The Algebra of Conditional Sets and the Concepts of Conditional Topology and Compactness

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1 The Algebra of Conditional Sets and the Concepts of Conditional Topology and Compactness Samuel Drapeau a,1,, Asgar Jamneshan c,2,, Martin Karliczek b,3,, Michael Kupper c,4, August 5, 2014 ABSTRACT We introduce the notions of conditional set and conditional inclusion. It is shown that the resulting conditional power set is a complete Boolean algebra which allows to develop a set theory, topology and analysis on conditional sets. Among others a conditional version of the following theorems are proved: Ultrafilter lemma, Tychonoff, Heine- Borel, Hahn-Banach and Banach-Alaoglu. KEYWORDS: Conditional set theory, conditional topology, conditional compactness, conditional real numbers, conditional topological vector spaces AUTHORS INFO a Technische Universität Berlin, Straße des 17. Juni 135, Berlin b Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin c Universität Konstanz, Universitätstraße 10, Konstanz 1 drapeau@math.hu-berlin.de 2 asgar.jamneshan@uni-konstanz.de 3 karliczm@math.hu-berlin.de 4 kupper@uni-konstanz.de Funding: MATHEON project E.11 Funding: Berlin Mathematical School Funding: Konsul Karl und Dr. Gabriele Sandmann Stiftung PAPER INFO We thank Fares Maalouf, Ieke Moerdijk, Stephan Müller and Martin Streckfuß for helpful comments and fruitful discussions. AMS CLASSIFICATION: 03B60,03C90,46S99,54J99 1 Introduction Conditional set theory is a novel method to study local and dynamic dependency. A conditional set encodes the information stored in a measure space, or more generally any complete Boolean algebra. Conditional set operations are constructed in such a way that they respect the structure of a conditional set and satisfy the Boolean laws. This allows to built thereupon conditional versions of basic topological notions and the conditional real numbers for which conditional versions of classical theorems are proved. These theorems then can be used as tools to deal with problems which depend locally on an uncertain information flow for example. Conditional set theory developed from the conditional analysis in topological L 0 -modules. Motivated by the duality of conditional risk measures, separation and duality for topological modules over the ring L 0 of measurable functions are established in Filipović et al. [8]. The proofs are adaptations of the respective proofs about real vector spaces combined with exhaustion arguments. In an exhaustion argument one constructs the greatest measurable set such that a certain property is satisfied. It relies strongly on the property of σ-stability of sets [8]. Furthermore, the conditional structure of finitely ranked L 0 -modules is studied in Cheridito et al. [5] and a conditional Brouwer fixed point theorem on (L 0 ) d is proved in Drapeau et al. [6] with similar ideas as in [8]. These results have already found numerous applications, for instance to risk measure theory in Filipović et al. [9], to utility maximization in Cheridito and Hu [3] and to backward stochastic differential equations in Cheridito and Stadje [4]. A version of the conditional set operations on (L 0 ) d, motivated by the construction of a conditional Lebesgue measure, was obtained in Streckfuß [19]. In Haydon et al. [10] randomly normed spaces, that is L 0 -modules endowed with vector space topologies, are introduced to study ultrapowers of Lebesgue-Bochner spaces. Moreover, see Schweizer and Sklar [16] for an comprehensive account on probabilistic metric spaces that is a linear space endowed 1

2 with a random metric yielding (ε, λ)-topologies. Recently, Hahn-Banach theorems for modules over Stonean algebras has been studied in Cerreia-Vioglio et al. [2]. Conceptually, a conditional set is related to a Boolean-valued set (see Bell [1] and the references therein), or equivalently to a sheaf over a complete Boolean algebra (see Lane and Moerdijk [14] and the references therein). We refer for a discussion and the connection to topos theory to Jamneshan [11]. The essential feature of L 0 -modules which allows to develop a conditional real analysis is the action of the measure algebra A associated to the underlying measure space (Ω, F, µ) by means of the indicator functions 1 A where A A. This action satisfies the following properties: Consistency: 1 A X = 1 A Y and B A imply 1 B X = 1 B Y. Stability: For every maximal disjoint family (A n ) n N A and every (X n ) n N L 0 there exists a unique X L 0 such that 1 An X n = 1 An X. More generally, we consider any complete Boolean algebra A and a structure - called a conditional set - given by a family of sets (X a ) and surjective functions γ a : X 1 X a both of which are parametrized by A satisfying consistency and stability. Here γ a is the substitute of the indicator 1 A. The conditional subset relation is a combination of set inclusion and comparing relative algebras of A. This relation yields on the conditional power set the structure of a complete Boolean algebra. The latter result is key to the definition of conditional topologies and the construction of the conditional real numbers. All conditional structures are defined such that they allow for an action of the underlying Boolean algebra satisfying consistency and stability. Often conditional structures can be recovered from classical structures with additional properties. For instance, we show how classical convergence implies conditional convergence and vice versa in Section 3. Of interest is the notion of conditional compactness, although weaker than the classical one, still yields the same consequences as the Tychonoff and the Heine-Borel theorem. The structure of this paper is as follows: In Section 2, we introduce conditional sets and the conditional set operations, and prove that the conditional power set has the structure of a complete Boolean algebra. Elementary properties of conditional set operations, relations and functions are given, conditional countability is introduced and the existence of a conditional choice function is proved. In Section 3, conditional topologies are defined together with the corresponding notions of convergence, continuity and compactness, and a conditional version of the Ultrafilter lemma and Tychonoff s theorem are proved. In Section 4, conditionally totally ordered fields are introduced and the conditional real numbers are constructed. The connection to L 0 -theory is discussed and a Heine-Borel theorem for conditional metric spaces is proved. Finally in Section 5, we introduce conditional topological vector spaces and prove among others a conditional version of the Banach-Alaoglu theorem. 2 Conditional set theory Let A = (A,,, c, 0, 1) be a complete Boolean algebra. For instance, a σ-complete Boolean algebra which satisfies the countable chain condition is complete [20], in particular the associated measure algebra of a σ-finite measure space (see e.g. [15]). Recall that A endowed with the order a b defined by a b = a is a complete lattice. For any (a i ) A the supremum and infimum is denoted by a i and a i respectively. Let A a = {b A : b a} denote the relative algebra of A with respect to a A. Furthermore, define p(a) = {(a i ) A : a i = a, a i a j = 0 whenever i j}, a A. 1 1 Note that (a i ) p(a) is not a partition of a, since a i = 0 for some i is allowed. 2

