aa r X i v : . [ m a t h . GN ] J u l MORE ON THE RINGS B ( X ) AND B ∗ ( X ) A. DEB RAY AND ATANU MONDAL
Abstract.
This paper focuses mainly on the bounded Baire one functions.The uniform norm topology arises from the sup-norm defined on the collection B ∗ ( X ) of all bounded Baire one functions. With respect to this topology, B ∗ ( X ) is a topological ring as well as a topological vector space. It is provedthat under uniform norm topology, the set of all positive units (respectively,negative units) form an open set and as a consequence of it, every maximalideal is closed in B ∗ ( X ). Since the natural extension of uniform norm topologyon B ( X ), when B ∗ ( X ) = B ( X ), does not show up these features, a topologycalled m B -topology is defined on B ( X ) suitably to achieve these results on B ( X ). It is proved that the relative m B topology coincides with the uniformnorm topology on B ∗ ( X ) if and only if B ( X ) = B ∗ ( X ). Moreover, B ( X )with m B -topology is 1st countable if and only if B ( X ) = B ∗ ( X ).The last part of the paper establishes a correspondence between the ideals of B ∗ ( X ) and a special class of Z B -filters, called e B -filters on a normal topologicalspace X . It is also observed that for normal spaces, the cardinality of thecollection of all maximal ideals of B ( X ) and those of B ∗ ( X ) are same. Introduction and Preliminaries
In [1], we have initiated the study of the ring B ( X ) of Baire one functions and haveachieved several interesting results. The bounded Baire one functions, denoted by B ∗ ( X ) is in general a subring of B ( X ), for any topological space X . It has beenproved in [1] that for a completely Hausdorff space X , total disconnectedness of X is a necessary condition for B ∗ ( X ) = B ( X ), however, the converse is not alwaystrue. It is therefore a question, when do we expect the converse to hold? In thispaper, introducing a suitable topology on B ( X ), called m B -topology, we establisha couple of necessary and sufficient conditions for B ∗ ( X ) = B ( X ). We observethat ( B ( X ) , m B ) is a 1st countable space if and only if B ( X ) = B ∗ ( X ), which isan analogue of the result proved in [4] for pseudocompact spaces.Defining zero sets of Baire one functions in the usual way, in [2], we have establishedthe duality between the ideals of B ( X ) with a collection of typical subfamilies ofthe zero sets of Baire one functions, called Z B -filters on X . The study of such corre-spondence has resemblance to the duality between ideals of C ( X ) and the Z -filterson X . It is of course a natural query whether such correspondence remains true ifwe confine the ideals within the class of ideals of B ∗ ( X ). It is not hard to observethat for any Z B -filter F on X , Z − B [ F ] T B ∗ ( X ) is an ideal in B ∗ ( X ). However, Z B fails to take an ideal in B ∗ ( X ) to a Z B -filter on X . Introducing a new type of Z B -filter, called e B -filter on X , we obtain in this paper a similar correspondence Mathematics Subject Classification.
Key words and phrases.
Uniform norm topology, u B -topology, m B -topology, Z B -filter, Z B -ultrafilter, e B -ideal, e B -filter, e B -ultrafilter. between the ideals of B ∗ ( X ) and the e B -filters on any normal topological space X .We now state some existing definitions and results that are required for this paper.The zero set Z ( f ) of a function f ∈ B ( X ) is defined by Z ( f ) = { x ∈ X : f ( x ) = 0 } and the collection of all zero sets in B ( X ) is denoted by Z ( B ( X )).It is evident that, Z ( f ) = Z ( | f | ), for all f ∈ B ( X ).For any real number r , r ∈ B ( X ) (cid:0) orB ∗ ( X ) (cid:1) will always indicate the real valuedconstant function defined on X whose value is r . So Z (0) = X and Z ( s ) = ∅ , forany s = 0.If a Baire one function f on X is a unit in the ring B ( X ) then { x ∈ X : f ( x ) =0 } = ∅ . The following result shows that in case of a normal space, this condition isalso sufficient. Theorem 1.1. [1]
For a normal space X , f ∈ B ( X ) is a unit in B ( X ) if andonly if Z ( f ) = { x ∈ X : f ( x ) = 0 } = ∅ . The following result provides a sufficient criterion to determine units in B ( X ),where X is any topological space. Theorem 1.2. [1]
Let X be a topological space and f ∈ B ( X ) be such that f ( x ) > orf ( x ) < , ∀ x ∈ X , then f exists and belongs to B ( X ) . Clearly, if a bounded Baire one function f on X is bounded away from zero, i.e.there exists some m ∈ R such that 0 < m ≤ f ( x ) , ∀ x ∈ X then f is also a boundedBaire one function on X . Theorem 1.3. [6] (i)
For any topological space X and any metric space Y , B ( X, Y ) ⊆ F σ ( X, Y ) , where B ( X, Y ) denotes the collection of Baire one functions from X to Y and F σ ( X, Y ) = { f : X → Y : f − ( G ) is an F σ set, for any open set G ⊆ Y } . (ii) For a normal topological space X , B ( X, R ) = F σ ( X, R ) . Theorem 1.4. [1]
For any f ∈ B ( X ) , Z ( f ) is a G δ set. The following results are the correspondences between the ring structure of B ( X )and the topological structure of X . Definition 1.5.
A nonempty subcollection F of Z ( B ( X )) is said to be a Z B -filteron X , if it satisfies the following conditions:(1) ∅ / ∈ F (2) if Z , Z ∈ F , then Z ∩ Z ∈ F (3) if Z ∈ F and Z ′ ∈ Z ( B ( X )) is such that Z ⊆ Z ′ , then Z ′ ∈ F . Definition 1.6. A Z B -filter is called Z B -ultrafilter on X , if it is not properlycontained in any other Z B -filter on X . Theorem 1.7. [2] If I be any ideal in B ( X ) , then Z B [ I ] = { Z ( f ) : f ∈ I } is a Z B -filter on X . Theorem 1.8. [2]
For any Z B -filter F on X , Z − B [ F ] = { f ∈ B ( X ) : Z ( f ) ∈ F } is an ideal in B ( X ) . Let, I B be the collection of all ideals in B ( X ) and F B ( X ) be the collection ofall Z B -filters on X . The map Z B : I B F B ( X ) defined by Z B ( I ) = Z B [ I ] , ∀ I ∈ ORE ON THE RINGS B ( X ) AND B ∗ ( X ) 3 I B is surjective but not injective in general. Although the restriction map Z B : M ( B ( X )) Ω B ( X ) is a bijection, where M ( B ( X )) and Ω B ( X ) are respectivelythe collection of all maximal ideals in B ( X ) and the collection of all Z B -ultrafilterson X [2].A criterion to paste two Baire one functions to obtain a Baire one function on abigger domain is proved for normal topological spaces in the following theorem: Theorem 1.9. ( Pasting Lemma ) Let X be a normal topological space with X = A ∪ B , where both A and B are G δ sets in X . If f : A R and g : B R areBaire one functions so that f ( x ) = g ( x ) , for all x ∈ A ∩ B , then the map h : X R defined by h ( x ) = ( f ( x ) if x ∈ Ag ( x ) if x ∈ B. is also a Baire one function.Proof. Let C be any closed set in R . Then h − ( C ) = f − ( C ) S g − ( C ). f − ( C )and g − ( C ) are G δ sets in A and B respectively, because f and g are Baire onefunctions [6]. Since A and B are G δ sets in X , f − ( C ) and g − ( C ) are G δ sets in X . We know finite union of G δ sets is G δ , so h − ( C ) is a G δ set in X . Hence h isa Baire one function [6]. (cid:3) The result remains true if the G δ sets are replaced by F σ sets. The proof beingexactly the same as that of Theorem 1.9, we only state the result as follows: Theorem 1.10.
