MynardAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 8, No. 2, 2007 pp. 171-185 Relations that preserve compact filters Frédéric Mynard Abstract. Many classes of maps are characterized as (possibly multi-valued) maps preserving particular types of compact filters. 1. Introduction A filter F on X is compact at A ⊂ X if every finer ultrafilter has a limit point in A. As a common generalization of compactness (in the case of a principal filter) and of convergence, it is not surprising that the notion turned out to be very useful in a variety of context (see for instance [5], [6], [2] under the name of compactoid filter, [16], [17], [18] under the name of total filter). The purpose of this paper is to build on the results of [5] and [6] to show that a large number of classes of single and multi-valued maps classically used in topology, analysis and optimization are instances of compact relation, that is, relation that preserves compactness of filters. It is well known (see for instance [5], [2]) that upper semi-continuous multivalued maps and compact valued upper semi-continuous maps are such instances. S. Dolecki showed [6] that closed, countably perfect, inversely Lindelöf and perfect maps are other examples of compact relations. In this paper, it is shown that continuous maps as well as various types of quotient maps (hereditarily quotient, countably biquotient, biquotient) are also compact relations. Moreover, I show that maps among these variants of quotient and of perfect maps with ranges satisfying certain local topological properties (such as Fréchetness, strong Fréchetness and bisequentiality) can be directly characterized in similar terms. This requires to work in the category of convergence spaces rather than in the category of topological spaces. Therefore, I recall basic facts on convergence spaces in the next section. The companion paper [15], which should be seen as a sequel to the present paper, uses these characterizations to present applications of product theorems for compact filters to theorems of stability under product of variants of com- pactness, of local topological properties (Fréchetness and its variants, among others) and of the classes of maps discussed above. 172 F. Mynard 2. Terminology and basic facts 2.1. Convergence spaces. By a convergence space (X, ξ) I mean a set X endowed with a relation ξ between points of X and filters on X, denoted x ∈ limξ F or F → ξ x, whenever x and F are in relation, and satisfying lim F ⊂ lim G whenever F ≤ G; {x}↑ → x (1) for every x ∈ X and lim (F ∧ G) = lim F ∩lim G for every filters F and G (2). A map f : (X, ξ) → (Y, τ ) is continuous if f (limξ F) ⊂ limτ f (F). If ξ and τ are two convergences on X, we say that ξ is finer than τ, in symbols ξ ≥ τ, if IdX : (X, ξ) → (X, τ ) is continuous. The category Conv of convergence spaces and continuous maps is topological (3) and cartesian-closed (4). Two families A and B of subsets of X mesh, in symbols A#B, if A ∩ B 6= ∅ whenever A ∈ A and B ∈ B. A subset A of X is ξ-closed if limξ F ⊂ A whenever F#A. The family of ξ-closed sets defines a topology T ξ on X called topological modification of ξ. The neighborhood filter of x ∈ X for this topology is denoted Nξ(x) and the closure operator for this topology is denoted clξ . A convergence is a topology if x ∈ limξ Nξ(x). By definition, the adherence of a filter (in a convergence space) is: (2.1) adhξ F = ⋃ G#F limξ G. In particular, the adherence of a subset A of X is the adherence of its principal filter {A}↑. The vicinity filter Vξ(x) of x for ξ is the infimum of the filters converging to x for ξ. A convergence ξ is a pretopology if x ∈ limξ Vξ(x). A convergence ξ is respectively a topology, a pretopology, a paratopology, a pseudotopology if x ∈ limξ F whenever x ∈ adhξ D, for every D-filter D#F where D is respectively, the class cl ♮ ξ (F1) of principal filters of ξ-closed sets ( 5), the class F1 of principal filters, the class Fω of countably based filters, the class F of all filters. In other words, the map AdhD [4] defined by (2.2) limAdhD ξ F = ⋂ D�D#F adhξ D 1If A ⊂ 2X , A↑ = {B ⊂ X : ∃A ∈ A, A ⊂ B}. 