3 We will often rely on the following fact: For every (a i ) i I A there exists (b i ) i I p( a i ) such that b i a i for all i I. Indeed, suppose a well-ordering on the index I and define b i := a i ( j<i b i ) c for each i I. Definition 2.1. Let (A,,, c, 0, 1) be a complete Boolean algebra, (X a ) a A be a family of sets and (γ a ) a A be a family of functions γ a : X 1 X a. Then is a conditional set on A if and only if (i) X 0 is a singleton, (ii) surjectivity: γ a is surjective for all a A, (iii) identity: γ 1 is the identity function, X := (X a, γ a ) a A (iv) consistency: γ a (x) = γ a (y) whenever γ b (x) = γ b (y), x, y X 1 and a b, (v) stability: for every (a i ) p(1) and for every (x i ) X ai there exists a unique x X 1 such that γ ai (x) = x i for all i. We call X 1 the primal set of X. In the class of all conditional sets we identify two conditional sets X and Y whenever the only difference is X 0 Y 0. Thus there exists one and only one conditional set on the degenerate Boolean algebra A = {0} which is called the conditional empty set and is denoted by 0. Examples 2.2. (i) Conditional sets on the trivial algebra A = {0, 1} correspond uniquely to non-empty sets X by means of X 1 = X and X 0 = { }. (ii) Every complete Boolean algebra A is a conditional set on A by setting X a = A a and γ a (b) = a b for every b A a and all a A. (iii) Let (Ω, F, µ) be a σ-finite measure space, A be its associated measure algebra and L 0 be the set of equivalence classes of real valued measurable functions. Set X a := {1 A X : X L 0, A a} L 0 and γ a (X) = 1 A X where 1 A is the indicator function of A a. 2 Then L 0 := (X a, γ a ) a A is the conditional set associated to L 0. (iv) Let X = (X a, γ a ) a A be a conditional set. For b A the restriction of X to b is defined as bx := (X a, γa) b a Ab. Remark 2.3. Fix a conditional set X = (X a, γ a ) a A. The surjectivity of γ a and X 0 imply that X a for all a A. Due to consistency there exists a unique function γ b a : X b X a such that γ b a γ b = γ a for every a, b A with a b and satisfying (i) γ a a is the identity for all a A, (ii) γ b a γ c b = γc a for all a, b, c A with a b c, (iii) for all a A, every (a i ) p(a) and each (x i ) X ai there exists a unique x X a such that γa a i (x) = x i for all i. 2 The associated measure algebra is the quotient algebra of F and its σ-ideal of null sets. The set A a is a representative of the equivalence class a A. Since 1 A L 0 is independent of representatives X a and γ a are well-defined. 3

4 We introduce the following notations. For each a A define ax := γ b a(x), for all b a and every x X b. Given (a i ) p(a) for some a A, a family (x i ) X ai is called matching and the unique element x X a satisfying a i x = x i for all i is called the amalgamation of (x i ). The combination of (a i ) and a matching family (x i ) is denoted by [a i, x i ] A a X. For the amalgamation of some [a i, x i ] we write ai x i and for a finite amalgamation we write a 1 x a n x n. For all a, b A with a b and every [b i, x i ] A b X it holds a b i x i = (a b i )x i. (2.1) Moreover, for all a A, every (a i ) i I p(a) and all [b ij, x ij ] j Ji A ai X for each i I it holds a i b ij x ij = (a i b ij )x ij. (2.2) i I j J i j J i, i I Conditional sets can be generated from classical sets as follows. Let E be a non-empty set and A be a complete Boolean algebra. For every (a i, x i ) A E where (a i ) p(a) for some a A, define formally a i x i := (a i, x i ) and denote by E a the collection of all such families where two of them ai x i and b j y j are identified whenever {a i : x i = z} = {b j : y j = z}, for all z E. Define γ a : E 1 E a by a i x i (a a i )x i. Inspection shows that E := (E a, γ a ) a A is a conditional set, and E is called the conditional set generated by E with respect to A. The conditional sets N, Z and Q generated by N, Z and Q, respectively, are called the conditional natural numbers, integers and rational numbers. Definition 2.4. Let (X i ) be a family of conditional sets on a complete Boolean algebra A. The conditional product of (X i ) is defined by ( X i := X i a, (γa)) i if the family is non-empty and 0 otherwise. Let X and Y be non-empty sets. Then the conditional product of the generated conditional sets X and Y coincides with the conditional set generated by the product set X Y. Definition 2.5. Let X = (X a, γ a ) a A and Y = (Y b, δ b ) b B be conditional sets. Then Y is a conditional subset of X, in symbols Y X, if and only if there exists b A such that (i) B = A b, (ii) Y a X a for all a A b, (iii) δ a is the restriction of γ b a to Y 1 for all a A b. In this case Y, as a conditional subset of X, is said to live on b. a A 4

5 Let X = (X a, γ a ) a A be a conditional set. A non-empty set Y X b for some b A is called stable if ai (a i x i ) Y, for all (a i ) p(b) and (x i ) Y. Every stable set Y corresponds uniquely to the conditional subset of X the primal set of which is Y. Every singleton x X b is stable and the corresponding conditional subset is called a conditional element of X. For every non-empty Y X b { ai (a i x i ) : (a i ) p(b), (x i ) Y } is stable by (2.2). We call the corresponding conditional subset the stable hull of Y, and denote it by cond(y ). Definition 2.6. The conditional power set of X is the conditional set P (X) = (P (X) a, π a ) a A where P (X) a := {Y a b : Y X, Y lives on b} and π a : P (X) 1 P (X) a maps Y b Y a b. Note that ap (X) = P (ax). Remark 2.7. If there is no risk of confusion, we will often identify a conditional set with its primal set. In particular, we write x X to denote a conditional element such that x 1 X 1 and Y P (X) to denote Y X. We understand then (x i ) X to be x i X for each i and (Y i ) P (X) to be Y i P (X) for each i. Fix for the remainder of this paper a non-degenerate complete Boolean algebra (A,,, c, 0, 1). Theorem 2.8. Let X be a conditional set. The conditional power set of X has the structure of a complete Boolean algebra. In particular, for Y, Y i X where Y i lives on a i for each i (i) the conditional union Y i is given by the stable set { } bi y i : (b i ) p( a i ), b i a i, y i Yb i i for all i whenever the family is non-empty and 0 otherwise; (ii) the conditional intersection Y i is given by the stable set Y i a where a := { a a i : Y i a } whenever the family is non-empty and X otherwise; (iii) the conditional complement is given by Y := {Z X : Y Z = 0}. Proof. We prove that (P (X), ) is a complete complemented distributive lattice. Inspection shows that (P (X), ) is a partially ordered set with least element 0 and greatest element X. 5