Let X be a normal topological space with X = A ∪ B , where both A and B are F σ sets in X . If f : A R and g : B R are Baire one functionsso that f ( x ) = g ( x ) , for all x ∈ A ∩ B , then the map h : X R defined by h ( x ) = ( f ( x ) if x ∈ Ag ( x ) if x ∈ B. is also a Baire one function. m B -topology - An extension of uniform norm topology Let X be any topological space. Defining the ‘sup’-norm as usual we get that B ∗ ( X )is a Banach space. The topology induced by the sup-norm is known as the uniformnorm topology and it is not hard to see that B ∗ ( X ) with uniform norm topologyis a topological ring as well as a topological vector space. For any f ∈ B ∗ ( X ), thecollection { B ( f, ǫ ) : ǫ > } , where B ( f, ǫ ) = { g ∈ B ∗ ( X ) : || f − g || ≤ ǫ } , forms abase for neighbourhood system at f in the uniform norm topology on B ∗ ( X ). Theorem 2.1.
The set U + B ∗ of all positive units of B ∗ ( X ) is an open set in theuniform norm topology on B ∗ ( X ) .Proof. Let f ∈ U + B ∗ . So, f is bounded away from 0, i.e., λ > f ( x ) > λ , for all x ∈ X . We observe that for any g ∈ B ( f, λ ), { h ∈ B ∗ ( X ) : || f − h || ≤ λ } , g ( x ) > λ for every x ∈ X . Hence g ∈ U + B ∗ and B ( f, λ ) ⊆ U + B ∗ [1].Therefore, U + B ∗ is an open set. (cid:3) Corollary 2.2.
The set U − B ∗ of all negative units in B ∗ ( X ) is open in the uniformnorm topology on B ∗ ( X ). A. DEB RAY AND ATANU MONDAL
Theorem 2.3. If I is a proper ideal in B ∗ ( X ) then I (Closure of I in the uniformnorm topology) is also a proper ideal in B ∗ ( X ) .Proof. Since B ∗ ( X ) is a topological ring, I is an ideal. We show that I is a properideal of B ∗ ( X ). Since I does not contain any unit, it does not contain any positiveunit as well. So, U + B ∗ ∩ I = ∅ = ⇒ I ⊆ B ∗ ( X ) \ U + B ∗ . U + B ∗ being an open setin uniform norm topology, B ∗ ( X ) \ U + B ∗ is a closed set containing I which implies I ⊆ B ∗ ( X ) \ U + B ∗ . Therefore I ∩ U + B ∗ = ∅ . Consequently, 1 / ∈ I , proving I a properideal of B ∗ ( X ). (cid:3) Corollary 2.4.
Each maximal ideal of B ∗ ( X ) is a closed set in the uniform normtopology. Proof.
Let M be a maximal ideal of B ∗ ( X ). Then M is also a proper ideal of B ∗ ( X ). By maximality of M , we get M = M . Hence M is closed. (cid:3) The natural extension of uniform norm topology on B ( X ) would be the one forwhich { U ( f, ǫ ) : ǫ > } is a base for neighbourhood system at f ∈ B ( X ), where U ( f, ǫ )= { g ∈ B ( X ) : | f ( x ) − g ( x ) | ≤ ǫ, for all x ∈ X } . We call this topology the u B -topology on B ( X ). Clearly, on B ∗ ( X ), the subspace topology obtained fromthe u B -topology coincides with the uniform norm topology. One may easily checkthat ( B ( X ) , u B ) is a topological group. However, it is neither a topological ringnor a topological vector space, unless B ( X ) = B ∗ ( X ). Theorem 2.5. If B ( X ) = B ∗ ( X ) then B ( X ) is neither a topological ring nor atopological vector space.Proof. Suppose, B ( X ) = B ∗ ( X ). Then there exists f ∈ B ( X ) \ B ∗ ( X ). Withoutloss of generality, assume that f ≥ X . We show that the multiplicationoperation B ( X ) × B ( X ) → B ( X ) is not continuous at ( , f ), for every f ∈ B ( X ).Let U ( ,
1) be a neighbourhood of . Each function in U ( ,
1) is bounded on X .Let U ( , ǫ ) and U ( f, δ ) be arbitrary basic neighbourhoods of and f in ( B ( X ), u B ). ǫ ∈ U ( , ǫ ) and f ∈ U ( f, δ ). As f ≥ ǫ is a constant function, ǫ f is anunbounded function which implies that ǫ f / ∈ U ( , , f ).The same argument leads to the fact that scalar multiplication is not continuousat ( , f ) if we consider B ( X ) as a vector space over R . (cid:3) In the next theorem we show that the collection of all positive units of B ( X ) isnot an open set in u B -topology, if B ( X ) = B ∗ ( X ). Theorem 2.6. If B ( X ) = B ∗ ( X ) then W + B , the collection of all positive units in B ( X ) is not an open set in u B -topology.Proof. Let f ∈ B ( X ) \ B ∗ ( X ) be an unbounded Baire one function with f ≥ X . We define g = f on X . Since f ( x ) > x ∈ X , g is a Baire one function.So g is a positive unit in B ( X ) which takes values arbitrarily close to 0. We showthat g is not an interior point of W + B . Indeed, for each ǫ > U ( g, ǫ ) * W + B . Wecan select a point a ∈ X , such that 0 < g ( a ) < ǫ . Taking h = g − g ( a ) we get that h ∈ U ( g, ǫ ) but h ( a ) = 0. So, h is not a unit in B ( X ), i.e., h / ∈ W + B . (cid:3) ORE ON THE RINGS B ( X ) AND B ∗ ( X ) 5 Corollary 2.7. If B ( X ) = B ∗ ( X ) then W − B , the collection of all negative unitsin B ( X ) is not an open set in u B -topology.We put together the outcome of the above discussion in the following theorem: Theorem 2.8.