2Several different variants of these axioms have been used by various authors under the name convergence space. 3In other words, for every sink (fi : (Xi, ξi) → X)i∈I , there exists a final convergence structure on X : the finest convergence on X making each fi continuous. Equivalently, for every source (fi : X → (Yi, τi))i∈I there exists an initial convergence: the coarsest conver- gence on X making each fi continuous. 4In other words, for any pair (X, ξ), (Y, τ ) of convergence spaces, there exists the coarsest convergence [ξ, τ ] -called continuous convergence- on the set C(ξ, τ ) of continuous functions from X to Y making the evaluation map ev : (X, ξ) × (C(ξ, τ ), [ξ, τ ]) → (Y, τ ) (jointly) continuous. 5More generally, if o : 2X −→ 2X and F ⊂ 2X then o♮F denotes {o(F ) : F ∈ F} and if D is a class of filters (or of family of subsets) then o♮ (D) denotes {F : ∃D ∈ D, F = o♮D}. Relations that preserve compact filters 173 is the (restriction to objects of the) reflector from Conv onto the full subcat- egory of respectively topological, pretopological, paratopological and pseudo- topological spaces when D is respectively, the class cl ♮ ξ (F1), F1, Fω and F. A convergence space is first-countable if whenever x ∈ lim F, there exists a countably based filter H ≤ F such that x ∈ lim H. Of course, a topological space is first-countable in the usual sense if and only if it is first-countable as a convergence space. Analogously, a convergence space is called sequentially based if whenever x ∈ lim F, there exists a sequence (xn)n∈ω ≤ F ( 6) such that x ∈ lim(xn)n∈ω. A class of filters D (under mild conditions on D) defines a reflective subcat- egory of Conv (and the associated reflector) via (2.2). Dually, it also defines (under mild conditions on D) the coreflective subcategory of Conv of D-based convergence spaces [4], and the associated (restriction to objects of the) core- flector BaseD is (2.3) limBaseD ξ F = ⋃ D�D≤F limξ D. For instance, if D = Fω is the class of countably based filter, then BaseD is the coreflector on first-countable convergence spaces. If D is the class E of filters generated by sequences, then BaseD is the coreflector on sequentially based convergences. 2.2. Local properties and special classes of filters. Recall that a topo- logical space is Fréchet (respectively, strongly Fréchet) if whenever x is in the closure of a subset A (respectively, x is in the intersection of closures of elements of a decreasing sequence (An)n of subsets of X) there exists a sequence (xn)n∈ω of elements of A (respectively, such that xn ∈ An) such that x ∈ lim(xn)n∈ω. In other words, if x is in the adherence of a principal (resp. countably based) filter, then there exists a sequence meshing with that filter that converges to x. These are special cases of the following general notion, defined for convergence spaces. Let D and J be two classes of filters. A convergence space (X, ξ) is called (J/D)-accessible if adhξ J ⊂ adhBaseD ξ J , for every J ∈ J. When D = Fω and J is respectively the class F, Fω and F1, then (J/D)-accessible topological spaces are respectively bisequential, strongly Fréchet and Fréchet spaces. Analogously, if D is the class of filters generated by long sequences (of arbitrary length) and J = F1 then (J/D)-accessible topo- logical spaces are radial spaces. We use the same names for these instances of (J/D)-accessible convergence spaces (see [4] for details). 6From the viewpoint of convergence, there is no reason to distinguish between a sequence and the filter generated by the family of its tails. Therefore, in this paper, sequences are identified to their associated filter and I will freely treat sequences as filters. Hence the notation (xn)n∈ω ≤ F. 174 F. Mynard A filter F is called J to D meshable-refinable, in symbol F ∈ (J/D)#≥, if J ∈ J, J #F =⇒ ∃D ∈ D, D#J and D ≥ F. It follows immediately from the definitions that a topological space is (J/D)- accessible if and only if every neighborhood filter is J to D meshable-refinable, and more generally that: Theorem 2.1. Let D and J be two classes of