BLAISE

Properties of the so called θ o -complete topological spaces are investigated. Also, necessary and suﬃcient conditions are given so that the space C ( X, E ) of all continuous functions, from a zero-dimensional topological space X to a non-Archimedean locally convex space E , equipped with the topology of uniform convergence on the compact subsets of X to be polarly barrelled or polarly quasi-barrelled.


Introduction
Let K be a complete non-Archimedean valued field and let C(X, E) be the space of all continuous functions from a zero-dimensional Hausdorff topological space X to a non-Archimedean Hausdorff locally convex space E. We will denote by C b (X, E) (resp.by C rc (X, E)) the space of all f ∈ C(X, E) for which f (X) is a bounded (resp.relatively compact) subset of E. The dual space of C rc (X, E), under the topology t u of uniform convergence, is a space M (X, E ) of finitely-additive E -valued measures on the algebra K(X) of all clopen , i.e. both closed and open, subsets of X.Some subspaces of M (X, E ) turn out to be the duals of C(X, E) or of C b (X, E) under certain locally convex topologies.In section 2 of this paper, we give some results about the space M (X, E ).The notion of a θ 0 -complete topological space was given in [2].In section 3 we study some of the properties of θ o -complete spaces.Among other results, we prove that a Hausdorff zero-dimensional space is θ o -complete iff it is homeomorphic to a closed subspace of a product of ultrametric spaces.In section 4, we give necessary and sufficient conditions for the space C(X, E), equipped with the topology of uniform convergence on the compact subsets of X, to be polarly barrelled or polarly quasi-barrelled,

Preliminaries
Throughout this paper, K will be a complete non-Archimedean valued field, whose valuation is non-trivial.By a seminorm, on a vector space over K, we will mean a non-Archimedean seminorm.Similarly, by a locally convex space we will mean a non-Archimedean locally convex space over K (see [9]).Unless it is stated explicitly otherwise, X will be a Hausdorff zero-dimensional topological space , E a Hausdorff locally convex space and cs(E) the set of all continuous seminorms on E. The space of all Kvalued linear maps on E is denoted by E , while E denotes the topological dual of E. A seminorm p, on a vector space G over K, is called polar if p = sup{|f | : f ∈ G , |f | ≤ p}.A locally convex space G is called polar if its topology is generated by a family of polar seminorms.A subset A of G is called absolutely convex if λx + µy ∈ A whenever x, y ∈ A and λ, µ ∈ K, with |λ|, |µ| ≤ 1.We will denote by β o X the Banaschewski compactification of X (see [3]) and by υ o X the N-repletion of X, where N is the set of natural numbers.We will let C(X, E) denote the space of all continuous E-valued functions on X and C b (X, E) (resp.C rc (X, E)) the space of all f ∈ C(X, E) for which f (X) is a bounded (resp.relatively compact) subset of E. In case E = K, we will simply write C(X), C b (X) and C rc (X) respectively.For A ⊂ X, we denote by χ A the K-valued characteristic function of A. Also, for X ⊂ Y ⊂ β o X, we denote by BY the closure of B in Y .If f ∈ E X , p a seminorm on E and A ⊂ X, we define The strict topology β o on C b (X, E) (see [4]) is the locally convex topology generated by the seminorms f → hf p , where p ∈ cs(E) and h is in the space B o (X) of all bounded K-valued functions on X which vanish at infinity, i.e. for every > 0 there exists a compact subset Y of X such that |h(x)| < if / ∈ Y .Let Ω = Ω(X) be the family of all compact subsets of β o X \ X.For H ∈ Ω, let C H be the space of all h ∈ C rc (X) for which the continuous extension h βo to all of β o X vanishes on H.For p ∈ cs(E), let β H,p be the locally convex topology on C b (X, E) generated by the seminorms The inductive limit of the topologies β H , H ∈ Ω, is the topology β.Replacing Ω by the family Ω 1 of all K-zero subsets of β o X, which are disjoint from X, we get the topology β 1 .Recall that a K-zero subset of β o X is a set of the form {x ∈ β o X : g(x) = 0}, for some g ∈ C(β o X).We get the topologies β u and β u replacing Ω by the family Ω u of all Q ∈ Ω with the following property: There exists a clopen partition The inductive limit of the topologies β H,p , as H ranges over Ω u , is denoted by β u,p , while β u is the projective limit of the topologies β u,p , p ∈ cs(E).For the definition of the topology β e on C b (X) we refer to [7].
Let now K(X) be the algebra of all clopen subsets of X.We denote by M (X, E ) the space of all finitely-additive E -additive measures m on K(X) for which the set m(K(X)) is an equicontinuous subset of E .For each such m, there exists a p ∈ cs(E) such that m p = m p (X) < ∞, where, for The space of all m ∈ M (X, E ) for which m p (X) < ∞ is denoted by In case E = K, we denote by M (X) the space of all finitely-additive bounded K-valued measures on K(X).
In this case we write V δ ↓ ∅.We denote by M τ (X) the space of all τ -additive members of M (X).Analogously, we denote by M σ (X) the space of all σ-additive m, i.e. those m with m(V n ) → 0 when V n ↓ ∅.For an m ∈ M (X, E ) and s ∈ E, we denote by ms the element of M (X) defined by (ms Next we recall the definition of the integral of an f ∈ E X with respect to an m ∈ M (X, E ).For a non-empty clopen subset A of X, let D A be the family of all α = {A 1 , A 2 , . . ., A n ; x 1 , x 2 , . . ., x n }, where {A 1 , . . ., A n } is a clopen partition of A and x k ∈ A k .We make D A into a directed set by defining α 1 ≥ α 2 iff the partition of A in α 1 is a refinement of the one in α 2 .For an α = {A 1 , A 2 , . . ., A n ; x 1 , x 2 , . . ., x n } ∈ D A and m ∈ M (X, E ), we define If the limit lim ω α (f, m) exists in K, we will say that f is m-integrable over A and denote this limit by A f dm.We define the integral over the empty set to be 0.For A = X, we write simply f dm.It is easy to see that if f is m-integrable over X, then it is m-integrable over every clopen subset A of X and A f dm = χ A f dm.If τ u is the topology of uniform convergence, then every m ∈ M (X, E ) defines a τ u -continuous linear functional φ m on C rc (X, E), φ m (f ) = f dm.Also every φ ∈ (C rc (X, E), τ u ) is given in this way by some m ∈ M (X, E ).
For p ∈ cs(E), we denote by M t,p (X, E ) the space of all m ∈ M p (X, E ) for which m p is tight, i.e. for each > 0, there exists a compact subset Y of X such that m p (A) , is an algebraic isomorphism.For m ∈ M τ (X) and f ∈ K X , we will denote by (V R) f dm the integral of f , with respect to m, as it is defined in [9].We will call (V R) f dm the (V R)-integral of f .For all unexplained terms on locally convex spaces, we refer to [8] and [9].