For a topological space X , the following statements are equivalent: (i) B ( X ) = B ∗ ( X ) . (ii) B ( X ) with u B -topology is a topological ring. (iii) B ( X ) with u B -topology is a topological vector space. (iv) The set of all positive units in B ( X ) forms an open set in u B -topology. (v) The set of all negative units in B ( X ) forms an open set in u B -topology. Corollary 2.9.
If a Completely Hausdorff space X is not totally disconnected then B ( X ) with u B -topology is neither a topological ring nor a topological vector space.Moreover, the collection of all positive units (respectively, negative units) does notform an open set in B ( X ). Proof.
Follows from Theorem 3.4 of [1]. (cid:3)
To overcome the inadequacy of u B -topology, we define another topology on B ( X )as follows: Define M ( g, u ) = { f ∈ B ( X ) : | f ( x ) − g ( x ) | ≤ u ( x ), for every x ∈ X } and f M ( g, u ) = { f ∈ B ( X ) : | f ( x ) − g ( x ) | < u ( x ), for every x ∈ X } ,where u is any positive unit in B ( X ). It is not hard to check that the collection B = { f M ( g, u ) : g ∈ B ( X ) and u is any positive unit in B ( X ) } is an open basefor some topology on B ( X ). Theorem 2.10. B ( X ) with m B -topology is a topological ring.Proof. Let f, g ∈ B ( X ) and M ( f + g, u ) be any neighbourhood of f + g in B ( X ).Clearly, M ( f, u ) + M ( g, u ) ⊆ M ( f + g, u ). So + : B ( X ) × B ( X ) → B ( X ) is acontinuous function.We show that the multiplication B ( X ) × B ( X ) → B ( X ) defined by ( f, g ) f.g is continuous.We select a positive unit u in B ( X ) and we want to produce a positive unit v ≤ B ( X ) such that f M ( f, v ) . f M ( g, v ) ⊆ f M ( f.g, u ).If v is to satisfy this relation then we should have,whenever | h ( x ) − f ( x ) | ≤ v ( x ) and | h ( x ) − g ( x ) | ≤ v ( x ), for all x ∈ X, then | h ( x ) h ( x ) − f ( x ) g ( x ) | ≤ u ( x ), for all x ∈ X .Now, if | h ( x ) − f ( x ) | ≤ v ( x ) and | h ( x ) − g ( x ) | ≤ v ( x ), for all x ∈ X | h ( x ) h ( x ) − f ( x ) g ( x ) | = | h ( x ) (cid:0) h ( x ) − g ( x ) (cid:1) + g ( x ) (cid:0) h ( x ) − f ( x ) (cid:1) | = | h ( x ) (cid:0) h ( x ) − g ( x ) (cid:1) + (cid:0) g ( x ) − h ( x ) (cid:1)(cid:0) h ( x ) − f ( x ) (cid:1) + h (cid:0) h ( x ) − f ( x ) (cid:1) | ≤ | h | v + v + | h | v = v (cid:2) | h | + v + | h | (cid:3) ≤ v (cid:2) | h | + 1 + | h | (cid:3) ≤ v (cid:2) | f | + v + | g | + v + 1 (cid:3) ≤ v (cid:2) | f | + | g | + 3 (cid:3) ≤ u (let). So forany given u if we choose v = (cid:0) u | f | + | g | +3 (cid:1) ∧
1, we get that multiplication operationis continuous. Hence, B ( X ) with m B -topology is a topological ring. (cid:3) Corollary 2.11. B ( X ) with m B -topology is a topological vector space. Theorem 2.12. W + B , the collection of all positive units (respectively, W − B , thecollection of all negative units) forms an open set in B ( X ) with m B -topology.Proof. Let u ∈ W + B . If we show that M ( u, u ) ⊆ W + B then u is an interior point of W + B . Indeed for any v ∈ M ( u, u ) = ⇒ | v ( x ) − u ( x ) | < u ( x )2 = ⇒ v ( x ) >
0, for all
A. DEB RAY AND ATANU MONDAL x ∈ X . So v is a positive unit, i.e., v ∈ W + B .That W − B is also an open set follows similarly. (cid:3) Theorem 2.13.
The closure I of a proper ideal I of B ( X ) with m B -topology isalso a proper ideal.Proof. B ( X ) with m B -topology being a topological ring, I is an ideal of B ( X ).It is enough to show that I is a proper ideal of B ( X ). Since W + B is an open set,proceeding as in Theorem 2.3 we obtain the result. (cid:3) Theorem 2.14. B ∗ ( X ) is a closed subset of B ( X ) with m B -topology.Proof. Choose f ∈ B ( X ) \ B ∗ ( X ). Then M ( f, ⊆ B ( X ) \ B ∗ ( X ) which provesthe desired result. (cid:3) Theorem 2.15.
The uniform norm topology on B ∗ ( X ) is weaker than the relative m B -topology on B ∗ ( X ) .Proof. For any f ∈ B ∗ ( X ) and ǫ > B ( f, ǫ ) = { g ∈ B ∗ ( X ) : | f ( x ) − g ( x ) | ≤ ǫ } = M ( f, ǫ ) ∩ B ∗ ( X ). (cid:3) Theorem 2.16.
Each maximal ideal in B ∗ ( X ) is closed in B ( X ) with m B -topology.Proof. Each maximal ideal M in B ∗ ( X ) is closed in B ∗ ( X ) with respect to relative m B -topology, since the uniform norm topology on B ∗ ( X ) is weaker than the relative m B -topology on B ∗ ( X ) and M is closed under uniform norm topology. Also B ∗ ( X )is closed in B ( X ) with m B -topology. Hence M is closed in B ( X ). (cid:3) Theorem 2.17.
