MEASURES OF LINDELOF AND SEPARABILITY IN APPROACH SPACES

In this paper we introduce the notions of separability and Lindelof in approach spaces and investigate their behaviour under products and subspaces.

Approach spaces, as explained in detail in [6] provide us with a common supercategory of the categories TOP of topological spaces with continuous maps, and p-MET of extended metric spaces and non-expansive maps.The advantage of doing this was not only that Kuratowski's rather peculiar measure of non-compactness was thus put into a canonical setting, and that this setting allowed for a unified treatment of compactness for topological spaces and of total boundedness for metric spaces, but also that it was thus possible to prove some fundamental relationships between the measure of compactness of a family of spaces and their productspace.
In further study of approach spaces, and their application, especially to the analytical study of spaces of probability measures [7], [1], it turns out to be indispensable also to have at our disposal a mechanism to measure the deviation an approach space may have from being Lindelhf and from being separable.The purpose of this paper is to introduce such canonical measures and to study their basic properties.We pay particular attention to subspaces and products of metric spaces. 2 PRELIMINARIES.
We shall use the following symbols R+ := [0, oo[, R. :=]0, oo[ and 15.+ := [0, oo].If A C X then E)A stands for the function X 15,.+ taking the value 0 in points of A and oo elsewhere.We put an (respectively T for an increasing (respectively a strictly increasing) function, system, sequence or whatever.We shall also use the symbols and ] respectively for strict decreasing respectively decreasing functions, system, sequence or whatever.
We shall denote an approach system by ((z)),e x or shortly if no confusion is possible.If is an approach system then A := (A(z)),ex is called a basis or base for if it fulfils (B1) Vz X :A(z) is a basis for an ideal (B2) Yx X: (x)= h(x) where: h() := {1v, N R, q A(a:) + >_ A N}.
Further [6] if is an approach system on X then the map 8:Xx2 x +:(x,A) sup inf b(a) is a distance on X.From 8 a distance on X we can construct the approach system 6 defined by: ,(z) := {IVA c X inf (a) < 8(a, A)} a.A (1) for all x X.Further we have 6. and 8 =.8.A space with an approach system or a distance is called an approach space.
Approach spaces and contractions constitute a topological category [6] denoted AP.TOP is reflectively and coreflectively embedded in AP by: (X, T) (X, where the approach system of At(T)is (bT(z))=ex {ulv(z 0, u.s.c, at x} for all z E X. The associated distance is given by r(z,A) 0 iff z E and 6T(z,A) oo iff z ' /i for all z E X, A C X. Given (X, ) IAPI its TOP-coreflection is given by: (X, T'()) (X, ), where T'() is the topology determined by the neighborhood system" N'(@)(z) {{v < e}lv c ,l',(-),e E R.,z G X}.
T" is left inverse, right adjoint to A,.We say that the approach space has property P iff the topological bicoreflection of this space has the topological property P (e.g.compact, LindelSf, ...).Analogously p-q-MET is bicoreflectively embedded in AP by: p-q-MET AP (x,d) (x, where An(a) is determinded by the approach system (Oa(z)),x with Oa(z) {vlv d(z,.)} for all z X.In this ce the sociated distance is given by a(z,A) inf,a d(z,a) for all z X, A C X. Given the approach space X with approach system its p-q-MET-coreflection is given by" (X,M()) (X,) where M(#) is the p-q metric ad by M()(.,u) s,(, {u)).M is of course left inverse, right adjoint to A.. Approach spaces for which (X x 2x) {0, } are topological [6].
If A is a basis for the approach space (X, #) then: 5(z,A)'= sup inf @(a).
3 MEASURES OF SEPARABILITY AND LINDELOF.We now introduce the measures of separability and Lindelff: DEFINITION 3.I If (X, ) is art approach space then we define the measure of LindeIgf (respectively separability) of (X, ) as (respectively as L(X) := sup inf sup inf (z)(z).
(2) S(X) inf supgCz, A)). ( The following result is a straightforward exercise in topology: A topological space X is Lindelb'f iff for every family (V=)=x where V= is a neighborhood of z there ezists a countable set {z, ln N} such that: U,eN V(z,,) X.
This result is not true for the Lindelbf measure, as the following example shows.COUNTEREXAMPLE 3.4 Conszder d< R x R ft whereby d<(a, b) b a zJ" a _< b and d<(a,b) := oo ira > b.II is easy to see that thzs defines a p-q-MET space on R Its topologzcal bzcoreflectzon ,s the RHO topology [3].We now consider the product of this approach space with ztself.It ,s not hard to see that L(R x R) 0. However since a coreflectwn preserves products, zt follows from [3] that the space is not Lindel6f TItEOREM 3.5 For a topologtcal approach space X we have: Further: L(X), S(X) {0, oo}.
() (x, -) It is clear that we also have: where A(z)is a basis of ,I,(z).If we take: A(z)--{OvlVis an neighborhood of x inT} then it is clear that L(X) can only have the values 0 or oo.Because S(z, A) can only have the values 0 or oo it is clear that S(X) can only take these values too.To prove (a) we only have to apply 3.3, and (b) follows easily from 3.2 and expression (5).
In metric spaces we know that separability and Lindelgf coincide.We shall prove that the measures of separability and Lindel&f also coincide in p-MET LEMMA 3.6 For the p-q-MET space (X,d) we have: ,:fix yEA S(X) sup inf d(z,y) (7) =X yEA for a certain A 2 ((x)) Proof.It is clear from the definition that for any A E 2 ((x))" L(X) > sup inf d(y,z).

