Monomorphisms in spaces with Lindelöf filters via some compact-open-like topologies on C(X)
aa r X i v : . [ m a t h . GN ] D ec Monomorphisms in spaces with Lindel¨offilters via some compact-open-liketopologies on C ( X ) Vasil Gochev ∗ Dept. of Math. and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria
Abstract
An object in the category
SpFi of spaces with filters, is a pair ( X , F ),where X is a compact Hausdorff space and F is a filter of dense open subsetsof X . A morphism f : ( Y , F ′ ) → ( X , F ) is a continuous map f : Y → X for which f − ( F ) ∈ F ′ whenever F ∈ F . In the present work we study thecategorical monomorphisms in the subcategory LSpFi of spaces with Lindel¨offilters, meaning filters with a base of Lindel¨of, or cozero sets. Of course, thesemonomorphisms need not be one-to-one. We extend the criterion derived byR. Ball and A. Hager in [3]. We define and study some compact-open-liketopologies on the set C ( X ) of continuous real-valued functions defined ona topological space X . We use them to give a new characterization of themonomorphisms in LSpFi . The category
SpFi of spaces with filters arises naturally from considerations inordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and fromconsiderations in general topology, e.g., the theory of the absolute and other covers, ∗ This paper was supported by the project No. 140/08.05.2014 ,,Function spaces and dualities”of the Sofia University ,,St. Kl. Ohridski”. Key words and phrases: C ( X ), ˇCech-Stone compactification, epimorphism, monomorphism,epi-topology, compact-open topology, space with filter, frame, lattice-ordered group. E-mail address: [email protected]fia.bg
SpFi is a pair ( X , F ), where X is acompact Hausdorff space and F is a filter of dense open subsets of X . A morphism f : ( Y , F ′ ) → ( X , F ) is a continuous map f : Y → X for which f − ( F ) ∈ F ′ whenever F ∈ F . The subcategory LSpFi of spaces with Lindel¨of filters is definedas follows: ( X, F ) ∈ LSpFi if and only if ( X, F ) ∈ SpFi and F has a base of cozerosets. Main references for topology are [7], [8], and [9]. All topological spaces arecompletely regular Hausdorff, usually compact. Comp is the category of compact(Hausdorff) spaces with continuous maps. For a space X , C ( X ) is the set (or l -group, vector lattice, ring, l -ring,...) of continuous real-valued functions on X . The cozero set of f ∈ C ( X ) is coz f ≡ { x | f ( x ) = 0 } , and coz X ≡ { coz f | f ∈ C ( X ) } .Each cozero set in a compact space X is an F σ , hence Lindel¨of.In a general category, a monomorphism is a morphism m which is left–cance-lable: mf = mg implies f = g . A one-to-one map is a monomorphism in SpFi since it is a monomorphism in the category
Comp of compact Hausdorff spaceswith continuous maps since it is a monomorphism in
Sets , but a monomorphism in
SpFi is not necessarily one-to-one. Such examples one can find in [3].This work focuses on the characterization of monomorphisms in the category
LSpFi via compact-open-like topologies defined on C ( X ). More precisely, we derivea new necessary and sufficient condition a morphism in LSpFi to be monomorphism(see Theorem 2.3(4) below). Based on this condition, for every object ( X , F ) ∈ SpFi are defined topologies τ F and τ F ε =0 on C ( X ) for which the following characterizationsare derived Corollary 1.1. (Corollary 4.1 below) f : ( X , F ) → ( Y , F ′ ) is a monomorphism in LSpFi if and only if e f ( C ( Y )) is dense in ( C ( X ) , τ F ε =0 ) . Corollary 1.2. (Corollary 4.2 below) f : ( X , F ) → ( Y , F ′ ) is a monomorphism in LSpFi if and only if e f ( C ( Y )) is dense in ( C ( X ) , τ F ) . Here, the map e f : C ( Y ) → C ( X ) is defined by e f ( g ) = g ◦ f for every g ∈ C ( Y ).We have shown in [10] that these topologies are always T , countably tight, andhomogeneous. We established necessary and sufficient conditions these topologies tobe Hausdorff or group topologies. We don’t include these results here because of thesignificant extra complications and length. There are different characterizations ofthe monomorphisms in SpFi and
LSpFi which are briefly summarized in [3], butfor more details one can see also [5], [6], [12], [13], [15], and [16].