For a normal topological space X , every closed ideal I in B ( X ) with m B -topology is a Z B -ideal.Proof. Suppose f, g ∈ B ( X ) with g ∈ I and Z ( f ) = Z ( g ). It is enough to showthat f ∈ I . Since I is closed, we show that for any positive unit u in B ( X ), M ( f, u ) T I = ∅ . Consider the map h : X R given by h ( x ) = | f ( x ) | ≤ u ( x ) f ( x ) − u ( x ) g ( x ) if f ( x ) ≥ u ( x ) f ( x )+ u ( x ) g ( x ) if f ( x ) ≤ − u ( x ) h is well defined. Observe that the collections { x ∈ X : | f ( x ) | ≤ u ( x ) } , { x ∈ X : f ( x ) ≥ u ( x ) } , { x ∈ X : f ( x ) ≤ − u ( x ) } are G δ sets in X and 0 , f ( x ) − u ( x ) g ( x ) , f ( x )+ u ( x ) g ( x ) are Baire one functions on their respective domains. So by Theorem 1.9 h is a Baireone function on X .For all x ∈ X, | h ( x ) g ( x ) − f ( x ) | ≤ u ( x ) implies hg ∈ M ( f, u ). But g ∈ I implies hg ∈ I . Therefore hg ∈ M ( f, u ) T I and f is a limit point of I . This completes theproof. (cid:3) In Theorem 2.15 we have seen that the uniform norm topology is weaker than therelative m B -topology on B ∗ ( X ). The following theorem shows that it is alwaysstrictly weaker unless B ( X ) = B ∗ ( X ). Put in other words, the equality of uniformnorm topology with relative m B -topology on B ∗ ( X ) characterizes B ( X ) = B ∗ ( X ). Theorem 2.18.
The uniform norm topology on B ∗ ( X ) is same as the relative m B -topology on B ∗ ( X ) if and only if B ( X ) = B ∗ ( X ) . ORE ON THE RINGS B ( X ) AND B ∗ ( X ) 7 Proof. If B ( X ) = B ∗ ( X ) then for any g ∈ B ( X ) and any positive unit u in B ( X )we get M ( g, u ) ∩ B ∗ ( X ) = M ( g, u ) = { f ∈ B ∗ ( X ) : | f ( x ) − g ( x ) | ≤ u ( x ), for every x ∈ X } . u is a unit in B ( X ) (cid:0) = B ∗ ( X ) (cid:1) implies that u is bounded away from 0.So u ( x ) ≥ ǫ , for all x ∈ X and for some ǫ >
0. Hence g ∈ B ( g, ǫ ) (cid:0) = M ( g, ǫ ) (cid:1) ⊆ M ( g, u ). Therefore M ( g, u ) is a neighbourhood of g in B ∗ ( X ) in the uniform normtopology. Hence this result along with Theorem 2.15 implies that uniform normtopology on B ∗ ( X ) = the relative m B -topology on B ∗ ( X ).For the converse, let B ( X ) = B ∗ ( X ). Then there exists f ∈ B ( X ) such that Z ( f ) = ∅ , f ( x ) > x ∈ X and f takes arbitrarily small values near 0. So f is a positive unit in B ( X ). Now, for any two real numbers r , s , it will neverhappen that | r − s | ≤ f . This means that M ( r, f ) ∩ { t : t ∈ R } = { r } . So the set { r : r ∈ R } is a discrete subspace of B ∗ ( X ) in the relative m B -topology. Fromthis it follows that, the scalar multiplication operation ψ : R × B ∗ ( X ) → B ∗ ( X ),defined by ψ ( α, g ) = α.g is not continuous at ( r, s ), where r, s ∈ R . Thus B ∗ ( X )with relative m B -topology is not a topological vector space and hence the relative m B -topology is not same as uniform norm topology on B ∗ ( X ). (cid:3) Theorem 2.19. B ( X ) with m B -topology is first countable if and only if B ( X ) = B ∗ ( X ) .Proof. If B ( X ) = B ∗ ( X ) then by Theorem 2.18 B ( X ) with m B -topology is firstcountable.Conversely, let B ( X ) = B ∗ ( X ). So there exists f ∈ B ( X ) \ B ∗ ( X ). Then g = f +1 is a positive unit in B ( X ). We can find a stictly increasing sequence { a n } of positive real numbers and a countable subset { p n } of X such that g ( p n ) = a n ,for all n ∈ N .Consider any countable collection of positive units in B ( X ), say { π n } .Let b n = min (cid:2) π ( p n ) , π ( p n ) , ..., π n ( p n ) (cid:3) . Then there always exists a real valuedcontinuous function σ : R → R such that σ ( x ) >
0, for all x ∈ R + and σ ( a n ) = b − n ,for all n ∈ N . Define ψ ( x ) = σ ( g ( x )) , ∀ x ∈ X . By Theorem 2.4 of [1] ψ is a positive unit in B ( X )and ψ ( p n ) = σ ( g ( p n )) = b n ≤ π n ( p n ), for all n ∈ N .Clearly π n ∈ M (0 , π n ) but π n / ∈ M (0 , ψ ), because at each p n , ψ ( p n ) ≤ π n ( p n ) < π n ( p n ). So the neighbourhood M (0 , ψ ) at 0 contains no M (0 , π n ) , n = 1 , , , ... .Which shows that at the point 0 in B ( X ) there is no countable neighbourhoodbase.Hence B ( X ) with m B -topology is not first countable. (cid:3) Corollary 2.20. B ( X ) with m B -topology is metrizable if and only if B ( X ) = B ∗ ( X ). 3. e B -ideals and e B -filters in B ∗ ( X )For any Z B -filter F on X , Z − B [ F ] T B ∗ ( X ) is an ideal in B ∗ ( X ). That Z B [ I ] isnot in general a Z B -filter is evident from the following example:Consider the ring B ∗ ( N ) and the ideal I as the collection of all sequences of realnumbers converge to 0. The sequence { n } ∈ I but Z ( { n } ) = ∅ . ∅ ∈ Z B [ I ] showsthat it is not a Z B -filter.In this section we introduce a special class of Z B -filters, called e B -filters. We locatea class of special ideals of B ∗ ( X ), called e B -ideals, which behave the same way A. DEB RAY AND ATANU MONDAL as the Z B -ideals in B ( X ). The e B -ideals and e B -filters play the pivotal role toestablish the desired correspondence.For f ∈ B ∗ ( X ) and ǫ >
0, we define E ǫB ( f ) = f − ([ − ǫ, ǫ ]) = { x ∈ X : | f ( x ) | ≤ ǫ } .Every set of this form is a member of Z ( B ( X )), as E ǫB ( f ) = Z (( | f | − ǫ ) ∨ Z ( h ), h ∈ B ∗ ( X ) is of the form E ǫB ( | h | + ǫ ). For every non empty set I (cid:0) ⊆ B ∗ ( X ) (cid:1) we define E ǫB [ I ] = { E ǫB ( f ) : f ∈ I } and E B ( I ) = S ǫ> E ǫB [ I ] = { E ǫB ( f ) : f ∈ I, ǫ > } . For any collection of zerosets F , i.e. F ⊆ Z ( B ( X )) we consider E ǫ − B [ F ] = { f ∈ B ∗ ( X ) : E ǫB ( f ) ∈ F } and define E − B ( F ) = T ǫ> E ǫ − B [ F ] = { f ∈ B ∗ ( X ) : E ǫB ( f ) ∈ F , ∀ ǫ > } .One may easily see that for any two ideals I , J and subcollections F and G of Z ( B ( X ))(i) I ⊆ J ⇒ E B ( I ) ⊆ E B ( J ).(ii) F ⊆ G ⇒ E − B ( F ) ⊆ E − B ( G ).We record the following facts in the following couple of theorems: Theorem 3.1.