filters. (1) A convergence space (X, ξ) is (J/D)-accessible if and only if ξ ≥ AdhJ BaseD ξ; (2) If ξ = Base(J/D)#≥ ξ, then ξ is (J/D)-accessible. If moreover ξ is pre- topological (in particular topological) then the converse is true. The following gathers the most common cases of (J/D)-accessible (topo- logical) spaces and (J/D)#≥-filters when D = Fω. Denote by F∧ω the class of countably deep (7) filters. The names for (J/Fω)#≥-filters come from the fact that a topological space is (J/Fω)-accessible if and only if every neighborhood filter is a (J/Fω)#≥-filter. class J (J/Fω)-accessible space (J/Fω)#≥-filter F bisequential [14] bisequential Fω strongly Fréchet or countably bisequential [14] strongly Fréchet (Fω/Fω)#≥ productively Fréchet[12] productively Fréchet F∧ω weakly bisequential [1] weakly bisequential F1 Fréchet [14] Fréchet Table 1 2.3. Compactness. Let D be a class of filters on a convergence space (X, ξ) and let A be a family of subsets of X. A filter F is D-compact at A (for ξ) [7] if (8) (2.4) D ∈D, D#F =⇒ adhξ D#A. Notice that a subset K of a convergence space X (in particular of a topo- logical space) is respectively compact, countably compact, Lindelöf if {K}↑ is D-compact at {K} if D is respectively, the class F of all, Fω of countably based, F∧ω of countably deep filters. On the other hand Theorem 2.2. Let D be a class of filters. A filter F is D-compact at {x} for ξ if and only if x ∈ limAdhD ξ F. In particular, if ξ is a topology, then x ∈ lim F if and only if F is compact at {x} if and only if F is F1-compact at {x}. For a topological space X, a subset K is compact if and only if every open cover of K has a finite subcover of K, if and only if every filter on K has 7A filter F is countably deep if ⋂ A ∈ F whenever A is a countable subfamily of F. 8Notice that (2.4) makes sense not only for a filter but for a general family F of subsets of X. Such general compact families play an important role for instance in [8]. Relations that preserve compact filters 175 adherent points in K. In contrast, for general convergence spaces, the definition of compactness in terms of covers (cover-compactness) and in terms of filters (compactness) are different. If (X, ξ) is a convergence space, a family S ⊂ 2X is a cover of K ⊂ X if every filter converging to a point of K contains an element of S. Hence a subset K of a convergence space is called cover-(countably ) compact if every (countable) cover of K has a finite subcover. It is easy to see that a cover-compact convergence is compact, but in general not conversely. For instance, in a pseudotopological but not pretopological convergence, points are compact, but not cover-compact. Notice that in this definition, we can assume the original cover S to be stable under finite union, in which case we call S an additive cover. The family Sc of complements of elements of an additive cover S is a filter-base on X with empty adherence. Hence K is cover-(countably) compact if every (countable) additive cover of K has an element that is a cover of K, or equivalently, if every (countable) filter-base with no adherence point in K has an element with no adherence point in K. In other words, K is cover-(countably) compact if every (countably based) filter whose every member has adherent points in K, has adherent points in K. More generally, we will need the following characterization of cover-compactness in terms of filters [6]. Let D and J be two classes of filters. A filter F is (D/J)- compact at B if D ∈ D, ∀J ∈ J, J ≤ D, adh J #F =⇒ adh D#B. It is clear than if F is (D/F1)-compact (at B), then it is D-compact (at B). More precisely, we have the following relationship between (D/F1)-compactness and D-compactness (which could be deduced from the results of [6, section 8]) Proposition 2.3. Let D be a class of filters on a convergence space (X, ξ). A filter F is (D/F1)-compact at B if and only if Vξ(F) = ⋃ F ∈F ⋂ x∈F Vξ(x) is D-compact at B. Proof. By definition, F is (D/F1)-compact at B if and only if D ∈ D, ( adh ♮ ξ D ) #F =⇒ adh D#B. It is easy to verify that ( adh ♮ ξ D ) #F if and only if D#Vξ(F), which