First we will recall some of the results established by R. Ball and A. Hager in [3].
Theorem 2.1. ([3])
Let ( Y , F ′ ) f ← ( X , F ) ∈ LSpFi . Then the following are equiva-lent. f is a monomorphism in LSpFi . (2) If x = x ∈ X then there are S ∈ F δ and neighborhoods U i of x i for which f [ U ∩ S ] ∩ f [ U ∩ S ] = ∅ . (3) If K and K are disjoint compact sets in X then there are S ∈ F δ andneighborhoods U i of K i for which f [ U ∩ S ] ∩ f [ U ∩ S ] = ∅ . (4) For each b ∈ C ( X ) there is S ∈ F δ for which x , x ∈ S and f ( x ) = f ( x ) imply b ( x ) = b ( x ) . Corollary 2.2. ([3])
If there is S ∈ F or S ∈ F δ such that f is one-to-one on S then f is monic.Proof. If there is S ∈ F or S ∈ F δ such that f is one-to-one on S then condition (2)of the previous theorem holds, so f is a monomorphism.In the following theorem, the equivalence of (1), (2), and (3) is from [3]. Wehave included condition (4). Theorem 2.3.
Let ( Y , F ′ ) f ← ( X , F ) ∈ LSpFi , and let A ≡ e f ( C ( Y )) . Then thefollowing are equivalent. (1) f is a monomorphism in LSpFi . (2) For every b ∈ C ( X ) there is an E ∈ F δ such that for every x , x ∈ E thereis a ∈ A with ax = bx and ax = bx . (3) For every b ∈ C ( X ) there is an E ∈ F δ such that for every finite F ⊆ E there is a ∈ A with a | F = b | F . (4) For every b ∈ C ( X ) there is an E ∈ F δ such that for every compact K ⊆ E there is a ∈ A with a | K = b | K .Proof. The equivalence of the first three conditions in the theorem is proven in [3].The implication (4) ⇒ (3) is clear. We will show that the condition (4) of Theorem2.1 implies (4) of this theorem. Let us fix b and S ∈ F δ as in Theorem 2.1(4). Thenfor every compact ∅ 6 = K ⊆ S , f ( K ) is a compact subset of Y . Let us define on f ( K ) a function g : f ( K ) → R as follows g ( y ) = b ( x ), where x ∈ K and y = f ( x ).The function g is well defined, because on f − ( y ) ∩ K the function b is equal to b ( x ).If g is continuous, then we can extend it to e g : Y → R by the normality of Y . Then a = e g ◦ f has the required property a | K = b | K . Thus we need to show that g iscontinuous function.Let us consider the following general situation: f : X → Y is a continuous ontomapping where X and Y are compact Hausdorff spaces. Let X /E ( f ) be the quotientspace under the equivalence relation E ( f ) induced by f . It is clear that f is closedmapping. Now from Corollary 2.4.8 from [8] it follows that f is a quotient mappingand by Proposition 2.4.3 from [8] it follows that the mapping f : X /E ( f ) → Y is3 homeomorphism. We have the following situation: X, Y, Z - topological spaces,
X, Y - compact Hausdorff, f : X → Y is a continuous onto mapping, g : X → Z is continuous and has the property g /f − ( y ) = constant = z ( y ) ∈ Z for every y ∈ Y . The map e g : Y → Z is defined by e g ( y ) = z ( y ), f : X /E ( f ) → Y is definedby f ( f − ( y )) = y , and q : X → X /E ( f ) is the natural quotient mapping. Fromthe previous notes it follows that f : X /E ( f ) → Y is a homeomorphism, so e g iscontinuous if and only if e g ◦ f is continuous if and only if (by proposition 2.4.2 from[8]) e g ◦ f ◦ q = e g ◦ f = g is continuous. This implies that the function g : f ( K ) → R defined above is continuous.Condition (4) from the previous theorem suggests on C ( X ) a topology τ suchthat A is a dense subset in ( C ( X ) , τ ). In the following section we define two topolo-gies on C ( X ) which have that property and characterize the monomorphisms in LSpFi . C ( X ) Let X be a topological space and S be a dense subset of X . Let τ S be the rel-ativization of the compact-open topology τ co on C ( S ) to C ( X ). (More details onthe definition end properties of the compact-open topology one can find in [8] and[17].) Here we consider C ( X ) as a subspace of ( C ( S ) , τ co ) via