6 Step 1 : Without loss of generality suppose that (Y i ) is a non-empty family. First, we show that Y i is the least upper bound of (Y i ). Due to (2.2) the set { } Z = bi y i : (b i ) p( a i ), b i a i, y i Yb i i for each i is stable, and thus defines a conditional subset of X living on a := a i. In order to see that Y i is an upper bound it suffices to show that Ya i i ( Y i ) ai for all i. To this end let y Ya i i and z ( Y i ) a a c. i Then a i y + (a a c i )z Z which implies that y = a i (a i y + (a a c i)z) ( Y i ) ai. Now let Ŷ P (X) be another upper bound. For every b i y i Z it holds b i bi y i = y i Y i b i Ŷb i, for all i. Thus b i y i Ŷb due to stability of Ŷ. Hence Y i Ŷ and therefore Y is the least upper bound. Second, we show that Y i is the greatest lower bound of (Y i ). Since a is attained due to stability of X it holds that W = Ya i is stable. Thus Y i is a conditional subset of X living on a. By definition Ya i Ya i for all i. Hence Y i is a lower bound. Let Ŷ P (X) be another lower bound which lives on c. Then Ŷc Yc i which implies c a since Ŷc. Hence Ŷ Y i and therefore Y i is the greatest lower bound. Step 2 : In order to prove distributivity let Y k X where Y k lives on a k for k = 1, 2, 3. It suffices to verify one of the two distributive laws, and we choose Since (P (X), ) is a lattice it holds (Y 1 Y 2 ) (Y 1 Y 3 ) = Y 1 (Y 2 Y 3 ). (Y 1 Y 2 ) (Y 1 Y 3 ) Y 1 (Y 2 Y 3 ). For the reverse, it suffices to convince oneself from the identity c 1 c 2 = c 3 where and where c 1 := {c a 1 a 2 : Yc 1 Yc 2 } c 2 := {c a 1 a 3 : Yc 1 Yc 3 } c 3 := {c a 1 (a 2 a 3 ) : Yc 1 Z c }, Z c = {b 1 x + b 2 y : (b 1, b 2 ) p(a 2 a 3 ), b 1 a 2, b 2 a 3, x Y 2 b 1, y Y 3 b 2 } is the primal set of Y 2 Y 3. Indeed, for every x Y 1 (Y 2 Y 3 ) there exists a decomposition into two pieces each of which belongs to Y 1 Y 2 and Y 1 Y 3, respectively, since c 1 c 2 = c 3. Step 3 : Finally, in order to verify complementation let Y P (X). Then 0 Y Y Y Y X. Suppose, for the sake of contradiction, that Y Y lives on some a > 0. This implies Y Y Y and (Y Y ) Y = Y Y 0 6

7 contradicting the definition of the conditional complement. Thus Y Y = 0. For X Y Y let x X and b = M where M = {a : ax Y a }. Then there exists (b i ) p(b) such that for each i there exists a i M with b i a i. Due to stability and consistency of Y it holds bx = b i x Y b. Since b is the greatest element of M, this implies b c x Y = 0. Thus b c x Yb and therefore x = bx + b c x c Y Y. Since the conditional power set is a conditional set we can form P (P (X)). We denote by S(X) P (X) the conditional set of conditional subsets of X living on 1, S(S(X)) the conditional set of conditional systems of conditional subsets living on 1. The systems S(X) and S(S(X)) are important since often properties of conditional sets can be expressed in terms of their primal sets. In particular, they allow to compare classical and conditional concepts. Note that a Y i = ay i and a Y i = ay i which implies ( a j i I Y ij) a j = i I a j Ya ij j j J j J and ( a j i I Y ij) a j = i I a j Ya ij j (2.3) where we suppose that each Y ij and each i Y ij lives on a condition greater than a j for each j. The conditional power set, like every complete Boolean algebra, fulfills the following properties: j J (i) De Morgan s law: ( i I Y i ) = i I (Y i ) for every index I, (ii) Distributivity: i I j J Y ij = { i I Y if(i) : f J I} for any index J and finite I, (iii) Associativity: i I ( j J Y ij ) = i I, j J Y ij for any index I and J. Remark 2.9. The conditional power set is atomic if and only if the underlying Boolean algebra A is so. Indeed, let A be the set of atoms of A and X be a conditional set on A. Then the set of atoms of P (X) is {bx : b A, x X}. Conversely, if A is atomless then for each a > 0 there exists 0 < b < a such that bx ax and bx ax. Similarly one can show that the distributive law of the conditional power set is the distributive law of A. Corollary Let X be a conditional set and Y 1, Y 2 S(X) and (Y i ) S(X). Then (i) Y 1 Y 2 implies Y 1 1 Y 2 1, (ii) ( Y i ) 1 = Y i 1 whenever Y i lives on 1, (iii) Y i = cond( Y i 1 ). Definition Let X and Y be conditional sets. A conditional binary relation is a conditional set R S(X Y ). A conditional relation R S(X X) is conditionally reflexive, symmetric, antisymmetric or transitive whenever R a X a X a is classically reflexive, symmetric, antisymmetric or transitive for every a A. Moreover, R is a conditional partial order or equivalence relation if R 1 is a classical partial order or equivalence relation. A conditionally partially ordered set (X, ) is conditionally totally ordered if for every x, y X there exists (a, b, c) p(1) such that ax < a ay, by < b bx and cx = c cy. 3 j J 3 For a conditionally partially ordered set (X, ) the ordered set (X 1, 1 ) is total if and only if A is trivial or X 1 a singleton. Indeed, suppose that x y and x y. Then there exists a > 0 such that x a y a. Thus ax a + a c y a c cannot be compared with ay a + a c x a c. 7