Some results on M (X, E )
Theorem 2.1.Let m ∈ M (X, E ) be such that ms ∈ M τ (X), for all s ∈ E, and let p ∈ cs(E) with m p < ∞.Then : is the smallest of all closed support sets for m.
(4) If V is a clopen set contained in the union of a family (V i ) i∈I of clopen sets, then Proof: (1).If x ∈ V , then N m,p (x) ≤ m p (V ) and so On the other hand, let m p (V ) > d.There exists a clopen set W , contained in V , and In fact, assume the contrary and let Z be a clopen neighborhood of x contained in W and such that m p (Z) < d.Now This contradiction proves (1). (2).
which proves that supp(m) is a support set for m.On the other hand, let Y be a closed support set for m.There exists a decreasing net and so supp(m) ⊂ V c , which implies that supp(m) ⊂ G. On the other hand, let x / ∈ supp(m).There exists a clopen neighborhood W of x disjoint from supp(m).Since supp(m) is a support set for m, we have that m p (W ) = 0 and thus N m,p = 0 on W , which proves that x / ∈ G. Thus G is contained in supp(m) and (3) follows.(4).Let m p (V ) > α > 0. There exists a clopen set Proof : We show first that, for µ ∈ M σ (X), then there exists an n with and the claim follows for µ.Suppose now that m p (V ) > r > 0. There exists a clopen subset W of V and s ∈ E such that |m(W )s| > r • p(s).Let µ = ms.Then µ ∈ M σ (X) and |µ|(V ) ≥ |m(W )s| > r • p(s).By the first part of the proof, there exists an n such that |µ|(V n ) > r • p(s).Hence, there exists a clopen subset We have the following easily established Theorem 2.3.Let m ∈ M (X, E ) be such that ms ∈ M τ (X) for all s ∈ E. Then : (1) supp(m βo ) = supp(m) βoX .
(4) For every Z ∈ Ω 1 there exists a clopen subset A on β o X disjoint from Z and such that supp(µ) ⊂ A. ( and so supp(µ) ⊂ n V c n βoX .In view of the compactness of supp(µ), there (2) ⇒ (3).Let V n ↓ ∅ and suppose that, for each n, there exists a clopen subset A of V n such that m(A) = 0. Claim.For each n, there exists k > n and a clopen set B with V k ⊂ B ⊂ V n and m(B) = 0. Indeed there exists a clopen subset Since A and D k are disjoint, we have that m(B) = m(A) = 0 and the claim follows.By induction, we choose for every k, we arrived at a contradiction.
(3) ⇒ (4).Let Z ∈ Ω 1 .There exists a decreasing sequence (V n ) of clopen sets with Z = V n βoX .By our hypothesis, there exists an n suich that There exists a decreasing sequence (V n ) of clopen sets with z ∈ Z = V n βoX .Clearly Z ∈ Ω 1 .Thus, there exists a clopen subset A of β o X disjoint from Z and containing supp(µ).Hence z is not in supp(µ).
. This completes the proof.