the natural embed-ding C ( X ) ∋ f f | S ∈ C ( S ). Since ( C ( S ) , τ co , + , ∨ , ∧ ) is a Hausdorff topological l -group with + , ∨ , ∧ defined pointwise, so is ( C ( X ) , τ S , + , ∨ , ∧ ) and the neighbor-hoods of the constant function 0 suffice to describe it.If S is a family of dense subsets of a topological space X , then for each S ∈ S wehave the corresponding τ s as above, and we consider the S -topology τ S ≡ ∧{ τ s | S ∈ S } on C ( X ). Here ∧ is taken in the lattice of topologies on C ( X ). It is now unclearwhat are the properties of the structures ( C ( X ) , τ S )? Topology τ S is always T , butnot necessarily Hausdorff or group topology. (See [11] for the definition of topologicalgroup and more.)The above is surely (or may be not) too general to say much. We limitthe generality to the following considerations: Let ( X, F ) ∈ LSpFi and F δ ≡{∩ F n | F , F , F . . . ∈ F } . For S ∈ F δ , let τ S be as above and K ( S ) is the familyof all compact subsets of S . We define τ F ≡ ∧{ τ s | S ∈ F δ } . A local base at a point f ∈ C ( X ) is the collection of all sets of the form S S ∈ F δ U s ( K s , ε s , f ), where K s ∈ K ( S )and ε s ∈ (0 ,
1] for every S ∈ F δ .The following proposition follows immediately from definitions. Proposition 3.1.
Let f : ( X , F ) → ( Y , F ′ ) be a morphism in LSpFi , τ F and τ F ′ be the corresponding topologies on C ( X ) and C ( Y ) . Let e f : C ( Y ) → C ( X ) be theinduced map defined by e f ( g ) = g ◦ f for every g ∈ C ( Y ) . Then e f : ( C ( Y ) , τ F ′ ) → ( C ( X ) , τ F ) is continuous. X be a Tychonoff space, andlet K ( X ) be the family of all compact subsets of X . We define the compact-zero topology τ ε =0 on C ( X ) as follows: Basic neighborhoods for f ∈ C ( X ) are of theform U ( K, f ) = { g ∈ C ( X ) | f /K = g/K } , where K ∈ K ( X ). It is straightforwardto check that { U ( K, f ) | f ∈ C ( X ) , K ∈ K ( X ) } form a base for a topology on C ( X ).This topology coincides with the discrete topology if and only it X is compact. It isclear that U ( K, f ) = { g ∈ C ( X ) | x ∈ K ⇒ | f ( x ) − g ( x ) | = 0 } , i.e. we allow ε = 0 inthe usual compact-open topology. Obviously this topology is finer than the compact-open topology on C ( X ). One can easily check that + , − , ∨ , ∧ are continuous, thus( C ( X ) , τ ε =0 , + , ∨ , ∧ ) is a Hausdorff topological l -group. Let ( X, F ) ∈ LSpFi and fora fixed S ∈ F δ let τ Sε =0 be the relativization of the compact-zero topology on C ( S )to C ( X ). We define the topology τ F ε =0 ≡ ∧{ τ Sε =0 | S ∈ F δ } . A local base at a point f ∈ C ( X ) is the collection of all sets of the form S S ∈ F δ U Sε =0 ( K s , f ), where K s ∈ K ( S )for every S ∈ F δ .The following proposition follows immediately from the definitions Proposition 3.2.
Let f : ( X , F ) → ( Y , F ′ ) be a morphism in LSpFi , τ F ε =0 and τ F ′ ε =0 be the corresponding topologies on C ( X ) and C ( Y ) . Let e f : C ( Y ) → C ( X ) be theinduced map defined by e f ( g ) = g ◦ f for every g ∈ C ( Y ) . Then e f : ( C ( Y ) , τ F ′ ε =0 ) → ( C ( X ) , τ F ε =0 ) is continuous. Propositions 3.1 and 3.2 show that the associations ( X , F ) → ( C ( X ) , τ F ( τ F ε =0 ))are functorial([14]).The space ( C ( X ) , τ F ε =0 ( τ F )) is always countably tight, homogeneous, and T ,but not necessarily Hausdorff or topological group. Necessary and sufficient condi-tions for τ F ε =0 or τ F to be Hausdorff, or group topology on ( C ( X ) , +) are derived in[10]. These considerations are not included here, because of their significant lengthand will be examined in a later paper. Of course, one can ask many questions aboutthe properties of these topologies. For an inspiration of such questions one could see[1] and [17]. τ F ε =0 and τ F The main results in this work are the following corollaries.