8 Let (X, ) be a conditionally partially ordered set. For x, y X define x < y by x y and ax = ay implies a = 0. A conditional subset Y S(X) has a conditional upper bound, lower bound, supremum, infimum, maximum, minimum, greatest and smallest element if Y 1 does so in a classical sense with respect to 1, and Y is conditionally bounded if it has a conditional upper and lower bound. If every pair x, y X has a conditional infimum (supremum), then (X, ) is conditionally downwards (upwards) directed. A conditionally downwards and upwards directed set is a conditional lattice. Examples (i) Let X, Y and R X Y be non-empty sets. Then R S(X Y) is called the conditional relation generated by R. It holds xry if and only if x i Ry j for all (i, j) where x = a i x i X and y = b j y j Y. Therefore if R is a classical partial order, equivalence, total order, direction or lattice, then the conditional relation R has the respective conditional properties. As such the conditionally totally ordered sets (N, ), (Z, ) and (Q, ) are constructed. (ii) The conditional subset of L 0 L 0 (see Example 2.2) which corresponds to the order of almost sure dominance on L 0 defines on L 0 a conditional total order. 4 Definition Let X and Y be conditional sets. A conditional function f : X Y is a conditional binary relation G S(X Y ) such that G a is the graph of a function f a : X a Y a for every a A. Note that if f = (f a ) a A is a conditional function, then af : ax ay is a conditional function given by af = (f b ) b Aa for each a A. The conditional set of all conditional functions from X to Y is denoted by M(X, Y ). A conditional element of M(X, Y ) is sometimes called a conditional family. A conditional function f is conditionally injective if f a : X a Y a is injective for all a A, conditionally surjective if f 1 is surjective and conditionally bijective if it is both conditionally injective and surjective. Let f : X Y be a conditional function, U X living on a and V Y living on b. The conditional image f(u) of U is given by the stable set {f a (x) : x U a } and the conditional preimage f 1 (V ) of V by the stable set {x X b : f b (x) V b }, b = { c b : f 1 c (V c ) }. Remark A conditional function f can be defined equivalently as a family of functions f a : X a Y a satisfying f 1 ( ai x i ) = a i f ai (x i ), for all [a i, x i ] A X. Examples (i) The conditional function generated by a classical injective function is conditionally injective. (ii) Let (X i ) be a family of conditional sets. The conditional j-th projection π j : X i X j is defined by π j := (π j a) a A where π j a : X i a X j a is the classical j-th projection for each a A. (iii) The conditional embedding Y X, where X is a conditional set and Y S(X), is a conditional function f where f a : Y a X a is a classical embedding for every a A. 4 For f, g L 0 define f g by f 1 (ω) g 1 (ω) µ-almost surely. This order is stable under amalgamations. Moreover, for every pair f, g L 0 the family ({f 1 < g 1 }, {f 1 > g 1 }, {f 1 = g 1 }) is a partition of Ω almost surely. 8

9 (iv) Let f : X Y be a conditional function and Z S(X). The conditional restriction of f to Z is the conditional function f Z where f Z a is the classical restriction of f a to Z a for each a A. Proposition Let f : X Y be a conditional function. Then U f(u) and V f 1 (V ) are conditional functions. Let U P (X), (U i ) P (X), V P (Y ), (V i ) P (Y ), U 1, U 2 P (X) be such that U 1 U 2 and V 1, V 2 P (Y ) be such that V 1 V 2. Then it holds f( U i ) = f(u i ) f 1 ( V i ) = f 1 (V i ) (2.4) f( U i ) f(u i ) f 1 ( V i ) = f 1 (V i ) (2.5) f(u) f(x) f(u ) f 1 (V ) = f 1 (V ) (2.6) f(u 1 ) f(u 2 ) f 1 (V 1 ) f 1 (V 2 ) (2.7) U f 1 (f(u)) f(f 1 (V )) V (2.8) and it is even an equality on the left-hand side of (2.8) if f is conditionally injective, and on the right-hand side if V f(x). Proof. Let (a i ) p(1) and (U i ) be a matching family in P (X). Due to stability of P (X) it holds ( ) f 1 ai U i = {f 1 (x) : x } { } a i U i = ai f ai (x i ) : x i U i = a i f ai (U i ). This shows that U f(u) is a conditional function from P (X) to P (Y ). The fact that V f 1 (V ) is a conditional function from P (Y ) to P (X) follows from { ( ) } a : fa 1 (a i a)v i = i I {a : fa 1 (av i ) } i I for every (a i ) i I p(1) and all matching families (V i ) i I in P (Y ). Since f and f 1 are conditional functions, the assertions in (2.4) follow from stability and (2.2). In order to show the left-hand side of (2.5), assume that U i lives on a i for each i. Since f a ( U i a) f a (U i a) for every a A, it holds { a a i : f a ( U i a) } {a a i : f a (U i a) }. Therefore f( U i ) f(u i ). Let a = { a a i : V i a } where V i lives on a i for each i. Then {a a : fa 1 ( Va) i } = {a a : fa 1 (Va) i } which implies f 1 ( V i ) = f 1 (V i ). The remaining assertions are consequences of the Boolean laws, and can be shown similarly to the respective classical properties. We prove the existence of a conditional choice function on the conditional power set. Theorem Let X be a conditional set and (Y i ) S(S(X)) be a conditional family. Then there exists a conditional family (y i ) of conditional elements of X such that y i Y i for every i. Proof. Let (Y i ) i I be a conditional family. By definition I is a conditional set. Let H := { (y j ) j J : J S(I), y j Y j for all j J } This set is non-empty since it contains all the conditional families of one element. Furthermore, it is stable due to (2.2) and thus defines a conditional element of S(S(X)) which we also denote by H. Order 9

10 H by conditional inclusion, that is (y j ) j J (z l ) l L for (y j ) j J, (z l ) l L H means that J L and y j = z j for every j J. Let (y j ) j J t, t T, be a chain in H. Then J := t J t S(I). For every j J define y j = a s y js where (a s ) p(1), j s Ja t s for some t T and y js a s Y js. By construction (y j ) j J H and (y j ) j J t (y j ) j J for all t T. By Zorn s Lemma H has a maximal element (y j ) j J. Suppose, for the sake of contradiction, that J I. Pick i I J and assume, without loss of generality, that i lives on 1. Let L := J {i}. Then L S(I) and for every l L choose y i Y i, y j Y j and define z l = ay j + a c y i where l = aj + a c i. By construction (z l ) l L H and (y j ) j J is conditionally strictly contained in (z l ) l L since J is conditionally strictly contained in L. This contradicts the maximality of (y j ) j J. Recall that by (N, ) we denote the conditionally totally ordered set of conditional natural numbers. Note that for m, n N with m n the conditional function is conditionally bijective. f : {1 l n m + 1} {m l n}, l f(l) = l + m 1 5 (2.9) Lemma Let Y S(N) be conditionally bounded. Then there exists a unique n N such that there exists a conditional bijection from Y to {1 l n}. Proof. Suppose that Y is conditionally bounded by m N where m = b j m j. Let I j be the set of all non-empty subsets of {1,..., m j } N. For each I I j let a I = {a : a b j, Y a = I}. Then (a I ) I Ij p(b j ) for each j. By construction there exists a conditional bijection from Y to {1 l n} where n = b j I I j a I I. 6 Uniqueness is implied by (2.9). Definition A conditional set X is conditionally countable if there exists a conditional injection X N. A conditional set X is conditionally finite if there exists n N such that there exists a conditional bijection from X into {1 k n}. As a convention 0 is conditionally finite. The conditional natural and rational numbers are conditionally countable. Let n = a i n i N be a conditional natural number. Then {1 k n} is an element of S(N) and it holds ({ }) {1 k n} = cond ai k i : 1 k i n i, k i N (2.10) Now let X be a conditional set and (Y k ) 1 k n S(S(X)) be a conditionally finite family of conditional subsets each of which lives on 1. Then (2.10) together with (2.1) and (2.2) yield the representations 1 k n Y k = a i ( ni j=1 Y kj ) ai and 1 k n Y k = a i ( ni j=1 Y kj ) ai (2.11) where we identified a conditional set with its primal set and supposed that each ni j=1 Y kj lives on a condition greater than a i. In particular, these representations imply that a conditional system X P (P (X)) is closed under finite conditional intersections or unions as soon as it is closed under conditionally finite intersections or unions. Proposition (i) If (Y k ) 1 k n is a conditionally finite family of conditionally finite subsets of X, then 1 k n Y k is conditionally finite. 5 The conditional addition is defined in Section 4. 6 I denotes the cardinality of I. 10

