PRECOMPACTNESS AND COMPACTNESS IN THE NAT-URAL FUNCTION SPACES OF THE CATEGORY OF SEMI-UNIFORM CONVERGENCE SPACES

Gerhard Preuß

Department of Mathematics, Free University of Berlin, Germany

Abstract. Semiuniform convergence spaces form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces) with many convenient properties such as cartesian closedness, hereditariness and the fact that products of quotients are quotients. Here necessary and sufficient conditions for precompactness in the natural function spaces of the category of semiuniform convergence spaces are studied. The results are applied to characterize compactness in these function spaces and to derive the classical Ascoli theorem.

Mathematics Subject Classifications (1991). 54A05, 54A20, 54D30, 54C35, 54E15, 18A40, 18D15

Keywords: Semiuniform convergence spaces, uniform convergence spaces (= uniform limit spaces), convergence spaces, local precompactness, local compactness, precompactness, compactness, uniform equicontinuity, equicontinuity, uniform convergence, cartesian closedness, bicoreflective subcategories.

1. Introduction

Semiuniform convergence spaces have been studied by the author (cf. [17], [18], [19] and [20]) as an alternative concept of space in topology. They form a strong topological universe, i.e. a cartesian closed and hereditary topological category such that products of quotients are quotients. They are the suitable framework for studying continuity, Cauchy continuity and uniform continuity as well as convergence structures in function spaces, namely simple convergence, continuous convergence and uniform convergence. Furthermore, the localization of compactness (when considered in the realm of semiuniform convergence spaces) leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the locally compact spaces are exactly the compactly generated spaces (even without the Hausdorff axiom!). Similarly, the localization of precompactness leads even to a topological universe (= cartesian closed and hereditary topological category) and the locally precompact spaces are exactly the precompactly generated spaces. Obviously, they form a generalization of locally compact spaces. Here it turns out that locally precompactness of the domain of the natural function spaces (= power objects) of the category of semiuniform convergence spaces is needed in order to characterize precompact subsets of these function spaces. The results of Wyler [23] who characterized precompactness in the power objects of the category of uniform limit spaces (= uniform convergence spaces) are partially enlarged to semiuniform convergence spaces. Furthermore, the results are applied to characterize compact subsets of the above mentioned function spaces and to derive the classical Ascoli theorem. In the last three decades many authors have worked in the field of generalizations of the Ascoli theorem. Among them are (in chronological order): Cook and Fischer [6], Poppe [14], Dubuc [7], Kneis [12], Gähler [9], Gray [11], Wyler [23], Bentley and Herrlich [3]. The termilogy of this article corresponds to [1] and [15].

Conventions. 1) A filter on a set X is not allowed to contain the empty set ϕ. 2) Subcategories are always assumed to be full and isomorphism–closed.

2. Some basic definitions and results

2.1 Definitions. 1) A semiuniform convergence space is a pair (X, J_{X}), where X is a set and J_{X} a set of filters on X × X such that the following are satisfied:

UC_{1}) ẋ × ẋ ∈ J_{X} for each x ∈ X, where ẋ = {A ⊂ X : x ∈ A},
UC_{2}) G ∈ J_{X} whenever F ∈ J_{X} and G ⊃ F,
UC_{3}) F ∈ J_{X} implies F^{−1} = {F^{−1} : F ∈ F} ∈ J_{X},

where F^{−1} = {(y, x) : (x, y) ∈ F }.

2) A semiuniform convergence space (X, J_{X}) is called diagonal pro-

vided that the filter (Δ_{X}) generated by the diagonal Δ_{X} of X × X belongs to J_{X}.

3) A semiuniform convergence space (X, J_{X}) is called a semiuniform

limit space provided that the following is satisfied:

UC_{4}) F ∈ J_{X} and G ∈ J_{X} imply F ∩ G ∈ J_{X}.

4) A semiuniform limit space (X, J_{X}) is called a uniform limit space

provided that the following is satisfied:

UC_{5}) F ∈ J_{X} and G ∈ J_{X} imply F ∘ G ∈ J_{X} (whenever F ∘ G exists, i.e. F ∘ G = {(x, y) : ∃z ∈ X with (x, z) ∈ G and (z, y) ∈ F } ≠ ϕ for every F ∈ F, G ∈ G), where F ∘ G is the filter generated by the filter base {F ∘ G : F ∈ F and G ∈ G}.