Corollary 4.1.
Let ( Y , F ′ ) f ← ( X , F ) ∈ LSpFi , and let A ≡ e f ( C ( Y )) . Then f is amonomorphism in LSpFi if and only if A is dense in ( C ( X ) , τ F ε =0 ) .Proof. The proof is straightforward and follows from Theorem 2.3(4) and the factthat basic neighborhoods of a function f ∈ C ( X ) are of the form S S ∈ F δ U Sε =0 ( K s , f ),where K s ∈ K ( S ) for every S ∈ F δ . 5 orollary 4.2. Let ( Y , F ′ ) f ← ( X , F ) ∈ LSpFi and let A ≡ e f ( C ( Y )) . Then f is amonomorphism in LSpFi if and only if A is dense in ( C ( X ) , τ F ) .Proof. ⇒ ) Obviously follows from Theorem 2.3(4). ⇐ ) For the converse we need some additional preliminary definitions and facts.Let ( X , F ) ∈ LSpFi and X ∗ = lim ← { βF | F ∈ F } , where βF is the ˇCech-Stonecompactification of F . For X ∗ the bonding maps are: when F, F ∈ F and F ⊆ F , βF π FF ← βF is the ˇCech - Stone extension of the inclusion. For F ∈ F , there is theprojection βF π F ← X ∗ , thus there exists a continuous map π : X ∗ → X . This π isirreducible, so inversely preserves dense sets ( π − [ F ] is dense in X ∗ for every F ∈ F ).Let F ∗ has base { π − [ F ] | F ∈ F } . Then π : ( X ∗ , F ∗ ) ։ ( X , F ) is a morphism in LSpFi and it is monic. (See [3].)
Proposition 4.3.
Let ( Y , F ′ ) f ← ( X , F ) ∈ LSpFi and π be as above. Then f ismonic if and only if f ◦ π is monic.Proof. If f is monic then f ◦ π is monic as a composition of two monomorphisms.For the converse we will use Theorem 2.3(4). We have( Y , F ′ ) f ← ( X , F ) π ← ( X ∗ , F ∗ ) , and let A ≡ e f ( C ( Y )) and B ≡ e π ( C ( X )). Let b ∈ C ( X ) then e π ( b ) ∈ C ( X ∗ ) andsince f ◦ π is monic, there exists D ∈ F ∗ δ such that for every compact K ⊆ D thereis c ∈ e π ( A ) = e π ( e f ( C ( Y ))) with c | K = e π ( b ) | K . Since { π − [ F ] | F ∈ F } is a base for F ∗ , we can take D = π − [ E ] where E ∈ F δ . Let K ∈ K ( E ), then π − [ K ] ∈ K ( D ),so there is a map c ∈ e π ( A ) such that c | π − [ K ] = e π ( b ) | π − [ K ]. Since c ∈ e π ( A ), thereis a map a ∈ A such that c = a ◦ π . Therefore a ◦ π | π − [ K ] = b ◦ π | π − [ K ], but thatmeans a | K = b | K .Let SY be the “ SpFi
Yosida functor” W SY → LSpFi ([5] and [18]). Here W is the category of archimedean l -groups with distinguished weak order unit, with l -group homomorphisms which preserve unit. This category includes all rings ofcontinuous functions C ( X ). Consider ( X , F ) π և ( X ∗ , F ∗ ) ∈ LSpFi and let C [ F ] =lim → { C ( F ) | F ∈ F } ∈ W . We have e π : C ( X ) → C [ F ] ∈ W , defined as e π ( f ) = f ◦ π .Let B ≡ e π ( C ( X )) ≤ C [ F ]. Since π is monic, ( X ∗ , F ∗ ) π ′ ։ ( X, { X } ) is monic too. π ′ = SY ( e π ), so e π is W -epic ([2]). Therefore B is τ F ∗ -dense in C [ F ] ([2]). Thus B is τ F ∗ -dense in C ( X ∗ ). Proposition 4.4.