11 (ii) If (Y n ) S(S(X)) is a conditionally countable family of conditionally countable subsets, then Y n is conditionally countable. Proof. Assertion (i) is implied by (2.11). As for (ii), denote by R n the graph of Y n N for every n N. Then (R n ) S(S(X N)) and thus R = R n Y n N. Inspection shows that R is the graph of a conditional injection from Y n into N. For a more detailed proof see Lemma in [12]. Definition Let X be a conditional set. A conditional filter on X is a conditional system F S(S(X)) satisfying (i) Z F whenever Y Z for some Y F, (ii) Y Z F for all Y, Z F. Given two conditional filters F and F, we say that F is conditionally finer than F if F F. A conditional ultrafilter is a conditionally maximal filter with respect to conditional inclusion. Given a conditional filter F, then af is a conditional filter on ax for each a A. For a conditional filter F it holds X F and 0 F, and since F S(S(X)) Corollary 2.10 implies that finite conditional intersections of elements of F live on 1, and so do conditionally finite intersections due to (2.11). Remark Let F P (P (X)) satisfy (i) 0 F, (ii) F is closed under finite conditional intersection, (iii) Z F whenever Y Z for some Y F. Then there exists a minimal condition m F > 0 such that m F F is a conditional filter on m F X. Indeed, let F be given by {Y i : i I} and suppose, for the sake of contradiction, that a i = 0 where Y i lives on a i for each i. By the de Morgan s law it holds a c i = 1. Let (b i) p(1) be such that b i a c i for all i. Then b i Yb i i = 0. Due to stability it follows that b i Yb i i F which contradicts 0 F. A conditional system B S(S(X)) is a conditional filter base if for every Y 1, Y 2 B there exists Y 3 B such that Y 3 Y 1 Y 2. For every conditional filter base B F B := {Z X : Y Z for some Y B} is a conditional filter, the conditional filter generated by B. Proposition If B is a conditional filter base, then B a is a classical filter base on X a for each a. Conversely, if B S(S(X)) is such that B 1 is a classical filter base on X 1, then B is a conditional filter base on X. Proof. The assertions follow from Corollary We prove a conditional version of the Ultrafilter lemma. Theorem For every conditional filter F there exists a conditional ultrafilter U such that F U. 11

12 Proof. Let { F 1 := ˆF1 : F ˆF, ˆF } conditional filter. Then F 1 is the primal set of some F S(S(S(X))). Let (F i ) F be a chain and define W := F i. By Corollary 2.10 it holds W S(S(X)), and by construction F W. The assertion follows by Zorn s lemma if we can show that W is a conditional filter. In order to verify the first property of a conditional filter, let Z X be such that Y Z for some Y = a i Y i W where (a i ) p(1) and Y i Fa i i for all i. It follows that Y i Z ai for every i and therefore a i Z a i F i, that is a i Z ai + a c i X a c W for every i. i By stability it follows that Z = ( a i ai Z ai + a c i X ) a c i W. To prove the second property, let Y, Z W with Y = a i Y i and Z = a i Z i where Y i, Z i Fa i i for each i. Then Y Z = a i (Y i Z i ) due to (2.3). Since F i is a conditional filter it follows that Y i Z i for each i, and thus Y Z S(X). Proposition Let U be a conditional filter. Then the following assertions are equivalent: (i) U is a conditional ultrafilter, (ii) if Y 1 Y 2 U for some Y 1, Y 2 X, then ay 1 a + a c Y 2 a c U, where either a = a 1 or a = a c 2 whereby Y i lives on a i for i = 1, 2, (iii) for every Y X, it holds ay a + a c Y a c U, where either a = a 1 or a = a c 2 whereby a 1 and a 2 are the conditions on which Y and Y live, respectively, (iv) for every Y S(X) such that Y U S(X) for every U U it holds Y U. Proof. Step 1: We show that (i) implies (ii). Let b 1 := a 1 a c 2, b 2 := a 1 a 2 and b 3 := a 2 a c 1. Then Y 1 Y 2 = b 1 Y 1 b 1 + b 2 (Y 1 Y 2 ) b2 + b 3 Y 2 b 3. Thus b 2 Y 1 b 2 Y 2 b 2 U. If b 2 Y 1 b 2 U then a = b 1 b 2 = a 1, and we are done. From b 2 Y 1 b 2 U it follows that F = {Z S(b 2 X) : Z b 2 Y 1 b 2 U} is a conditional filter on b 2 X, and contains b 2 Y 2 which lives on b 2. Furthermore, b 2 U F and therefore from U being a conditional ultrafilter it follows that b 2 Y 2 b 2 U. Thus a c = b 2 b 3 = a 2 does the job. Step 2: To show that (ii) implies (iii), it suffices to set Y 1 := Y and Y 2 := Y since Y Y = X U. Step 3: We show that (iii) implies (i). Let V be a conditional filter conditionally finer than U. For every Y V it holds Y S(X). By assumption either Y U or ay a + a c Ya U where Y lives on a c > 0. c The latter case implies a c Y a c U a c V. Since Y V both a c Y and a c Y are in a c V. Yet a c being strictly positive this implies that a c V is a conditional filter on a c X. However then a c Y a c Y = 0 which is impossible. Hence a c = 0 and therefore Y U showing that V U. Step 4: We show that (i) is equivalent to (iv). Assume (i) and let Y S(X) be such that Y U S(X) for every U U. By assumption B := {Y U : U U} is in S(S(X)). Inspection shows that B is a conditional filter base on X. It further holds that U F B. Hence U = F B and thus Y U. Conversely, let V be a conditional ultrafilter of U and let Y V. From U V it follows that Y U S(X) for every U U, and therefore Y U showing that V U. Proposition Let f : X Y be a conditional function. For every conditional filter F the family f(f) := {f(u) : U F} is a conditional filter base on Y. If U is a conditional ultrafilter on X, then f(u) generates a conditional ultrafilter on Y. 12