5) A map f : (X, J_{X}) → (Y, J_{Y}) between semiuniform convergence

spaces is called uniformly continuous provided that (f × f)(J_{X}) ⊂ J_{Y}.

6) The category of semiuniform convergence spaces (and uniformly

continuous maps) is denoted by SUConv, whereas its full subcategories of semiuniform limit spaces and uniform limit spaces are denoted by SULim and ULim respectively.

2.2 Remarks. 1) The category Unif of uniform spaces (and uniformly continuous maps) in the sense of A. Weil [22] can be nicely embedded into the category SUConv, namely Unif is isomorphic to the full subcategory PrULim of ULim whose object class consists of all principal uniform limit spaces (a uniform limit space (X, J_{X}) is called a principal uniform limit space provided that there is a filter F on X × X such that J_{X} = [F], where [F] = {G : G is a filter on X × X with G ⊃ F}). Since PrULim is a bireflective subcategory of ULim (cf. [15;4.2.8.1]) and ULim is a bireflective subcategory of SU-Conv (cf. [17;2.2]), Unif can be embedded into SUConv as a bireflective subcategory. Obviously, a principal uniform limit space (= uniform space) is diagonal.

2) SUConv is a topological category, in which initial and final structures

can be easily described (cf. e.g. [18;2.2.2]).

2.3 Definitions. 1) A Kent convergence space is a pair (X, q), where X is a set and q ⊂ F (X)×X (provided F (X) denotes the set of all filters on X), such that the following are satisfied:

C_{1}) (ẋ, x) ∈ q for each x ∈ X,

C_{2}) (G, x) ∈ q whenever (F, x) ∈ q and G ⊃ F,

C_{3}) (F ∩ ẋ, x) ∈ q, whenever (F, x) ∈ q.

If (X, q) is a Kent convergence space, then we say that F ∈ F (X) converges to x ∈ X provided that (F, x) ∈ q.

2) A Kent convergence space (X, q) is called symmetric provided that

the following is satisfied:

(F, x) ∈ q and y ∈ ⋂_{F ∈F}F imply (F, y) ∈ q .

3) A map f : (X, q) → (X′, q′) between Kent convergence spaces is called

continuous provided that (f(F), f(x)) ∈ q′ for each (F, x) ∈ q.

2.4 Remarks. 1) a) Every semiuniform convergence space (X, J_{X}) has an underlying (symmetric) Kent convergence space (X, q_{γ_{J_{X}}}) defined as follows:

(F, x) ∈ q_{γ_{J_{X}}} ⟺ F ∩ ẋ ∈ γ_{J_{X}}

where γ_{J_{X}} = {H ∈ F (X) : H × H ∈ J_{X}}.

b) A semiuniform convergence space (X, J_{X}) is called a convergence

space provided that

J_{X} = {F ∈ F (X × X) : there are some x ∈ X and G ∈ F (X) with
(G, x) ∈ q_{γ_{J_{X}}} such that F ⊃ G × G},

in other words: A convergence space is a semiuniform convergence space which is generated by all convergent filters. The category Conv of convergence spaces (and uniformly continuous maps) is isomorphic to the category KConv_{s} of symmetric Kent convergence spaces (and continuous maps); it is a bicoreflective subcategory of SUConv (cf. [18;3.2, 3.3 and 3.4]) and thus closed under formation of quotients and coproducts (formed in SUConv).

2) If (X, q) is a Kent convergence space, then for each subset A of X we

define the closure \overline{A} of A by

\overline{A} = \{x ∈ X : there is some G ∈ F (X) with (G, x) ∈ q and A ∈ G\}

3) Conv is closed under formation of closed subspaces and products

(formed in SUConv) [18;3.13].

2.5 Definition. A semiuniform convergence spaces (X, J_{X}) is called regular (resp. t– regular) provided that for each F ∈ J_{X} the subfilter generated by the filter base \{\overline{F} : F ∈ F\} (resp. \{F ∈ F : F = \overline{F} \}) belongs to J_{X}, where \overline{F} denotes the closure of F with respect to the underlying Kent convergence space of the product space (X, J_{X})×(X, J_{X}).