concludes the proof. � Calling a convergence ξ pretopologically diagonal, or P -diagonal, if limξ F ⊂ limξ Vξ(F) for every filter F, we obtain the following result, which is a particular case of a combination of Propositions 8.1 and 8.3 and of Theorem 8.2 in [6], even though the assumption that adh ♮ ξ D ⊂ D seems to be erroneously missing in [6]. Corollary 2.4. If ξ is P -diagonal (in particular if ξ is a topology) and if adh ♮ ξ D ⊂ D, then (D/F1)-compactness amounts to D-compactness for ξ. 176 F. Mynard Proof. Assume that F is D-compact (at B). To show that it is (D/F1)-compact (at B), we only need to show that Vξ(F) is D-compact (at B). But D#Vξ(F) if and only if ( adh ♮ ξ D ) #F. Therefore, adhξ(adh ♮ ξ D)#B because adh ♮ ξ D ∈ D. Now, x ∈ adhξ(adh ♮ ξ D) if there exists a filter G# adh ♮ ξ D with x ∈ limξ G. Note that Vξ(G)#D and that, by P -diagonality, x ∈ limξ Vξ(G). Hence adhξ(adh ♮ ξ D) ⊂ adhξ D and adhξ D#B. � In some sense, the converse is true: Proposition 2.5. If ξ = AdhD ξ and D-compactness implies (D/F1)-compactness in ξ, then ξ is P -diagonal. Proof. If x ∈ limξ F then F is D-compact at {x}, hence (D/F1)-compact at {x}. Since ξ = AdhD ξ, we only need to show that x ∈ adhξ D whenever D is a D-filter meshing with Vξ(F). For any such D, we have adh ♮ ξ D#F so that x ∈ adhξ D because F is (D/F1)-compact at {x}. � 2.4. Contour filters. If F is a filter on X and G : X → FX then the contour of G along F is the filter on X defined by ∫ F G = ∨ F ∈F ∧ x∈F G(x). This type of filters have been used in many situations, among others by Froĺık under the name of sum of filters for a ZFC proof of the non-homogeneity of the remainder of βN [11], by C. H. Cook and H. R. Fisher [3] under the name of compression operator of F relative to G, by H. J. Kowalsky [13] under the name of diagonal filter, and after them by many other authors to characterize topologicity and regularity of convergence spaces. To generalize this construc- tion, I need to reproduce basic facts on cascades and multifilters. Detailed information on this topic can be found in [9]. If (W, ⊑) is an ordered set, then we write W (w) = {x ∈ W : w ⊑ x}. An ordered set (W, ⊑) is well-capped if its every non empty subset has a maxi- mal point (9). Each well-capped set admits the (upper) rank to the effect that r(w) = 0 if w ∈ max W , and for r(w) > 0, r(w) = rW (w) = sup v=w (r(v) + 1). A well-capped tree with least element is called a cascade; the least element of a cascade V is denoted by ∅ = ∅V and is called the estuary of V . The rank of a cascade is by definition the rank of its estuary. A cascade is a filter cascade if its every (non maximal) element is a filter on the set of its immediate successors. 9In other words, a well-capped ordered set is a well-founded ordered set for the inverse order. Relations that preserve compact filters 177 A map Φ : V \ {∅V } → X, where V is a cascade, is called a multifilter on X. We talk about a multifilter Φ : V → X under the understanding that Φ is not defined at ∅V . A couple (V, Φ0) where V is a cascade and Φ0 : max V → A is a called a perifilter on A. In the sequel we will consider V implicitly talking about a perifilter Φ0. If Φ|max V = Φ0, then we say that the multifilter Φ is an extension of the perifilter Φ0. The rank of a multifilter (perifilter) is, by definition, the rank of the corresponding cascade. If D is a class of filters, we call D-multifilter a multifilter with a cascade of D-filters as domain. The contour of a multifilter Φ : V → X depends entirely on the underlying cascade V and on the restriction of Φ to max V , hence on the corresponding perifilter (V, Φ|max V ). Therefore we shall not distinguish between the contours of multifilters and of the corresponding perifilters. The contour of Φ : W → X is defined by induction to the effect that ∫ Φ = Φ♮(∅W ) if r(Φ) = 1, and ( 10) ∫ Φ = ∫ ∅W ( ∫ Φ|W (.) ) otherwise. With each class D of filters we associate the class ∫ D of all D-contour filters, i. e., the contours of D-multifilter. If D and J are two classes of filters, we say that J is D-composable if for every X and Y, the (possibly degenerate) filter HF = {HF : H ∈ H, F ∈ F}↑ (11) belongs to J(Y ) whenever F ∈ J(X) and H ∈ D(X × Y ), with the convention that every class of filters contains the degenerate filter. If a class D is D- composable, we simply say that D is composable. Notice that (2.5) H# (F × G) ⇐⇒ HF#G ⇐⇒ H−G#F, where H−G={H−G = {x ∈ X : (x, y) ∈ H and y ∈ G} : H ∈ H, G ∈ G}↑. Lemma 2.6. Let D and J be two classes of filters. If D is a J-composable class of filters, then ∫ D is also J-composable. Proof. We proceed by induction on the rank of a D-multifilter. The case of rank 1 is simply J-composability of D. Assume that for each D-multifilter Φ on X of rank β smaller than α and each J-filter J on X × Y, the filter J ( ∫ Φ) is the contour of some D-multifilter on Y. Consider now a D-multifilter (Φ, V ) on X of rank α and a J-filter J on X × Y. Then ∫ Φ = ∫ ∅V ( ∫ Φ|V (.) ) = ∨ F ∈∅V ∧ v∈F ∫ Φ|V (v), and J ( ∫ Φ ) = ∨ F ∈∅V ∧ v∈F J ( ∫ Φ|V (v) ) . 10Φ(v) is the image by Φ of v treated as a point of V , while Φ♮(v) is the filter generated by {Φ(F ) : F ∈ v}. 11HF = {y ∈ Y : (x, y) ∈ H and x ∈ F }. 178 F. Mynard As each Φ|V (v) is a multifilter of rank smaller than α, each J (∫ Φ|V (v) ) is a (∫ D ) -filter. Moreover ∅V is a D-filter, so that J (∫ Φ ) is a contour of (∫ D ) - filters along a D-filter, hence a (∫ D ) -filter. � 3. Compact relations A relation R : (X, ξ) ⇉ (Y, τ ) is D-compact if for every subset A of X and every filter F that is D-compact at A, the filter RF is D-compact at RA. Proposition 3.1. If D is F1-composable, then R : (X, ξ) ⇉ (Y, τ ) is D-compact if and only if RF is D-compact at Rx whenever x ∈ limξ F. Proof. Only the ”if” part needs a proof, so assume that RF is D-compact at Rx whenever x ∈ limξ F, and consider a filter G on X which is D-compact at A. Let D#RG be a D-filter on Y . Then R−D#G so that there exists x ∈ A∩adhξ R −D. Therefore, there exists U#R−D such that x ∈ limξ U. By assumption, RU is D-compact at Rx ⊂ RA. Since D#RU, the filter D has adherent points in Rx hence in RA. � Corollary 3.2. Let D be an F1-composable class of filters and let f : (X, ξ) → (Y, τ ) with τ = AdhD τ . The following are equivalent: (1) f is continuous; (2) f is a compact relation; (3) f is a D-compact relation. Proof. (1 =⇒ 2). If x ∈ limξ F, then f (x) ∈ limτ f (F) so that f (F) is compact at f (x) and f is a compact relation by Proposition 3.1. (2 =⇒ 3) is obvious and (3 =⇒ 1) follows from Proposition 2.2. � In particular, F1-compact (equivalently compact) maps between pretopolog- ical spaces (in particular between topological spaces) are exactly the continuous ones. Notice that when D contains the class of principal filters, then a D-compact relation R is F1-compact and Rx is D-compact for each x in the domain of R, because {x}↑ is D-compact at {x}. When the cover and filter versions of compactness coincide (in particular, in a topological space), the converse is true: Proposition 3.3. Let D be an F1-composable class of filters. If R : (X, ξ) ⇉ (Y, τ ) is an F1-compact relation and if Rx is (D/F1)-compact in τ for every x ∈ X, then R is D-compact. Proof. Using Proposition 3.1, we need to show that RF is D-compact at Rx whenever x ∈ limξ F. Consider a D-filter D#RF. Then, adhτ D#Rx for every D ∈ D so that adhτ D#Rx, because Rx is D F1 -compact. � In view of Corollary 2.4, we obtain: Relations that preserve compact filters 179 Corollary 3.4. Let D be an F1-composable class of filters such that adh ♮ τ D(τ ) ⊂ D(τ ) and let τ be a P -diagonal convergence (for instance a topology). Then R : (X, ξ) ⇉ (Y, τ ) is D-compact if and only if it is F1-compact and Rx is D- compact in τ for every x ∈ X. An immediate corollary of [9, Theorem 8.1] is that for a topology, D-compactness amounts to (∫ D ) -compactness, provided that D is a composable class of fil- ters. However, the proof of [9, Theorem 8.1] only uses F1-composability of D. Consequently, Corollary 3.5. Let D be an F1-composable class of filters and let τ be a topol- ogy such that adh♮τ D(τ ) ⊂ D(τ ). Let R : (X, ξ) ⇉ (Y, τ ) be a relation. The following are equivalent: (1) R is