13 Proof. By Proposition 2.16 and since F S(S(X)) it holds f(f) S(S(Y )). Let V 1 = f(u 1 ), V 2 = f(u 2 ) and V 3 = f(u 1 U 2 ) where U 1, U 2 F. Since F is a conditional filter and (2.5) it follows V 1 V 2 V 3 f(f). Thus f(f) is a conditional filter base. As for the second claim, suppose that U is a conditional ultrafilter and denote by V the conditional filter generated by f(u). Let V S(a 1 Y ) and V S(a 2 Y ). By (2.4) and (2.6) it holds f 1 (V ) f 1 (V ) = f 1 (V ) f 1 (V ) = f 1 (V V ) = f 1 (Y ) = X U. By Proposition 2.25 it follows that U := af 1 (V ) + a c f 1 (V ) U where either a = a 1 or a = a c 2. Due to (2.4), (2.6) and (2.8) it holds f(u) f(u) = af ( f 1 (V ) ) + a c f ( f 1 (V ) ) av + a c V. Hence for every V Y it holds av a + a c Va V where either V S(aY ) or V S(a c Y ). Now c Proposition 2.25 implies that V is a conditional ultrafilter. As in the classical case there exists a one-to-one relation between conditional nets and filters. Let X be a conditional set and (I, ) be a conditional direction. A conditional net is a conditional family (x i ) i I X. A conditional net (y j ) j J is a conditional subnet of (x i ) i I if there exists a conditional function φ : J I such that x φ(j) = y j, and for any i there exists j 0 such that j j 0 implies φ(j) i. A conditional sequence is a conditional family (x n ) n N. A conditional net in a conditionally partially ordered set (X, ) is conditionally increasing (decreasing) if i j implies x i x j (x i x j ). Given a conditional net (x i ) and a(x i ) is a conditional net for every a A. 3 Conditional Topology Definition 3.1. Let X be a conditional set. A conditional topology on X is a conditional system T P (P (X)) satisfying (i) 0, X T, (ii) closed under finite intersections, (iii) and closed under arbitrary unions. The pair (X, T ) is called a conditional topological space. Given two conditional topologies T 1 and T 2, we say T 1 is conditionally weaker than T 2 whenever T 1 T 2. Due to consistency at is a conditional topology on ax for every a A. Due to stability a conditional topology is closed under conditionally finite intersections and conditional unions over arbitrary conditional families. A conditional element O T is called conditionally open and Y X is conditionally closed if Y T. The conditional interior of some Y X is and the conditional closure of Y is int(y ) := {O : O Y, O conditionally open} cl(y ) := {F : Y F, F conditionally closed}. 13

14 For every a A it holds cl(ay ) = acl(y ) and int(ay ) = aint(y ). Due to the duality principle in Boolean algebras the conditional set of all conditionally closed sets satisfies the dual properties of the conditional set of all conditionally open sets. In particular, cl(y ) = int ( Y ) and int(y ) = cl ( Y ). (3.1) Let (X, T ) be a conditional topological space. A conditional set U S(X) is a conditional neighborhood of x X if there exists O T such that x O U. Let U(x) denote the set of all conditional neighborhoods of x. A conditional neighborhood base of x is a conditional system V S(S(X)) such that for every U U(x) there exists V V with x V U. A conditional system B S(S(X)) is a conditional topological base on X if (i) B = X, (ii) for every O 1, O 2 B and x O 1 O 2 there is O 3 B such that x ao 3 O 1 O 2 where O 1 O 2 lives on a. Note that fore every x X the conditional set U(x) is a conditional filter on X. Moreover, it holds au(x) = U(ax) for all a A. Given a conditional topological base B, due to consistency also ab is a conditional topological base on ax for every a. The conditional topology conditionally generated by some G P (P (X)) is defined as Inspection shows that T G := {T : G T, T conditional topology}. T G = {a i I j Ji O ij : a A, I conditional index, J i conditionally finite, O ij G, i I, j J i } where 0 := 0 and 0 := X. For a conditional topological base B S(S(X)) it holds Indeed, for O 1,..., O n B T B = {a O i : a A, O i B each i}. (3.2) O 1... O n = a H, H = {O B : ao O 1... O n for some a A}. Hence for every conditionally finite (O k ) 1 k n there exist H S(B) such that 1 k n O k = a H due to the stability of B and (2.11). Finally, the associativity of conditional unions yields (3.2). A conditional topological space (X, T ) is conditionally first countable if every x X has a conditionally countable neighborhood base. It is conditionally second countable if T is conditionally generated by a conditionally countable topological base. It is conditionally Hausdorff if for every pair x, y X with x y = 0 there exists U U(x) and V U(y) such that U V = 0. A conditional subset Y S(X) is conditionally dense if cl(y ) = X, and (X, T ) is conditionally separable if there exists a conditionally countable dense Y S(X). Lemma 3.2. Let X be a conditional set and (Y i ) S(S(X)). It holds ( Y i ) 1 = Y i 1 Proof. Let x Y i. Then x = b j x j where (b j ) p(1) and x j conditional family it holds x = b j x j b j Y ij b j since all Y i live on 1 and by Corollary Y ij b j for each j. Since it is a = Y b ji j. Thus x Y1 i. The converse is true 14

