardinal numbers are preserved under open mappings, are inverse invariant of a class of closed irreducible mappings. Moreover, it is showed that the functor of probability measures of finite supports preserves the local density of compacts. AMS Subject Classification: 54A25, 54B20


Introduction
The local properties play an important role in general topology. For instance, compactness in R n is equivalent to total boundedness and closedness. In the case of local compactness boundedness is not necessary. In a case of locality some properties can be lost or some new properties may appear. For example, an open subspace of a compact space may often be non-compact. But any open subspace of a locally compact space is locally compact.
In the paper we consider the local density and the local weak density of topological spaces. We investigate what properties are preserved in a local cases or what properties may appear.
It is known that the density (the weak density) is preserved under continuous onto mappings, is hereditary with respect to F σ -sets and closed domains. The continuum product of separable (weakly separable) spaces is separable (weakly separable, respectively), any compact extension of a weakly separable space is weakly separable.
It turns out that the local density (the local weak density) is not preserved under a continuous mapping (example 2.1), a compact extension of a locally separable (locally weakly separable) space can be not locally separable (locally weakly separable, respectively) (example 2.2), the product of infinitely many locally separable (locally weakly separable) spaces is not always locally separable (respectively, locally weakly separable) (example 2.3).
Recall some definitions and propositions related to the work.
The density of a topological space is defined with following way: If d(X) ≤ ℵ 0 for a space X we say that X is separable. Definition 1.2. (see [2]) 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 ∈ X}, 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.
It is clear that every metrizable space is stratifiable.

Main Results
Definition 2.1. We say that a topological space X is locally τ -dense at a point x ∈ X if τ is the smallest cardinal number such that x has a τ -dense neighborhood in X.
The local density at a point x is denoted by ld(x). The local density of a space X is defined as the supremum of all numbers ld(x) for x ∈ X; this cardinal number is denoted by ld(X).
If ld(X) ≤ ℵ 0 for a space X, we say that X is locally separable.
Definition 2.2. A topological space X is locally weakly τ dense at a point x ∈ X if τ is the smallest cardinal number such that x has a neighborhood of weak density τ in X.
The local weak density at a point x is denoted by lwd(x). The local weak density of a topological space X is defined with following way: If lwd(X) ≤ ℵ 0 for a space X, then we say that X is locally weakly separable [4].
From the work [4] it is easy to obtain the followings 3) Linear ordered.
Theorem 2.2. Let X be a locally weakly τ -dense space and let G be its subspace satisfying one of the following conditions: Proof. 1) It is clear that the inequality lwd(X) ≤ ld(X) holds for any topological T 1 -space X.
2) We shall show ld(X) ≤ lwd(X). Let lwd(X) = τ . Then for any point x ∈ X there exists a neighborhood Ox of x such that wd(Ox) = τ. Since X is stratifiable, by theorem 1.2 the subspace Ox is also stratifiable. Then by theorem 1.2 we have d(Ox) = wd(Ox) = τ. For an arbitrary point x ∈ X we have found the neighborhood Ox of density τ . This means that ld(X) ≤ τ = lwd(X). Theorem 2.3 is proved.
Remark 2.1. The following example shows that in proposition 1.1 openness of a mapping f is important. There is a continuous mapping f which does not preserve the local weak density of topological spaces.
Example 2.1. Let (R, τ 1 ) be the real line with the discrete topology and let (R, τ 2 ) be the real line with the natural topology. Consider identity mapping from Since (R, τ 1 ) is discrete space, every mapping, in particular, id is continuous. Moreover, each point x ∈ R has one-point neighborhood {x}. This implies Theorem 2.4. Let f be a perfect mapping of the space X onto the space Y . Then: . The set f (X\G) is closed as the image of a closed set under a closed mapping. Hence, the set Y \f (X\G) is an open set containing y.
As it was shown in 1), Y \f (X\G) is an open set containing y and the inclusion Y \f (X\G) ⊂ f (G) holds. Since d(G) ≤ τ , by theorem 1.4 we see that d(f (G)) ≤ τ . Then by proposition 1.1 we have d(Y \f (X\G)) ≤ τ and we have found the neighborhood of density ≤ τ for an arbitrary chosen point y ∈ Y . Therefore ld(Y ) ≤ τ . The inequality ld(Y ) ≤ ld(X) is proved. Theorem 2.4 is proved.
Definition 2.4. (see [5]) A continuous mapping f : Theorem 2.5. Let f be a closed and irreducible mapping of a space X onto a space Y . Then: Indeed Hence wd(f −1 (Of (x))) ≤ τ . Since the point x was chosen arbitrarily, we obtain lwd(X) ≤ τ .
2) Let us prove now ld(X) ≤ ld(Y ). Let ld(Y ) = τ ≥ ℵ 0 . We have to show ld(X) ≤ τ . Take an arbitrary point x ∈ X, then f (x) ∈ Y . Since ld(Y ) = τ , there exists a neighborhood Of (x) of f (x) in Y such that d(Of (x)) ≤ τ . This means that there is a dense subset M = {y α : α ∈ A} of the subspace Of (x) such that |A| ≤ τ . Let us choose a point x α from each set f −1 (y α ) and form the set M 1 = {x α : α ∈ A}. It is clear that |M 1 | = τ . Let us show that M 1 is dense in f −1 (Of (x)).
Let G be an arbitrary nonempty subset of f −1 (Of (x)). Then f (G) ⊂ f (Of (x)). Since f is closed and irreducible, the set f (X\G) is closed in Y and f (X\G) = Y . As it was shown in 1), Y \f (X\G) is a nonempty open set of the space Y and . Therefore d(f −1 (Of (x))) ≤ τ and -the point x being arbitrary -ld(X) ≤ τ . Theorem 2.5 is proved.