For any subset I of B ∗ ( X ) , I ⊆ E − B (cid:0) E B ( I ) (cid:1) , where E − B (cid:0) E B ( I ) (cid:1) = { f ∈ B ∗ ( X ) : E ǫB ( f ) ∈ E B ( I ) , ∀ ǫ > } . Proof.
Let f ∈ I . Then E ǫB ( f ) ∈ E ǫB [ I ], ∀ ǫ >
0. By definition f ∈ E − B (cid:0) E B ( I ) (cid:1) .Hence we have the required inclusion. (cid:3) Theorem 3.2.
For any subcollection F of Z ( B ( X )) , E B (cid:0) E − B ( F ) (cid:1) ⊆ F , where E B (cid:0) E − B ( F ) (cid:1) = S ǫ> { E ǫB ( f ) : E δB ( f ) ∈ F for all δ > } .Proof. The proof follows trivially from the definitions of E B and E − B . (cid:3) It is interesting to note that, in Theorem 3.1, the inclusion may be a strict one evenif we consider I as an ideal in B ∗ ( X ).For example, consider the ring B ∗ ( N ) and the function f ( n ) = n in B ∗ ( N ). Suppose I = < f > , i.e., the ideal in B ∗ ( N ) generated by f . It’s quite clear that f / ∈ I . Nowwe will show that f ∈ E − B (cid:0) E B ( I ) (cid:1) or equivalently, E ǫB ( f ) ∈ E B ( I ), ∀ ǫ >
0. For any ǫ > E ǫB ( f ) = { n ∈ N : | f ( n ) | ≤ ǫ } = { n ∈ N : f ( n ) ≤ ǫ } = E ǫ B ( f ) ∈ E B ( I ).So, I $ E − B (cid:0) E B ( I ) (cid:1) .In Theorem 3.2 too, the inclusion may be proper even when F is a Z B -filter.As an example in support of our claim, we consider the ring B ∗ ( R ) and the Z B -filter F = { Z ∈ Z ( B ( R )) : 0 ∈ Z } . Now we consider a function f : R R defined by f ( x ) = | x || x | +1 . Clearly { } = Z ( f ) ∈ F . We show that { } / ∈ E B (cid:0) E − B ( F ) (cid:1) . Let g = f + ǫ , for any arbitrary ǫ >
0. It is easy to observe that E ǫB ( g ) = { } but ifwe take any positive number δ < ǫ then E δB ( g ) = ∅ , which does not belong to F . Hence { } / ∈ E B (cid:0) E − B ( F ) (cid:1) . Definition 3.3.
An ideal I in B ∗ ( X ) is called an e B -ideal if I = E − B (cid:0) E B ( I ) (cid:1) . I is an e B -ideal if and only if for all ǫ > , E ǫB ( f ) ∈ E B ( I ) implies f ∈ I .From the definition it is clear that intersection of e B -ideals is an e B -ideal. Definition 3.4. A Z B -filter F is said to be an e B -filter if F = E B (cid:0) E − B ( F ) (cid:1) .Equivalently, F is an e B -filter if and only if whenever Z ∈ F there exist ǫ > f ∈ B ∗ ( X ) such that Z = E ǫB ( f ) and E δB ( f ) ∈ F , ∀ δ > ORE ON THE RINGS B ( X ) AND B ∗ ( X ) 9 In what follows, we consider X to be always a normal topological space. Theorem 3.5. If I is any proper ideal in B ∗ ( X ) then E B ( I ) is an e B -filter.Proof. We prove this in two steps. At first we show that E B ( I ) is a Z B -filter andthen we establish E B ( I ) = E B (cid:0) E − B ( E B ( I ) (cid:1) . To show E B ( I ) is a Z B -filter we needto checka) ∅ / ∈ E B ( I ).b) E B ( I ) is closed under finite intersection.c) E B ( I ) is closed under superset.We assert that ∅ / ∈ E B ( I ). If our assertion is not true then ∅ = E ǫB ( f ), for some f ∈ I and some ǫ >
0. So | f ( x ) | > ǫ , ∀ x ∈ X , which implies that f is boundedaway from zero and so f is a unit in B ∗ ( X ) [1]. This contradicts that I is proper.For b) it is enough to show that for any two members in E B ( I ) their intersec-tion contains a member in E B ( I ). Let E ǫB ( f ) , E δB ( g ) ∈ E B ( I ). Therefore thereexist some f , g ∈ I and ǫ , δ > E ǫB ( f ) = E ǫ B ( f ) and E δB ( g ) = E δ B ( g ). Choosing ǫ , δ in such a way that δ < ǫ <
1. Clearly, E δ B ( f + g ) ⊆ E ǫ B ( f ) T E δ B ( g ) = E ǫB ( f ) T E δB ( g ). Since I is an ideal f + g ∈ I and that provesour claim.Now let E ǫB ( f ) ∈ E B ( I ) and Z ( f ′ ) ( f ′ ∈ B ∗ ( X )) be any member in Z ( B ( X )) sothat E ǫB ( f ) ⊆ Z ( f ′ ). We shall show that Z ( f ′ ) ∈ E B ( I ). Since E ǫB ( f ) = E ǫ B ( f )and Z ( f ′ ) = Z ( | f ′ | ), we can start with f, f ′ ≥ P = { x ∈ X : | f ( x ) | ≥ ǫ } .We define a function g : X R by g ( x ) = ( x ∈ E ǫB ( f ) f ′ ( x ) + ǫf ( x ) if x ∈ P .