15 Recall that a classical topological base on some set X is a family B of subsets of X such that B = X, and for every O 1, O 2 B and x O 1 O 2 there exists O 3 B such that x O 3 O 1 O 2. We write T B for the classical topology generated by B. Denote by cl(y ) the classical closure of some Y X and by int(y ) its classical interior. Proposition 3.3. Let X be a conditional set and B S(S(X)). Then B is a conditional topological base on X if and only if B a is a classical topological base on X a for all a. It holds (T B S(X)) 1 = T B1 S(X) 1. Proof. Let B be a conditional topological base. Then X 1 = ( O B O) 1 = O1 B 1 O 1 by Lemma 3.2. Let O1, 1 O1 2 B 1 and x 1 O1 1 O1. 2 Then O1 1 O1 2 and thus O 1 O 2 lives on 1. Therefore x O 1 O 2 and by assumption there exists O 3 B such that x O 3 O 1 O 2. Corollary 2.10 yields x 1 O1 3 O1 1 O1. 2 This shows that B 1 is a classical topological base on X 1. By consistency it follows that B a is a classical topological base on X a for all a A. Conversely, since B S(S(X)) and B 1 = X 1 it follows from Lemma 3.2 that B = X. Let x O 1 O 2 where x lives on a. Then there exists Oa 3 B a such that x a Oa 3 Oa 1 Oa. 2 Hence x ao 3 O 1 O 2 and thus B is a conditional topological base on X. As for the second assertion, let O S(X) be such that O 1 T B1. Then there exists a family (O i ) B such that O 1 = O1. i Since O S(X) it even holds O1 i = ( O i ) 1 by Lemma 3.2, and therefore O T B. Conversely, let O T B S(X). Then O = O i S(X) for some family (O i ) B. Due to Corollary 2.10 it holds ( O i ) 1 = O1, i and thus O T B1 S(X) 1. Proposition 3.4. Let (X, T ) be a conditional topological space and Y S(X). Then it holds where a = {a : O Y, O lives on a}. int(y ) = {x a X : x U Y for some U U(x)} (3.3) cl(y ) = {x X : U Y S(X) for all U U(x)} (3.4) Proof. Denote the right-hand side of (3.3) and (3.4) by M and N, respectively. We prove (3.3). On the one hand, for every x M there exists a conditional open set O U(x) such that x O Y, and thus x int(y ). On the other hand, every x int(y ) is of the form x = c i x i where x i O i Y and O i T for every i. Hence x O = c i O i Y, and thus x M. For the second identity we show N = int(y ) due to duality. By definition x N if au ay = 0 implies a = 0 for all U U(x). This is equivalent to that au ay implies a = 0 for all U U(x). This is the case if and only if x aint(y ) implies a = 0, which by definition means x int(y ). Proposition 3.5. Let X be a conditional set, B be a conditional topological base for some conditional topology T on X and Y S(X). Then it holds cl(y ) 1 = cl(y 1 ) and int(y ) a = int(y a ) where the classical closure is with respect to T B1 and the interior with respect to T Ba where a is as in Proposition 3.4. Proof. Both assertions are consequences of Corollary 2.10, Propositions 3.3 and 3.4. Examples 3.6. (i) For n N and q Q define and let B 1/n (q) = {p Q : q p < 1/n} B = { B 1/n (q) : q Q, n N }. Then B is a conditional topological base of a conditionally Hausdorff topology on Q and Q endowed with T B is conditionally separable. The conditional topology T B is called the conditional Euclidean topology on Q. 15

16 (ii) Let (Ω, F, µ) be a σ-finite measure space, A be the associated measure algebra and L 0 be the set of equivalence classes of real measurable functions. Recall from Example 2.2 that L 0 denotes the conditional set associated to L 0. For every ε L 0 with ε > 0 and f L 0 define Then B ε (f) = {g L 0 : f g < ε}. B 1 = {B ε (f) : ε L 0, ε > 0, f L 0 } is a base of a classical topology on L 0, called the L 0 -topology in [8]. For every partition (A n ) of Ω in F and every (f n ) B ε (f) it holds 1 An f n B ε (f) and for every (B εn (f n )) B 1 it holds 1An B εn (f n ) = B 1 εn( An 1 An f n ) B 1. Due to Proposition 3.3 the conditional set B defined by B 1 is a conditional topological base on L 0. The conditional topology generated by B is conditionally Hausdorff and L 0 endowed with it is conditionally separable (for a proof see Lemma in [12]). Definition 3.7. Let (X, T ) and (X, T ) be two conditional topological spaces. A conditional function f : X Y is conditionally continuous at x X if f 1 (U) is a conditional neighborhood of x for every conditional neighborhood U of f(x). A conditional function is said to be conditionally continuous on X if it is conditionally continuous at every x X. If f is conditionally continuous at x X, then so is af : ax ax at ax ax for every a. Proposition 3.8. The following statements are equivalent: (i) f is conditionally continuous, (ii) f 1 (O) is conditionally open in X for every conditionally open set O in X, (iii) f 1 (F ) is conditionally closed in X for every conditionally closed set F in X, (iv) f 1 (int(z)) int(f 1 (Z)) for every Z X, (v) f(cl(z)) cl(f(z)) for every Z X. Proof. The equivalence of (i)-(v) can be proved from Proposition 2.16 similarly to the respective classical equivalences. For a detailed proof see Proposition in [12]. Proposition 3.9. Let (X, B, T ) and (X, B, T ) be two conditional topological spaces endowed with conditional topological bases, respectively. A conditional function f : X X is conditionally continuous if and only if f 1 is continuous with respect to the classical topologies T B1 and T B 1. Proof. By Proposition 3.3 the systems B 1 and B 1 are classical topological bases on X 1 and X 1. Assume that f is continuous in a classical sense. Let O B and suppose that f 1 (O) lives on a. By Proposition 3.3 it holds af 1 a (O a ) + a c X a c = f 1 1 (ao a + a c Y a c) T B1 S(X) 1 = (T B S(X)) 1. Thus f 1 (O) = af 1 (ao a + a c Y a c) = af 1 (ao a + a c Y a c) + a c 0 T. Conversely, assume that f is conditionally continuous and let O 1 B 1. If f1 1 (O 1) = we are done. Suppose that f1 1 (O 1). Then f1 1 (O 1) is an element of (T B S(X)) 1. Thus f1 1 (O 1) T B1 due to Proposition 3.3. Definition Let X be a conditional set, (X i, T i ) be a family of conditional topological spaces and (f i ) be a family of conditional functions f i : X X i. The conditional initial topology on X for the family (f i ) is the conditional topology generated by cond(g) where G := {f 1 i (O i ) : O i T i for some i}. 16