θ o -Complete Spaces
Recall that θ o X is the set of all z ∈ β o X with the following property: For each clopen partition (V i ) of X there exists i such that z ∈ V i βoX (see [2]).
Then the family of all W α , α a clopen partition of X, is a base for a uniformity U c = U X c , compatible with the topology of X, and (θ o X, U θoX c ) coincides with the completion of (X, U c ).We will say that Note that several authors use the term bounded set instead of bounding.But in this paper we will use the term bounding to distinguish from the notion of a bounded set in a topological vector space.A set A ⊂ X is bounding iff A υoX is compact.In this case (as it is shown in [2, Theorem

4.6]) we have that A
Clearly a continuous image of a bounding set is bounding.Let us say that a family F of subsets of X is finite on a subset A of X if the family {f ∈ F : F ∩ A = ∅} is finite.We have the following easily established Lemma 3.1.For a subset A of X, the following are equivalent : (1) A is bounding.
(2) Every continuous real-valued function on X is bounded on A.
(3) Every locally finite family of open subsets of X is finite on A.
(4) Every locally finite family of clopen subsets of X is finite on A.

Theorem 3.2. Every complete Hausdorff locally convex space
Proof: Let U be the usual uniformity on E, i.e. the uniformity having as a base the family of all sets of the form Given W p, , we consider the clopen partition α = (V i ) i∈I of E generated by the equivalence relation x ∼ y iff p(x − y) ≤ .Then W p, = W α and so U is coarser that U c .Since (E, U) is complete and U c is compatible with the topology of E, it follows that (E, U c ) is complete and the result follows.Proof: Assume first that G is Hausdorff.Let Ĝ be the completion of G.The closure B of A in Ĝ is bounding and hence B is totally bounded, which implies that A is totally bounded.If G is not Hausdorff, we consider the quotient space F = G/{0} and let u : G → F be the quotient map.Since u is continuous, the set u(A) is bounding, and hence totally bounded, in F .Let now V be a convex neighborhood of zero in G.Then, u(V ) is a neighborhood of zero in F .Let S be finite subset of A such that u(A) ⊂ u(S) + u(V ).But then