D-compact; (2) R is F1-compact and Rx is D-compact in τ for every x ∈ X; (3) R is F1-compact and Rx is (∫ D ) -compact in τ for every x ∈ X; (4) R is (∫ D ) -compact. Proof. (1 ⇐⇒ 2) and (3 ⇐⇒ 4) follow from Corollary 3.4 and (1 ⇐⇒ 4) follows from [9, Theorem 8.1]. � The observation that perfect, countably perfect and closed maps can be characterized as D-compact relations is due to S. Dolecki [6, section 10]. Recall that a surjection f : X → Y between two topological spaces is closed if the im- age of a closed set is closed and perfect (resp. countably perfect, resp. inversely Lindelöf ) if it is closed with compact (resp. countably compact, resp. Lindelöf) fibers. Once the concept of closed maps is extended to convergence spaces, all the other notions extend as well in the obvious way. As observed in [6, section 10], preservation of closed sets by a map f : (X, ξ) → (Y, τ ) is equivalent to F1-compactness of the inverse map f − when (X, ξ) is topological, but not if ξ is a general convergence. More precisely, calling a map f : (X, ξ) → (Y, τ ) adherent [6] if y ∈ adhτ f (H) =⇒ adhξ H ∩ f −y 6= ∅, we have: Lemma 3.6. (1) A map f : (X, ξ) → (Y, τ ) is adherent if and only if f − : (Y, τ ) ⇉ (X, ξ) is an F1-compact relation; (2) If f : (X, ξ) → (Y, τ ) is adherent, then it is closed; (3) If f : (X, ξ) → (Y, τ ) is closed and if adherence of sets are closed in ξ (in particular if ξ is a topology), then f is adherent. Proof. (1) follows from the definition and is observed in [6, section 10]. (2) If f (H) is not τ -closed, then there exists y ∈ adhτ f (H)�f (H). Since f is adherent, there exists x ∈ adhξ H ∩ f −y. But x /∈ H because f (x) = y /∈ f (H).Therefore H is not ξ-closed. (3) is proved in [6, Proposition 10.2] even if this proposition is stated with a stronger assumption. � 180 F. Mynard Hence, a map f : (X, ξ) → (Y, τ ) with a domain in which adherence of subsets are closed (in particular, a map with a topological domain) is adherent if and only if it is closed if and only if f − : (Y, τ ) ⇉ (X, ξ) is an F1-compact relation. If the domain and range of a map are topological spaces, it is well known that closedness of the map amounts to upper semicontinuity of the inverse relation. It was observed (for instance in [5]) that a (multivalued) map is upper semicontinuous (usc) if and only if it is an F1-compact relation. A surjection f : X → Y is D-perfect if it is adherent with D-compact fibers. In view of Corollary 3.5, compact valued usc maps between topological spaces, known as usco maps, are compact relations. Another direct consequence of Lemma 1 and of Corollary 3.5 is: Theorem 3.7. Let f : (X, ξ) → (Y, τ ) be a surjection, let D be an F1- composable class of filters, and let ξ be a topology such that adh ♮ ξ D ⊂ D. The following are equivalent: (1) f is D-perfect; (2) f − : Y ⇉ X is D-compact; (3) f − : Y ⇉ X is (∫ D ) -compact; (4) f is (∫ D ) -perfect. The equivalence between the first two points was first observed in [6, Propo- sition 10.2] but erroneously stated for general convergences as domain and range. Indeed, if f : (X, ξ) → (Y, τ ) is a surjective map between two con- vergence spaces and if f − : (Y, τ ) ⇉ (X, ξ) is D-compact, then f is adherent and has D-compact fibers; if on the other hand f is adherent and has (D/F1)- compact fibers then f − : (Y, τ ) ⇉ (X, ξ) is D-compact. Hence, the two con- cepts are equivalent only when D-compact sets are (D/F1)-compact in ξ, for instance if adh ♮ ξ (D) ⊂ D and ξ is a P -diagonal convergence (in particular if ξ is a topology). S. Dolecki offered in [4] a unified treatment of various classes of quotient maps and preservation theorems under such maps in the general context of convergences. He extended the usual notions of quotient maps to convergence spaces in the following way. A surjection f : (X, ξ) → (Y, τ ) is D-quotient if (3.1) y ∈ adhτ H =⇒ f −(y) ∩ adhξ f −H 6= ∅, for every H ∈ D(Y ). When D is the class of all (resp. countably based, principal, principal of closed sets) filters, then continuous D-quotient maps between topo- logical spaces are exactly biquotient (resp. countably