17 The conditional initial topology is the conditionally weakest topology for which every f i is conditionally continuous. Examples (i) Let (X, T ) be a conditional topological space, Y S(X) and f : Y X the conditional embedding (see Example 2.15). The conditional initial topology of T for f is the conditional relative topology with respect to Y. (ii) Let (X i, T i ) be a family of conditional topological spaces. The conditional product topology on Xi is the conditional initial topology for the family of conditional projections (π i ). Proposition Let X, Y be conditional sets, (X i, B i, T i ) be a conditional family of conditional topological spaces endowed with a conditional topological base where X i S(Y ) for each i. Let further (f i ) be a conditional family of conditional functions f i : X X i. Let T be the classical initial topology on X 1 for the family (f i 1) and T be the conditional initial topology on X for the family (f i ). Then it holds T = T B and T = T B = T B1 where B = cond(b) and B = {(f i 1) 1 (O i 1) : O i 1 B i 1 for some i}. Proof. The conditional initial topology can be generated by the stable hull of the conditional preimages of a conditional topological base of each of the T i. Due to Proposition 3.3 it suffices to show that B is stable. Indeed, since (f i ) and (T i ) are conditional families aj (fa ij j ) 1 (Oa ij j ) = (f a ji j ) 1 (O a ji j ) B 1. Definition Let (X, T ) be a conditional topological space, F be a conditional filter and (x i ) X be a conditional net. A conditional element x X is a conditional (i) limit point of F if U(x) F, (ii) cluster point of F if x cl(y ) for every Y F, (iii) limit point of (x i ) if for every U U(x) there exists i 0 such that (x i ) i i0 U, (iv) cluster point of (x i ) if for every U U(x) and every i there exists j i such that x j U. One can replace in the previous definition a conditional filter by a conditional filter base. Due to the one-to-one relation between conditional filters and nets (see Proposition in [12]) we formulate the following statements just for conditional filters. Let Lim F := {cl(y ) : Y F}. If Lim F lives on a, it is the collection of all conditional cluster points of af. If x is a conditional limit or cluster point of a conditional filter F, then ax is a conditional limit or cluster point of the conditional filter af. We indicate by F x that x is the conditional limit point of F. For a classical topology T and a classical filter F on X we denote by F T x the convergence of F to x and by LimF the set of cluster points of F with respect to T. Proposition Let (X, T ) be a conditional topological space. If F is a conditional filter, then the following assertions are equivalent: (i) x Lim F, (ii) there exists a conditional filter G F such that G x. 17

18 Proof. To show that (i) implies (ii) let x Lim F. Then {V U : V U(x) and U F} is a conditional filter base of some G F. By construction G x. To show that (ii) implies (i) let G be a conditional filter conditionally finer than F and G x. Hence V Y S(X) for every V U(x) and every Y F since both are elements of G. Thus (3.4) implies that x cl(y ) showing that x Lim F. Proposition Let X be a conditional set, G S(S(X)) be a conditional filter base, B S(S(X)) be a conditional topological base and B = B 1 and G = G 1. Then it holds x Lim G if and only if x 1 LimG, and G B x 1 if and only if G x. In particular, Y S(X) is conditionally closed if and only if the conditional limit of every conditionally converging filter is in Y. Proof. The first equivalence is a consequence of the Propositions 2.23 and 3.5. The equivalence between classical convergence and conditional convergence follows from the Propositions 2.23 and 3.3. The latter together with Proposition 3.5 yields the characterization of conditionally closed sets. Proposition Let (X, T ) and (X, T ) be two conditional topological spaces and f : X X be a conditional function. Then the following assertions are equivalent: (i) f is conditionally continuous at x, (ii) f(f) f(x) for every conditional filter F x. In particular the composition of conditionally continuous functions is conditionally continuous. Proof. By Proposition 3.9 conditional continuity is equivalent to classical continuity. The claim follows from the respective classical result and Proposition Let (X, T ) be a conditional topological space. A conditional open covering of X is a conditional family (O j ) T such that X O j. A conditional family (Y j ) j J P (X) has the conditional finite intersection property if j N Y j 0 for every conditionally finite N J. Definition A conditional topological space X is conditionally compact if for every conditional open covering (O j ) there exists a conditionally finite subfamily (O j k ) 1 k n such that X 1 k n O j k. (3.5) Moreover, Y X is conditionally compact if Y is conditionally compact with respect to the conditional relative topology on Y (see Example 3.11). Due to (2.3) the equation (3.5) is equivalent to X i I a i j J i O j for some (a i ) i I p(1) and J i finite for each i I. Proposition Let (X, T ) be a conditional topological space. The following statements are equivalent: (i) X is conditionally compact, (ii) every conditional filter on X has a conditional cluster point, 18

19 (iii) every conditional ultrafilter on X has a conditional limit point, (iv) every conditional family (F j ) of conditionally closed sets with the conditional finite intersection property fulfills F j 0. Proof. The equivalence between (ii) and (iii) follows from Proposition In order to show that (i) implies (iv) let (F j ) be a conditional family of conditionally closed sets with the conditional finite intersection property. Suppose, for the sake of contradiction, that F j = 0. By the de Morgan s law it follows that O j = X where O j = (F j ) T. Hence the conditional family (O j ) is a conditional open covering of X. Therefore there exists n N such that X = 1 k n O j k which yields 0 = 1 k n F j k contradicting the hypothesis. Conversely, suppose that (iv) holds and let (O j ) be a conditional open covering of X. It follows that F j = 0 for the conditional family of conditionally closed sets F j = (O j ). Hence there exists a conditionally finite family N such that j N F j = 0 since the assumptions are violated. Taking the conditional complement yields (i). To show that (ii) implies (iv) let (F j ) be a conditional family of conditionally closed sets with the conditional finite intersection property. Let G = { j N F j : N J conditionally finite}. Since (F j ) is a conditional family it follows that G P (P (X)). Furthermore, by the conditional finite intersection property it follows that 0 G. Let a > 0 be the minimal condition of G (see Remark 2.22) and Y 1 = j N F j, Y 2 = j M F j G. By (2.3) it holds that N M is conditionally finite. This implies Y 1 Y 2 = j N M F j G. Hence G is a conditional filter base of a conditional filter F on ax. Since af + a c X is a conditional filter on X, there exists x Lim(F + a c X) and therefore ax Lim F = {cl(y ) : Y F} j F j which shows that (iv) holds. Conversely, suppose that (iv) holds and let F be a conditional filter on X. Since cl( Y i ) cl(y i ), it follows that {cl(y ) : Y F} is a conditional family of conditionally closed sets fulfilling the conditional finite intersection property by (2.3). Thus Lim F = {cl(y ) : Y F} 0. Let a > 0 be the condition on which Lim F lives. Assume, for the sake of contradiction, that a < 1. Since a c F is a conditional filter on a c X, it follows that {cl(y ) : Y a c F} is a conditional family of conditionally closed sets in a c X fulfilling the conditional finite intersection property. Thus Lim a c F = {cl(y ) : Y a c F} 0. If Lim a c F lives on b > 0, then Lim F lives on a b due to stability. However this contradicts the maximality of a. Thus Lim F lives on 1, and therefore F has a conditional cluster point. Proposition Let (X, T ) and (X, T ) be two conditional topological spaces. The following conditional subsets of X and X are conditionally compact: every conditionally finite subset, conditionally finite unions of conditionally compact subsets, a conditionally closed subset of a conditionally compact subset and the conditional image under every conditionally continuous function f : Y X of every conditionally compact subset Y X. Moreover, if X is a conditionally Hausdorff space, then every conditionally compact subset is conditionally closed. 19