We observe that E ǫB ( f ) T P = { x ∈ X : f ( x ) = ǫ } and ∀ x ∈ E ǫB ( f ) T P , f ′ ( x ) + ǫf ( x ) = 0 + ǫǫ = 1 . It is clear that both E ǫB ( f ) and P are G δ sets and 1, f ′ ( x ) + ǫf ( x ) are Baire one functions on E ǫB ( f ) and P respectively. Therefore by Theorem 1.9, g is a Baire one function on X , in fact g ∈ B ∗ ( X ).Now consider the function f g : X R given by( f g )( x ) = ( f ( x ) if x ∈ E ǫB ( f )( f f ) ′ ( x ) + ǫ if x ∈ P . .Since I is an ideal f g ∈ I and it is easy to check that Z ( f ′ ) = E ǫB ( f g ). Hence Z ( f ′ ) ∈ E B ( I ). So E B ( I ) is a Z B -filter. By Theorem 3.1 we get I ⊆ E − B (cid:0) E B ( I ) (cid:1) .Since the map E B preserves inclusion, we obtain E B ( I ) ⊆ E B (cid:0) E − B (cid:0) E B ( I ) (cid:1)(cid:1) . Also E B ( I ) is a Z B -ideal, so by Theorem 3.2 E B (cid:0) E − B (cid:0) E B ( I ) (cid:1)(cid:1) ⊆ E B ( I ). Combiningthese two we have E B (cid:0) E − B (cid:0) E B ( I ) (cid:1)(cid:1) = E B ( I ). Hence E B ( I ) is a e B -filter. (cid:3) Theorem 3.6.
For any Z B -filter F on X , E − B ( F ) is an e B -ideal in B ∗ ( X ) .Proof. We show that E − B ( F ) is an ideal in B ∗ ( X ). Let f, g ∈ E − B ( F ). Thereforefor any arbitrary ǫ > E ǫ B ( f ), E ǫ B ( g ) ∈ F . F being a Z B -filter E ǫ B ( f ) T E ǫ B ( g ) ∈ F . Also we know E ǫ B ( f ) T E ǫ B ( g ) ⊆ E ǫB ( f + g ). Hence E ǫB ( f + g ) ∈ F , or equiva-lently f + g ∈ E − B ( F ). Consider f ∈ E − B ( F ) and h be any bounded Baire one function on X with anupper bound M > ǫ be any arbitrary positive real number. So | h ( x ) | ≤ M ,for all x ∈ X . For any point x ∈ E ǫM B ( f ) = ⇒ | f ( x ) | ≤ ǫM = ⇒ | M f ( x ) | ≤ ǫ = ⇒| f ( x ) h ( x ) | ≤ ǫ = ⇒ x ∈ E ǫB ( f h ). This implies E ǫM B ( f ) ⊆ E ǫB ( f h ). So E ǫB ( f h ) ∈ F ,for any arbitrary ǫ >
0. Therefore by definition of E − B ( F ), f h ∈ E − B ( F ). Hence E − B ( F ) is an ideal in B ∗ ( X ).By Theorem 3.1, E − B ( F ) ⊆ E − B (cid:0) E B ( E − B ( F )) (cid:1) . Also by Theorem 3.2, E B ( E − B ( F )) ⊆ F . Since E − B preserves inclusion, E − B (cid:0) E B ( E − B ( F )) (cid:1) ⊆ E − B ( F ). Hence E − B ( F ) = E − B (cid:0) E B ( E − B ( F )) (cid:1) and so, E − B ( F ) is an e B -ideal. (cid:3) Corollary 3.7.
The correspondence I E B ( I ) is one-one from the set of all e B -ideals in B ∗ ( X ) onto the set of all e B -filters on X . Theorem 3.8. If I is an ideal in B ∗ ( X ) then E − B (cid:0) E B ( I ) (cid:1) is the smallest e B -idealcontaining I .Proof. It follows from Theorem 3.5 and Theorem 3.6 that E − B (cid:0) E B ( I ) (cid:1) is an e B -ideal. Also from Theorem 3.1 we have I ⊆ E − B (cid:0) E B ( I ) (cid:1) . If possible let J be any e B -ideal containing I . So I ⊆ J and since E B and E − B preserve inclusion, we canwrite E − B (cid:0) E B ( I ) (cid:1) ⊆ E − B (cid:0) E B ( J ) (cid:1) = J (cid:0) since J is an e B -ideal (cid:1) . It completesthe proof. (cid:3) Corollary 3.9.
Every maximal ideal in B ∗ ( X ) is an e B -ideal. Proof.
Follows immediately from the theorem. (cid:3)
Corollary 3.10.
Intersection of maximal ideals in B ∗ ( X ) is an e B -ideal. Proof.
Clear. (cid:3)
Theorem 3.11.
For any Z B -filter F on X , E B (cid:0) E − B ( F ) (cid:1) is the largest e B -filtercontained in F .Proof. Theorem 3.5, Theorem 3.6 and Theorem 3.2 show that E B (cid:0) E − B ( F ) (cid:1) is an e B -filter contained in F . If possible let E be any e B -filter contained in F . So E ⊆ F = ⇒ E B (cid:0) E − B ( E ) (cid:1) ⊆ E B (cid:0) E − B ( F ) (cid:1) = ⇒ E ⊆ E B (cid:0) E − B ( F ) (cid:1) . Thiscompletes the proof. (cid:3) Lemma 3.12.
Let I and J be two ideals in B ∗ ( X ) with J an e B -ideal. Then I ⊆ J if and only if E B ( I ) ⊆ E B ( J ) .Proof. I ⊆ J = ⇒ E B ( I ) ⊆ E B ( J ) follows from the definition of E B ( I ).For the converse, let f ∈ I . To show that f ∈ J .Suppose ǫ > f ∈ I = ⇒ E ǫB ( f ) ∈ E B ( I ) = ⇒ E ǫB ( f ) ∈ E B ( J ) = ⇒ f ∈ J (since J is an e B -ideal ). Therefore I ⊆ J . (cid:3) Lemma 3.13.
For any two Z B -filters F and F on X , with F an e B -filter, F ⊆ F if and only if E − B ( F ) ⊆ E − B ( F ) Proof.
Straightforward. (cid:3)
Lemma 3.14.
Let A be any Z B -ultrafilter. If a zero set Z meets every member of A then Z ∈ A . ORE ON THE RINGS B ( X ) AND B ∗ ( X ) 11 Proof.
Consider the collection A ∪ { Z } . By hypothesis, it has finite intersectionproperty. So it can be extended to a Z B -filter with Z as one of its member. As this Z B -filter contains a maximal Z B -filter, it must be A . So Z ∈ A . (cid:3) Theorem 3.15.