=EX yEA
Further it is clear that for each n E N we can choose a set A,, 2 ((x)) such that: sup inf d(y, x) > L(X) =E X yEA.
And thus" L(X) sup inf d(y,z) where A U,eN A,,.The equality for S(X) is shown in a similar way.true.COROLLARY 3.7 For a p-MET space (X,d) we have: L(X) S(X) 1.In countercxample 5.4 we see that for a general p-q-MET space the previous result is not COROLLARY 3.8 An eztended pseudometrtc space X is separable iff S(X)=L(X)=O.
It is easy to see that for contractions we have the following: THEOREM 3.9 For X,X' E IAPI and f" X X' a contraction (a) L(f(X)) <_ L(X) () s(f(x)) < s(x).
In topological spaces we can state the Lindel6f property by means of filters with the countable intersection property [3].We can do the same here for approach spaces.We shall put F(X) (respectively F,(X)) for the set of filters (respectively the set of filters with the countable intersection property) and analogously as for a filter we shall say that a base/ of a filter " E F(X) has the countable intersection property if for all (B,,),,eN 6 B" Obviously if a filter .T has a base with the countable intersection property then the filter Y itself has the countable intersection property.
We are now ready to prove the following theorem: THEOREM 3.10 For (X, 4) e lAP with base (h(z))xex we have: Proo L(X) sup inf a.T(z).
From the arbitrariness of e we deduce that: Second to prove the other inequality we first prove the'following assertion: Assertion: For every F Fw(X) and Y 2((x)): inf sup if (y)(z)_< sup inf (y)(z).

EX I/EY
I/Y FE.'EF Since this is true for every > 0 this proves the assertion.
In this section we shall discuss the relations between the measures of LindelSf and separability of a product space and its component spaces.REMARK 4.1 Since the projections are contractions it is clear from Theorem 3.9 that the measures of separability and Lindelb'f of the components are always less than or equal to the cor- responding measure for the product space.So we only have to prove one equality for each of the measures.

MEASURE OF LINDELSF.
Since the real line with the right half-open topology is LindelSf and the product with itself is not LindelSf [3], Theorem 3.5 allows us to conclude that the product of approach spaces can have measure of LindelSf equal to co while the measures of LindelSf for the components are 0. MEASURE OF SEPARABILITY.
For topological spaces [2] the product of separable spaces is separable iff the cardinality of the index set is less than or equal to the continuum.Along the same lines we obtain: THEOREM ,.For the approach spaces Xi IAPI, I where 1I < I2NI we have: s(IIx,) up s(z,).iI I Proof.Consider the set A, := {ai(n)ln e N} C Xi" sup sup inf i(a,(n)) < S(Xi) + e.
/EL, This theorem cannot be improved.Indeed take ]X] > ]2[ and consider the set N with the product topology.The set N is clearly a separable topological space but it is well known ( [9] example 103) that N is not separable.From Theorem 3.5 we now conclude that S(N)=0 but S(N) oo.
In this section we discuss some properties of the measures of separability and Lindel6f in p-q-MET spaces, especially for subsets and products.
THEOREM 5.1 For a p-MET space X and Y C X we have: (a) L(Y) <_ 2L(X) () s(Y) <_ s(x).
for a certain A e 2 ((x)) and we write A Y {b, ln < inN} and A 91 Y {c,,[n e N} [if one of the sets is empty or finite we adjust the indexset ].Suppose that there is a y Y and an m fi N: infeN d(b,y) > L(X) + l/m, then there exists a c,: d(,y) < L(X) + 1/m.Consider now the sets: B, {y Y[ inf d(b,y) > L(X) + 1/m and d(c,,y) < L(X) + 1/m}.
Because this is true for every m we have: (0) inf d(b,y) < 2L(X).Sometimes we can sharpen our inequalities, as the following theorem shows.THEOREM 5.For a p-MET space X with an ultrametric d and Y C X we have: (a) L(Y) < L(X) () s(r) '_< s(x).
Then L(Ro)= but L(R)= 1/2.So the foregoing results are not true for general p-(q)-MET" spaces.
For p-q-MET" spaces we have: PROPOSITION 6.1 Let (Xj)iE be an at most countable family of p-q-MET spaces.Then: L(1-I x) sup L(X). E.S Proof.The first inequality follows from Remark 4.1.We now prove the second inequality.
Put X := l]eJ X, and for each Xj we consider Aj countable such that: L(X) sup inf di(a,x).(12) z.X$ aqA Follows from Corollary 3.7 and (a).