Consider ( X ∗ , F ∗ ) π ′ ։ ( X, { X } ) . Then e π : ( C ( X ) , τ F ) → ( C ( X ∗ ) , τ F ∗ ) is a topological and algebraic embedding. roof. The proof is straightforward and follows from the following facts:(1) π has the property: ∀ E ∈ F ∗ δ ∃ S E ∈ F δ ( π − [ S E ] ⊆ E ) . (2) e π : ( C ( X ) , τ S E ) → ( C ( X ∗ ) , τ E ) is a topological and algebraic embedding. (Seealso [7] and [17].) Corollary 4.5.
Let ( Y , F ′ ) f և ( X , F ) ∈ LSpFi and A ≡ e f ( C ( Y )) be dense in ( C ( X ) , τ F ) . Then f is monic.Proof. By Proposition 4.3 it suffices to show that ( Y , F ′ ) f ← ( X , F ) π ← ( X ∗ , F ∗ ) ismonic. B ≡ e π ( C ( X )) is τ F ∗ -dense in C [ F ] and by Proposition 4.4 ( C ( X ) , τ F ) ishomeomorphic to B . Therefore A is τ F ∗ -dense in C [ F ]. Thus A λ ≤ C [ F ] is W -epic.Then SY ( λ ) is LSpFi -monic. But SY ( λ ) = f ◦ π .This completes the proof of Corollary 4.2.Let us note, that the condition (3) from Theorem 2.3 suggests another topologyon C ( X ). More precisely, for any S ∈ F δ one could consider the topology of pointwiseconvergence (see [1]) on C ( S ) and the relativization of this topology to C ( X ). If itis denoted by σ s , then define the topology σ F ≡ ∧{ σ S | S ∈ F δ } . This topology iscoarser than the topology τ F . For a topological space Y , let F in ( Y ) be the familyof all finite subsets of Y . For a function f ∈ C ( X ) the basic neighborhoods are ofthe form S S ∈ F δ U S ( K S , ε S , f ), where K S ∈ F in ( S ) and ε S ∈ (0 ,
1] for every S ∈ F δ .Sets of the form U S ( K S , ε s , f ) ≡ { g ∈ C ( X ) | x ∈ K S ⇒ | f ( x ) − x ( x ) | < ε } , where K S ∈ F in ( S ) and ε S ∈ (0 , f in ( C ( X ) , σ S ). Fromthis fact and Theorem 2.3(3) the following corollary is immediate. Corollary 4.6.
Let ( Y , F ′ ) f և ( X , F ) ∈ LSpFi be a monomorphism in
LSpFi and A ≡ e f ( C ( Y )) . Then A is dense in ( C ( X ) , σ F ) . In a similar way one can define the topology σ F ε =0 as the meet of the topolo-gies { σ Sε =0 | S ∈ F δ } . Here in the definition of the topology of pointwise convergencewe again alow ε = 0, i.e. sets of the form U S ( K S , f ) ≡ { g ∈ C ( X ) | x ∈ K S ⇒| f ( x ) − x ( x ) | = 0 } , where K S ∈ F in ( S ), are the basic neighborhoods of f in( C ( X ) , σ Sε =0 ). Thus, for a function f ∈ C ( X ) the basic neighborhoods are of theform S S ∈ F δ U S ( K S , f ), where K S ∈ F in ( S ) for every S ∈ F δ . Now, it is clear that thefollowing corollary is true. Corollary 4.7.
Let ( Y , F ′ ) f և ( X , F ) ∈ LSpFi and A ≡ e f ( C ( Y )) . Then f is amonomorphism in LSpFi if and only if A is dense in ( C ( X ) , σ F ε =0 ) . Let us mention, that the authors of [3] said some words about a possible topol-ogy on C ( X ), but left this problem open. We do not know is the converse of Corollary4.6 also true? 7 eferences [1] A. V. Arkhangel’skii : Topological Function Spaces, Mathematics and its Apli-cations, Vol.78, Kiuwer Academic Publishers, Dordrecht, Boston, London,1992.[2]
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