Let A be any Z B -ultrafilter. A zero set Z in Z ( B ( X )) belongsto A if and only if Z meets every member of E B (cid:0) E − B ( A ) (cid:1) .Proof. A is a Z B -ultrafilter, so E B (cid:0) E − B ( A ) (cid:1) ⊆ A . If we assume Z ∈ A then Z meets every member of A [2] and hence Z meets every member of E B (cid:0) E − B ( A ) (cid:1) .Conversely, suppose Z meets every member of E B (cid:0) E − B ( A ) (cid:1) . We claim that Z intersects every member of A . If not, there exists Z ′ ∈ A for which Z T Z ′ = ∅ .Therefore Z and Z ′ are completely separated in X by B ( X ) [1] and so thereexists f ∈ B ∗ ( X ) such that f ( Z ) = 1 and f ( Z ′ ) = 0. Clearly for all ǫ > ,Z ′ ⊆ Z ( f ) ⊆ E ǫB ( f ). Now Z ′ ∈ A implies that E ǫB ( f ) ∈ A , for all ǫ >
0. If wechoose ǫ < then Z T E ǫB ( f ) = ∅ , which contradicts that Z meets every memberof E B (cid:0) E − B ( A ) (cid:1) . Hence Z meets every member of A and therefore Z ∈ A . (cid:3) Theorem 3.16. If A is any Z B -ultrafilter on X then E − B ( A ) is a maximal idealin B ∗ ( X ) .Proof. We know from Theorem 3.6 that E − B ( A ) is an ideal in B ∗ ( X ). Let M ∗ bea maximal ideal in B ∗ ( X ) which contains E − B ( A ). By using Lemma 3.12 we canwrite E B (cid:0) E − B ( A ) (cid:1) ⊆ E B ( M ∗ ). Theorem 3.5 asserts that E B ( M ∗ ) is a Z B -filteron X , therefore every member of E B ( M ∗ ) meets every member of E B (cid:0) E − B ( A ) (cid:1) .By Theorem 3.15 every member of E B ( M ∗ ) belongs to A . Hence E B ( M ∗ ) ⊆ A .Every maximal ideal is an e B -ideal, so M ∗ = E − B (cid:0) E B ( M ∗ ) (cid:1) ⊆ E − B ( A ). Thisimplies M ∗ = E − B ( A ) which completes the proof. (cid:3) Corollary 3.17.
For any Z B -ultrafilter on X , E − B ( A ) = E − B (cid:0) E B (cid:0) E − B ( A ) (cid:1)(cid:1) . Definition 3.18. An e B -filter is called an e B -ultrafilter if it is not contained inany other e B -filter. In other words a maximal e B -filter is called an e B -ultrafilter.Using Zorn’s lemma one can establish that, every e B -filter is contained in an e B -ultrafilter. Theorem 3.19. If M ∗ is a maximal ideal in B ∗ ( X ) and F is an e B -ultrafilter on X thena) E B ( M ∗ ) is an e B -ultrafilter on X .b) E − B ( F ) is a maximal ideal in B ∗ ( X ) .Proof. a) Since M ∗ is a maximal ideal in B ∗ ( X ) then by Corollary 3.9 we have M ∗ = E − B (cid:0) E B ( M ∗ ) (cid:1) . Suppose the e B -filter E B ( M ∗ ) is contained in an e B -ultrafilter F ′ . So E B ( M ∗ ) ⊆ F ′ = ⇒ E − B (cid:0) E B ( M ∗ ) (cid:1) = M ∗ ⊆ E − B ( F ′ ) = ⇒ M ∗ = E − B ( F ′ ) (since M ∗ is maximal ideal). Therefore E B ( M ∗ ) = E B (cid:0) E − B ( F ′ ) (cid:1) = F ′ . Hence E B ( M ∗ ) is an e B -ultrafilter.b) Let M ∗ be any maximal extension of the ideal E − B ( F ). Now E − B ( F ) ⊆ M ∗ = ⇒ F = E B (cid:0) E − B ( F ) (cid:1) ⊆ E B ( M ∗ ). By part (a) we can conclude that F = E B ( M ∗ ), which gives us E − B ( F ) = M ∗ . Hence we are done. (cid:3) Corollary 3.20.
Let M ∗ be an e B -ideal, then M ∗ is maximal in B ∗ ( X ) if andonly if E B ( M ∗ ) is an e B -ultrafilter. Proof.
Straightforward. (cid:3)
Corollary 3.21. An e B -filter F is an e B -ultrafilter if and only if E − B ( F ) is amaximal ideal in B ∗ ( X ). Proof.
Straightforward. (cid:3)
Corollary 3.22.
The correspondence M ∗ E B ( M ∗ ) is one-one from the set ofall maximal ideals in B ∗ ( X ) onto the set of all e B -ultrafilters. Theorem 3.23. If A is a Z B -ultrafilter then it is the unique Z B -ultrafilter con-taining E B (cid:0) E − B ( A ) (cid:1) . In fact, E B (cid:0) E − B ( A ) (cid:1) is the unique e B -ultrafilter containedin A .Proof. Let A ∗ be a Z B -ultrafilter containing E B (cid:0) E − B ( A ) (cid:1) and Z ∈ A ∗ . Clearly Z meets every member of A ∗ and so it meets every member of E B (cid:0) E − B ( A ) (cid:1) . ByTheorem 3.15 Z ∈ A and A ∗ ⊆ A . Since both A and A ∗ are maximal Z B -filterhence we have A = A ∗ . So A is unique one containing E B (cid:0) E − B ( A ) (cid:1) .For the second part, E B (cid:0) E − B ( A ) (cid:1) is e B -ultrafilter follows from Theorem 3.16 andTheorem 3.19. To prove the uniqueness we suppose E be any e B -ultrafilter con-tained in A . Then E ⊆ A = ⇒ E B (cid:0) E − B ( E ) (cid:1) = E ⊆ E B (cid:0) E − B ( A ) (cid:1) = ⇒ E = E B (cid:0) E − B ( A ) (cid:1) . This completes the proof. (cid:3) Corollary 3.24.
Every e B -ultrafilter is contained in a unique Z B -ultrafilter. Corollary 3.25.
There is a one to one correspondence between the collection ofall Z B -ultrafilter on X and the collection of all e B -ultrafilter on X . Proof.
Let us define a map λ from the set of all Z B -ultrafilters on X to the setof all e B -ultrafilters by λ ( A ) = E B (cid:0) E − B ( A ) (cid:1) . We shall show that the map isone-one and onto. To show the injectivity, suppose λ ( A ) = λ ( B ), which implies E B (cid:0) E − B ( A ) (cid:1) = E B (cid:0) E − B ( B ) (cid:1) . By Theorem 3.23 we can say E B (cid:0) E − B ( A ) (cid:1) = E B (cid:0) E − B ( B ) (cid:1) is contained in both A and B . Now let Z ∈ A . So it intersects everymember of E B (cid:0) E − B ( B ) (cid:1) and by Theorem 3.15 Z ∈ B . This gives us A ⊆ B . Bya similar argument we can show that B ⊆ A . Hence A = B and λ is one-one.Again let E be any e B -ultrafilter. By Corollary 3.24 there is a unique Z B -ultrafilter A containing E . Therefore E = E B (cid:0) E − B ( A ) (cid:1) and this implies λ is onto with λ ( A ) = E . (cid:3) In [2] we have shown that there is a bijection between the collection of all maximalideals in B ( X ) and the collection of all Z B -ultrafilters. Using Corollary 3.22 andCorollary 3.25 we therefore obtain the following: Theorem 3.26.