Corollary 3.3. A subset B, of a complete
which proves that A is totally bounded.Theorem 3.5.We have: (1) Closed subspaces of θ o -complete spaces are θ o -complete.
(4) θ o X is the smallest of all θ o -complete subspaces of β o X which contain X.
(3) ⇒ (1).Assume that z / ∈ θ o X.Then, there exists a clopen partition Then d is a continuous ultrapseudometric on X.Let Y = X d be the corresponding ultrametric space and let π : X → Y d be the quotient map, x → xd = x.Since π is continuous, there exists (by ( 3)) an x ∈ X such that π βo (z) = xd .Let (x δ ) be a net in X converging to z.Then xδ = π βo (x δ ) → π βo (z) = x, and so d(x δ , x) → 0. If x ∈ A i , then |f i (x δ ) − 1| → 0, and so there exists contradiction.This completes the proof.
Theorem 3.7.Let X be a dense subspace of a Hausdorff zero-dimensional space Y .The following are equivalent : (1) Y ⊂ θ o X (more precisely, Y is homeomorphic to a subspace of θ o X).
(2) Each continuous function, from X to any ultrametric space Z, has a continuous extension to all of Y .
Proof: (1) implies (2) by the preceding Theorem.(2) ⇒ (1).We will prove first that, for each clopen subset V of X, we have that where Then d is a continuous ultrapseudometric on X.Let π : X → X d be the quotient map.By our hypothesis, there exists a continuous extension h : There are nets (x δ ), (y γ ), in V , V c respectively, such that x δ → z, and y γ → z.Let d be the ultrametric of X d and let δ o , γ o be such that This, being true for each clopen subset V of X, implies that Proof: Consider the ultrametric space X d and let d be its ultrametric.Let h be the coninuous extension of the quotient map π : X → X d to all of θ o X. Define It is easy to see that d θo is a continuous ultrapseudometric which is an extension of d.Finally, let be any continuous ultrapseudometric on θ o X, which is an extension of d, and let y, z ∈ θ o X.There are nets (y δ ) δ∈∆ , (z γ ) γ∈Γ ) in X which convergence to y, z, respectively.Let Φ = ∆ × Γ and consider on Φ the order (δ 1 , γ 1 ) ≥ (δ 2 , γ 2 ) iff δ 1 ≥ δ 2 and γ 1 ≥ γ 2 .For φ = (δ, γ) ∈ Φ, we let and hence = d θo , which completes the proof.
Theorem 3.9.Let (H n ) be a sequence of equicontinuous subsets of C(X).
Then d is a continuous ultrapseudometric on X.Take Y = X d and let π : X → Y be the quotient map.Then π βo (z) = u ∈ Y .Choose x ∈ X with π(x) = u, and let (x δ ) be a net in X converging to z in β o X.Now f (x δ ) → f βo (z) for all f ∈ H. Since π(x δ ) → π(x), we have that d(x δ , x) → 0, and so |f (x δ ) − f (x)| → 0 for all f ∈ H. Thus, for f ∈ H, we have f (x) = lim f (x δ ) = f βo (z), and the result follows.
Theorem 3.10.If H ⊂ C(X) is equicontinuous, then the family Moreover, if H is pointwise bounded, then the same holds for H θo Proof: Define Let π θo : θ o X → X d be the continuous extension of the quotient map π : X → X d .Let z ∈ θ o X and > 0. There exists x ∈ X such that π θo (z) = π(x).Let (x δ ) be a net in X converging to z.Then π(x δ ) → π θo (z) = π(x) and so d(x δ , x) → 0. Thus, for f ∈ H, we have This proves that H θo is equicontinuous on θ o X.The last assertion follows from the preceding Theorem.
Theorem 3.11.U c = U X c is the uniformity U generated by the family of all continuous ultrapseudometrics on X.
where Theorem 3.12.Let (Y i , f i ) i∈I be the family of all pairs (Y, f ), where Y is an ultrametric space and f : X → Y a continuous map.Then Proof: It follows from Theorem 3.6.
Theorem 3.13.A Hausdorff zero-dimensional space X is θ o -complete iff it is homeomorphic to a closed subspace of a product of ultrametic spaces.
Proof: Every ultrametric space is θ o -complete.Thus the sufficiency follows from Theorem 3.5.Conversely, assume that X is θ o -complete and let (Y i , f i ) i∈I be as in the preceding Theorem.Then X The quotient map π : X → X d is continuous and π(x) = π(y), which implies that u(x) = u(y).Clearly u is continuous.Also u −1 : u(X) → X is continuous.Indeed, let V be a clopen subset of X containing x o and consider the pseudometric d(x, y) = |χ V (x) − χ V (y)|.Let π : X → X d be the quotient map.There exists a i ∈ I such that Y i = X d and f for all i, and hence z ∈ θ o X = X, by the preceding Theorem.Thus y i = f i (z), for all i, and hence y = u(z).This proves that X is homeomorphic to a closed subspace of Y and the result follows.