biquotient, hereditarily quotient, quotient) maps. Now, I present a new characterization of D-quotient maps as D-compact relations, in this general context of convergence spaces. As mentioned before, the category of convergence spaces and continuous maps is topological, hence if f : (X, ξ) → Y, there exists the finest convergence —called final convergence and denoted f ξ — on Y making f continuous. Analogously, if f : X → (Y, τ ), there exists the coarsest convergence — called initial conver- gence and denoted f −τ — on X making f continuous. If τ is topological, so is f −τ. In contrast, f ξ can be non topological even when ξ is topological. Relations that preserve compact filters 181 Theorem 3.8. Let D be an F1-composable class of filters. Let f : (X, ξ) → (Y, τ ) be a surjection. The following are equivalent: (1) f : (X, ξ) → (Y, τ ) is D-quotient; (2) τ ≥ AdhD f ξ; (3) f : (X, f −τ ) → (Y, f ξ) is a D-compact relation. Proof. The equivalence (1 ⇐⇒ 2) is [4, Theorem 1.2]. (1 ⇐⇒ 3) . Assume f is D-quotient and let x ∈ limf −τ F. Then f (x) ∈ limτ f (F), so that f (x) ∈ adhτ D whenever D ∈D(Y ) and D#f (F). By (3.1) , f −(f (x)) ∩ adhξ f −D 6= ∅ so that f (x) ∈ f (adhξ f −D) . In view of [6, Lemma 2.1], f (x) ∈ adhf ξ D. Conversely, assume that f : (X, f −τ ) → (Y, f ξ) is D-compact and let y ∈ adhτ D. There exists G#D such that y ∈ limτ G. By definition of f −τ, the filter f −G is converging to every point of f −y for f −τ. In view of Proposition 3.1, the filter f f −G is D-compact at {y} for f ξ. Since f is surjective, f f −G ≈ G and G#D so that y ∈ adhf ξ D = f (adhξ f −D) , by [6, Lemma 2.1]. Therefore, f −(y) ∩ adhξ f −D 6= ∅. � Notice that even if the map has topological range and domain, the notions need to be extended to convergence spaces to obtain such a characterization. 4. Compactly meshable filters and relations In view of the characterizations above of various types of maps as D-compact relations, results of stability of D-compactness of filters under product would particularize to product theorems for D-compact spaces, but also for various types of quotient maps, for variants of perfect and closed maps, for usc and usco maps. Product theorems for D-compact filters and their applications is the purpose of the companion paper [15]. A (complicated but extremely useful) notion fundamental to this study of products is the following: A filter F is M-compactly J to D meshable at A, or F is an M-compactly (J/D)#-filter at A, if J ∈ J, J #F =⇒ ∃D ∈ D, D#J and D is M-compact at A. While the importance of this concept will be best highlighted by how it is used in the companion paper [15], I show here that the notion of an M- compactly (J/D)#-filter is instrumental in characterizing a large number of classical concepts. The notion of total countable compactness was first introduced by Z. Froĺık [10] for a study of product of countably compact and pseudocompact spaces and rediscovered under various names by several authors (see [19, p. 212]). A topological space X is totally countably compact if every countably based filter has a finer (equivalently, meshes a) compact countably based filter. The name comes from total nets of Pettis. Obviously, a topological space is totally countably compact if and only if {X} is compactly Fω to Fω meshable. In [19], J. Vaughan studied more generally under which condition a product of D-compact spaces is D-compact, under mild conditions on the class of filters 182 F. Mynard D. He used in particular the concept of a totally D-compact space X, which amounts to {X} being a compactly (D/D)#-filter. On the other hand, Theorem 2.1 can be completed by the following im- mediate rephrasing of the notion of M-compactly (J/D)#-filters relative to a singleton in convergence theoretic terms. Proposition 4.1. Let D, J and M be three classes of filters, and let ξ and θ be two convergences on X. The following are equivalent: (1) θ ≥ AdhJ BaseD AdhM ξ; (2) F is an M-compactly (J/D)#-filter at {x} in ξ whenever x ∈ limθ F. In particular, ξ = AdhM ξ is (J/D)-accessible if and only if F is an M-compactly (J/D)#-filter at {x} whenever