For any normal topological space X , M (cid:0) B ( X ) (cid:1) and M (cid:0) B ∗ ( X ) (cid:1) have the same cardinality, where M (cid:0) B ( X ) (cid:1) , M (cid:0) B ∗ ( X ) (cid:1) are the collections of allmaximal ideals in B ( X ) and B ∗ ( X ) respectively.Proof. Let us define a map ψ : M (cid:0) B ( X ) (cid:1)
7→ M (cid:0) B ∗ ( X ) (cid:1) by ψ ( M ) = E − B (cid:0) Z B [ M ] (cid:1) .From [2] we know that Z B [ M ] is a Z B -ultrafilter on X and by Theorem 3.16 E − B (cid:0) Z B [ M ] (cid:1) is a maximal ideal in B ∗ ( X ). We claim that ψ is one-one and onto.Suppose ψ ( M ) = ψ ( N ), for some M, N ∈ M (cid:0) B ( X ) (cid:1) . Clearly E B (cid:0) E − B (cid:0) Z B [ M ] (cid:1)(cid:1) = E B (cid:0) E − B (cid:0) Z B [ N ] (cid:1)(cid:1) contained in Z B [ M ] as well as Z B [ N ]. Since each member of Z [ M ] intersects every member of E B (cid:0) E − B (cid:0) Z B [ N ] (cid:1)(cid:1) , by Theorem 3.15 Z B [ M ] ⊆ ORE ON THE RINGS B ( X ) AND B ∗ ( X ) 13 Z B [ N ]. Similarly, Z B [ N ] ⊆ Z B [ M ]. Hence Z B [ M ] = Z B [ N ] and this implies M = N [2]. i.e., ψ is one-one.Let M ∗ ∈ M (cid:0) B ∗ ( X ) (cid:1) . By Theorem 3.19 E B ( M ∗ ) is an e B -ultrafilter. Let A be the unique Z B -ultrafilter containing E B ( M ∗ ). Z − B [ A ] belongs to M (cid:0) B ( X ) (cid:1) [2]. We know E B (cid:0) E − B ( A ) (cid:1) is the unique e B -ultrafilter contained in A . There-fore E B (cid:0) E − B ( A ) (cid:1) = E B ( M ∗ ) and E − B ( A ) = M ∗ (since both E − B ( A ) and M ∗ are maximal ideals in B ∗ ( X ) and maximal ideals are e B -ideals). Consider M = Z − B [ A ], clearly M ∈ M (cid:0) B ( X ) (cid:1) . Now ψ ( M ) = E − B (cid:0) Z B [ M ] (cid:1) = E − B (cid:0) Z B [ Z − B [ A ]] (cid:1) = E − B ( A ) = M ∗ [2]. i.e., Z − B [ A ] is the preimage of M ∗ . (cid:3) The following property characterizes maximal ideals of B ∗ ( X ). Theorem 3.27.
An ideal M ∗ in B ∗ ( X ) is a maximal ideal if and only if whenever f ∈ B ∗ ( X ) and every E ǫB ( f ) intersects every member of E B ( M ∗ ) , then f ∈ M ∗ .Proof. Suppose f ∈ B ∗ ( X ) and every E ǫB ( f ) meets E B ( M ∗ ), where M ∗ is a max-imal ideal. We claim f ∈ M ∗ . If not, then the ideal < M ∗ , f > generated by M ∗ and f must be equal to B ∗ ( X ). Hence 1 = h + f g , where g ∈ B ∗ ( X ) and h ∈ M ∗ .Let u be an upper bound of g and ǫ (0 < ǫ < ) be any pre assigned positivenumber.Then we have ∅ = E ǫB (1) = E ǫB ( h + f g ) ⊇ E ǫ B ( h ) T E ǫ B ( f g ) ⊇ E ǫ B ( h ) T E ǫ u B ( f ).This is a contradiction to our hypothesis as E ǫ B ( h ) T E ǫ u B ( f ) = ∅ . Hence f ∈ M ∗ .Conversely, suppose M ∗ is any ideal in B ∗ ( X ) with the given property. Let M ∗∗ bea maximal ideal containing M ∗ in B ∗ ( X ) and f ∈ M ∗∗ . As E B ( M ∗∗ ) is an e B -filter, E ǫB ( f ) meets every member of E B ( M ∗∗ ), for any ǫ >
0. Since E B ( M ∗ ) ⊆ E B ( M ∗∗ ), E ǫB ( f ) meets every member of E B ( M ∗ ) also. By our hypothesis f ∈ M ∗ . Therefore M ∗ = M ∗∗ . (cid:3) Remark . One may observe that each e B -ideal of B ∗ ( X ) is closed in B ∗ ( X )with uniform norm topology and hence, closed in B ∗ ( X ) with respect to the relative m B -topology. Since B ∗ ( X ) is closed in B ( X ) with respect to m B -topology, it thenfollows that every e B -ideal of B ∗ ( X ) is a closed set in B ( X ) with m B -topology. References [1] A. Deb Ray and Atanu Mondal,
On Rings Of Baire One Functions , Appl. Gen. Topol.,20(1) (2019), 237-249.[2] A. Deb Ray and Atanu Mondal,
Ideals In B ( X ) And Residue Class Rings Of B ( X ) Modulo An Ideal , Appl. Gen. Topol., 20(2) (2019), 379-393.[3] L. Gillman and M. Jerison,
Rings of Continuous Functions.
New York: Van NostrandRein- hold Co., 1960.[4] E. Hewitt,
Rings of real-valued continuous functions. I , Trans. Amer. Math. Soc., 64(1948), 4599[5] Amir Veisi, e c -Filters and e c -ideals in the functionally countable subalgebra of C ∗ ( X ),Appl. Gen. Topol., 20(2) (2019), 395-405.[6] Libor Vesely, Characterization of Baire-One Functions Between Topological Spaces , ActaUniversitatis Carolinae. Mathematica et Physica, 33 (2) (1992), 143-156.
Department of Pure Mathematics, University of Calcutta, 35, Ballygunge CircularRoad, Kolkata - 700019, INDIA
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