Thus, considering M (X) as the dual of the Banach space F = (C rc (X), τ u ), D is w -bounded of F and so sup m∈H ms = d s < ∞.Hence, |m(V )s| ≤ d s for all V ∈ K(X).It follows that the set is a w -bounded subset of E .Since E is polarly barrelled, there exists p ∈ cs(E) such that |u(s)| ≤ 1 for all u ∈ M and all s ∈ E with p(s) ≤ 1. Hence sup m∈H m p < ∞.We may choose p so that m p ≤ 1 for all m ∈ H. Let Then Z is bounding.In fact, assume that Z is not bounding.Then, by [6, Proposition 6.6], there exists a sequence (m n ) in H and f ∈ C(X, E) such that < m n , f >= λ n , for all n, where |λ| > 1, which contradicts the fact that H is w -bounded.By our hypothesis now, Z is compact.Since {f ∈ G : f Z,p ≤ 1} ⊂ H o , the result follows.Let now G, E be Hausdorff locally convex spaces.We denote by L s (G, E) the space L(G, E) of all continuous linear maps, from G to E, equipped with the topology of simple convergence.Theorem 4.4.Assume that E is polar and let G be polarly barrelled.If E is a µ o -space (e.g. when E is metrizable or complete), then L s (G, E) is a µ o -space.
Proof: Let Φ be a bounding subset of L s (G, E).For x ∈ G, the set Φ(x) = {φ(x) : φ ∈ Φ} is a bounding subset of E and hence its closure M x in E is compact.Φ is a topological subspace of E G and it is contained in the compact set M = x∈G M x .Since the closure of Φ in E G is compact, it suffices to show that this closure is contained in L(G, E).To this end, we prove first that, given a polar neighborhood W of zero in E, there exists a neighborhood show first that there exists p ∈ cs(E) such that sup m∈H m p < ∞.In fact, let B be a bounded subset of E and consider the set Then for A ∈ K(X), s ∈ B, m ∈ H, we have |m(A)s| ≤ ms ≤ d.Hence Φ is a strongly bounded subset of E .By our hypothesis, Φ is an equicontinuous subset of E .Thus, there exists p ∈ cs(E) such that |m(A)s| ≤ 1 for all m ∈ H and all s ∈ E with p(s) ≤ 1.It follows from this that sup m∈H m p = r < ∞.We may choose p so that r ≤ 1.Let now Then Y is w o -bounded.Assume the contrary.Then, there exists a sequence (V n ) of distinct clopen subsets of X, such that V n ∩Y = ∅ for all n and (V n ) is finite on each compact subset of X. .For each n there exists m n ∈ H with V n ∩ supp(m n ) = ∅.Then (m n ) p (V n ) > 0. There are a clopen subset W n of V n and s n ∈ E, with p(s n ) ≤ 1, such that m(W n )s n = γ n = 0. Let |λ| > 1 and take M = {γ −1 n λ n χ Wn s n : n ∈ N}.Since (W n ) is finite on each compact subset of X, it follows that M is a bounded subset of G and so M is absorbed by H o .Let λ o = 0 be such that M ⊂ λ o H o .But then
Then d is a continuous ultrapseudometric on X.Since W = {(x, y) : d(x, y) < 1/2}, it follows that U c is coarser than U. Conversely, let d be a continuous ultrapseudometric on X, > 0 and D = {(x, y) : d(x, y) ≤ }.If α is the clopen partition of X corresponding to the equivalence relation x ∼ y iff d(x, y) ≤ , then D = W α and the result follows.

D
= {ms : m ∈ H, s ∈ B}.If h ∈ C rc (X), then the set {hs : s ∈ B} is a bounded subset of G and so sup m∈H hs dm = sup m∈H h d(ms) < ∞.Considering D a a subset of the dual of the Banach space F = (C rc (X), τ u ), we see that D is a w -bounded subset of F and hence equicontinuous.Thus d = sup m∈H,s∈B ms < ∞.
Hausdorff locally convex space E, is bounding iff it is totally bounded.
Proof: If B is bounding, then B = B θoE is compact and hence totally bounded, which implies that B is totally bounded.Conversely, if B is totally bounded, then B is totally bounded.Thus B is compact and hence B is bounding.Theorem 3.4.If G is a locally convex space (not necessarily Hausdorff ), then every bounding subset A of G is totally bounded.
Now our hypothesis (2) and the preceding Theorem imply that Y ⊂ θ o X, and the result follows.
Theorem 3.8.For each continuous ultrapseudometric d on X, there exists a continuous ultrapseudometric d θo on θ o X which is an extension of d.Moreover, d θo is the unique continuous extension of d.