x ∈ lim F. In view of Table 1, this applies to a variety of classical local topological properties. A relation R : (X, ξ) ⇉ (Y, τ ) is M-compactly (J/D)-meshable if F → ξ x =⇒ R(F) is M-compactly (J/D) -meshable at Rx in τ. Theorem 4.2. Let M ⊂ J, let τ = AdhM τ and let f : (X, ξ) → (Y, τ ) be a continuous surjection. The map f is M-quotient with (J/D)-accessible range if and only if f : (X, f −τ ) → (Y, f ξ) is an M-compactly (J/D)-meshable relation. Proof. Assume that f is M-quotient with (J/D)-accessible range and let x ∈ limf −τ F. Then y = f (x) ∈ limτ f (F). Let J be a J-filter such that J #f (F). Since y ∈ adhτ J and τ is (J/D)-accessible, there exists a D-filter D#J such that y ∈ limτ D. To show that f (F) is M-compactly (J/D)-meshable at y in f ξ, it remains to show that D is M-compact at {y} for f ξ, that is, that y ∈ limAdhM f ξ D, which follows from the M-quotientness of f. Conversely, assume that f : (X, f −τ ) → (Y, f ξ) is an M-compactly (J/D)- meshable relation, and let y ∈ limτ G. Then f −(G) converges to any point x ∈ f −y for f −τ. Therefore, f (f −G) is M-compactly (J/D)-meshable at {y} in f ξ. Because f is a surjection, f (f −G) = G. Consider M∈ M ⊂ J such that M#G. There exists a D-filter D#M which is M-compact at {y} in f ξ. Hence, y ∈ adhf ξ M, so that y ∈ limAdhM f ξ G. Therefore, f is M-quotient. Moreover, if y ∈ adhτ J for a J-filter J , then there exists G#J such that y ∈ limτ G. By the previous argument, G is M-compactly (J/D)-meshable at {y} in f ξ. In particular, there exists a D-filter D#J which is M-compact at {y} in f ξ. In other words, y ∈ limAdhM f ξ D. Since f : (X, ξ) → (Y, τ ) is continuous, τ ≤ f ξ so that AdhM τ = τ ≤ AdhM f ξ. Hence y ∈ limτ D and τ is (J/D)-accessible. � Relations that preserve compact filters 183 M J D map f as in Theorem 4.2 F1 F F1 hereditarily quotient with finitely generated range F1 F1 Fω hereditarily quotient with Fréchet range F1 Fω Fω hereditarily quotient with strongly Fréchet range F1 F Fω hereditarily quotient with bisequential range F1 F F hereditarily quotient Fω Fω F1 countably biquotient with finitely generated range Fω Fω Fω countably biquotient with strongly Fréchet range Fω F Fω countably biquotient with bisequential range Fω F F countably biquotient F F F1 biquotient with finitely generated range F F Fω biquotient with bisequential range F F F biquotient Theorem 4.3. Let M ⊂ J and D be three classes of filters, where J and D are F1-composable. Let τ = AdhM τ and let ξ be a P -diagonal convergence such that adh ♮ ξ(M) ⊂ M . Let f : (X, ξ) → (Y, τ ) be a continuous surjection. The map f is M-perfect with (J/D)-accessible range if and only if f − : (Y, τ ) ⇉ (X, ξ) is an M-compactly (J/D)-meshable relation. Proof. Assume that f is M-perfect with (J/D)-accessible range and let y ∈ limτ G. Consider a J-filter J #f −G. By F1-composability, f (J ) is a J-filter. Moreover f (J )#G so that y ∈ adhτ f (J ). Since τ is (J/D)-accessible, there exists a D-filter D#f (J ) such that y ∈ limτ D. In view of Corollary 3.5, f − : (Y, τ ) ⇉ (X, ξ) is M-compact because f is M-perfect. Therefore, f −D is M -compact at f −y in ξ. Moreover, f −D ∈ D because D is F1-composable and f −D#J . Hence, f −G is M-compactly (J/D)-meshable at f −y in ξ. Conversely, assume that f − : (Y, τ ) ⇉ (X, ξ) is an M-compactly (J/D)- meshable relation. It is in particular an M-compact relation because M ⊂ J. In view of Corollary 3.5, f is M-perfect. Now assume that y ∈ adhτ J where J ∈ J. There exists G#J such that y ∈ limτ G. Therefore, f −G is M-compactly (J/D)-meshable at f −y in ξ. 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Ninth Annual Spring Topology Conf., Memphis State Univ., Memphis, Tenn., 1975), Lecture Notes in Pure and Appl. Math., Vol. 24 (New York), Dekker, 1976, pp. 259–265. [19] , Products of topological spaces, Gen. Topology Appl. 8 (1978), 207–217. Relations that preserve compact filters 185 Received November 2005 Accepted May 2006 Frédéric Mynard (fmynard@georgiasouthern.edu) Dept. Mathematical Sciences, Georgia Southern University, PO Box 8093, Statesboro, Ga 30460-8093, U.S.A.