From Theorems 2.4 and 2.5 we directly obtain the following theorem.
Theorem 2.6. Let f be a perfect and irreducible mapping of a space X onto a space Y . Then lwd(Y ) = lwd(X) and ld(X) = ld(Y ).
Let X be a compact space. By C(X) denote the set of all continuous maps φ : X → R with the usual sup-norm φ = sup{|φ(x)| : x ∈ X}. A continuous functional µ : C(X) → R is called a measure on the compact X. A measure is positive (notation µ ≥ 0), if µ(φ) ≥ 0 for any φ ≥ 0. A measure is normed, if µ = 1. 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 where φ 1 , φ 2 , ..., φ k ∈ C(X) and ε > 0.
Proof. Let X be an infinite compact space and let ld(X) = τ ≥ ℵ 0 . Then ld(X n ) = τ by proposition 2.2. In [7] V.Basmanov showed that the space P n (X) can be represented as a continuous image of the space X × σ n−1 , where σ n−1 is the (n − 1)-dimensional simplex. The mapping π : X × σ n−1 → P n (X) is defined by the formula π(x 1 , ..., 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 each i ∈ N , δ x i are Dirak's measures at points x i respectively. The mapping π is perfect, since π is continuous mapping defined on compact X ×σ n−1 . Then by theorem 2.4 we have ld(P n (X)) ≤ τ . Theorem 2.7 is proved.
Let us recall now P.S. Alexandroff's one-point compactification. Let X be an uncountable set and ξ is a point which does not belong to X. Define a topology on the set X ∪ {ξ} with a base. The family of all the one-point subsets of X and all the sets of the form X\A, where A is a finite subset of X, forms a base of topology on X ∪ {ξ}. This space is denoted by αX. Clearly, the space αX is compact and Hausdorff.
Remark 2.1. There exist a topological space Y and its dense subspace X such that ld(X) < ld(Y ) and lwd(X) < lwd(Y ).
Example 2.2. Let X be the discrete space of cardinality τ = c (continuum). It is clear that ld(X) = 1 < ℵ 0 (ℵ 0 is the infinite countable cardinal). Consider P.S. Alexandroff's one-point compactification of X. This compactification denote by Y . Then Y is compact and ld(Y ) ≥ τ .
Theorem 2.8. (see [2]) Let X be a topological T 1 -space. If wd(X s ) ≤ τ for every s ∈ S and |S| ≤ 2 τ , then wd( s∈S X s ) ≤ τ . Theorem 2.9. Let τ be an infinite cardinal number. Consider a family of topological spaces {X s : s ∈ S}, where |S| ≤ 2 τ . If every space X s is locally weakly τ -dense and there exists a finite subset S 0 of the index set S such that X s is weakly τ -dense for all s ∈ S\S 0 , then the product s∈S X s is locally weakly τ -dense.
Proof. Take an arbitrary point x = {x s : s ∈ S} from the product s∈S X s . Since all the spaces X s are locally weakly τ -dense, the point x s ∈ X s has a neighborhood U s of weak density ≤ τ for every s ∈ S 0 . The set It seems, the inverse statement is also true Theorem 2.10. Suppose that the product s∈S X s is locally weakly τ -dense.
Then all the spaces X s are locally weakly separable, moreover, there exists a finite subset S 0 of the index set S such that X s is weakly τ -dense for every s ∈ S\S 0 .
Proof. Since all the projection p s : s∈S X s → X s are open, from proposition 2.1 it directly follows that all the spaces X s are locally weakly τ -dense. Let us now prove the second statement of the theorem. Take an arbitrary point x = (x s ) s∈S from the product s∈S X s . Since the product s∈S X s is locally weakly τ -dense, the point x has a weakly τ -dense neighborhood s∈S 0 U s × s∈S\S 0 X s (since the weak density is hereditary with respect to open subsets, we can assume that the neighborhood is from the canonical base of s∈S X s ), where S 0 is a finite subset of S such that x s ∈ U s for s ∈ S 0 , x s ∈ X s for s ∈ S \ S 0 . Since the weak density is preserved under open mappings and the projections are open maps, we see that X s is weakly τ -dense for every s ∈ S\S 0 . Theorem 2.10 is proved.
Note that in the inverse statement the condition |S| ≤ 2 τ is omitted. Then all the spaces X s are locally separable, moreover, there exists a finite subset S 0 of S such that X s is separable for every s ∈ S\S 0 . Remark 2.2. The following example shows that the local separability and the local weak separability are not preserved under the infinite product. Example 2.3. Let X be the discrete space of cardinality c. It is obvious that ld(X) = lwd(X) = 1 < ℵ 0 , i.e. it is locally separable and locally weakly separable. Now let us consider the product X N of countably many copies of the space X. Take an arbitrary point x = (x i ) i∈N ∈ X N and consider it arbitrary canonical neighborhood Ox = {x i 1 } × ... × {x i k } × X N ′ , where N ′ = N \{i 1 , i 2 , ..., i k }. It is clear that Ox is homeorphic to X N . This implies that d(Ox) = d(X N ) = c and wd(Ox) = wd(X N ) = c. We obtain the fact that the point x has neither separable nor weakly separable neighborhood. Therefore ld(X N ) > ℵ 0 and lwd(X N ) > ℵ 0 , i.e. the product of countably many copies of the space X is neither locally separable, nor locally weakly separable.