2.6 Remark. Every topological R_{0}–space (X, X), i.e. every topological space (X, X) such that x ∈ \overline{\{y\}} implies y ∈ \overline{\{x\}} for each (x, y) ∈ X × X, defines a symmetric Kent convergence space (X, q_{X}) as follows:

(F, x) ∈ q_{X} iff F ⊃ U(x) ,

where U(x) denotes the neighborhood filter of x with respect to X. Thus it may be considered to be a convergence space in the sense of 2.4 1)b). Having this explanation in mind, we obtain:

a) Every regular topological space (in the usual sense) is t–regular.

b) Every uniform space (= principal uniform limit space) is t–regular.

c) Every t–regular semiuniform convergence space is regular.

2.7 Definitions. 1) A filtermerotopic space (shortly: a filter space) is a pair (X, γ), where X is a set and γ ⊂ F (X) such that the following are satisfied:

(1) ẋ ∈ γ for each x ∈ X,

(2) G ∈ γ whenever F ∈ γ and F ⊂ G.

2) A map f : (X, γ) → (X′, γ′) between filter spaces is called Cauchy

continuous provided that the filter f(F) generated by {f[F] : F ∈ F} belongs to γ′ for each F ∈ γ.

3) The category of filter spaces (and Cauchy continuous maps) is

denoted by Fil.

2.8 Remarks. 1) Every semiuniform convergence space (X, J_{X}) has an underlying filter space (X, γ_{J_{X}}) defined as follows

γ_{J_{X}} = {F ∈ F (X) : F × F ∈ J_{X}} .

The elements of γ_{J_{X}} are called J_{X}–Cauchy filters.

2) If a semiuniform convergence space (X, J_{X}) is called Fil–determined

provided that J_{X} = {F ∈ F (X × X) : there is some G ∈ γ_{J_{X}} with G × G ⊂ F} (i.e. a semiuniform convergence space (X, J_{X}) is Fil–determined iff it is ’generated’ by all J_{X}– Cauchy filters), then the category Fil–D–SUConv of all Fil–determined semiuniform convergence spaces (and uniformly continuous maps) is isomorphic to Fil and a bireflective and bicoreflective subcategory of SUConv (cf. [18;2.4]).

3) The category Prox of proximity spaces (and δ–maps) in the sense

of Efremovic [18] can be embedded into Fil as a bireflective subcategory (cf. [17;2.4.1)] and [15]), whereas the category Conv can be embedded into Fil as a bicoreflective subcategory (cf. [18;2.8, 2.11.3) and 3.2]).

2.9 Proposition. Let (X, J_{X}) be a semiuniform limit space. A J_{X}–Cauchy filter F on X converges to x ∈ X in (X, q_{γ_{J_{X}}}) iff F × ẋ ∈ J_{X}.

Proof. ′′ ⟹′′. Obvious.

′′ ⟸′′. F ×ẋ ∈ J_{X} implies (F ×ẋ)^{−1} = ẋ×F ∈ J_{X}. Furthermore, F ×F ∈ J_{X}, since F is a J_{X}–Cauchy filter. By UC_{1}), ẋ × ẋ ∈ J_{X} for each x ∈ X. Then it follows from UC_{4}) that

(F ∩ ẋ) × (F ∩ ẋ) = (F × F) ∩ (F × ẋ) ∩ (ẋ × F) ∩ (ẋ × ẋ) ∈ J_{X} ,

i.e. F converges to x in (X, q_{γ_{J_{X}}}).

2.10 Definitions. A semiuniform convergence (X, J_{X}) is called

1) precompact (or totally bounded) provided that each ultrafilter U on X is a J_{X}– Cauchy filter,

2) compact provided that each ultrafilter U on X converges in (X, q_{γ_{J_{X}}}),

3) complete provided that each J_{X}–Cauchy filter converges in (X, q_{γ_{J_{X}}}),

4) locally precompact (resp. locally compact) provided that each F ∈ J_{X} contains a precompact (resp. compact) subset of the product space (X, J_{X}) × (X, J_{X}), where a subset of a semiuniform convergence space is called precompact (resp. compact) provided that it is precompact (resp. compact) as a subspace.

2.11 Proposition. (cf. [18;3.7]). Every convergence space is complete.

2.12. The concepts defined under 2.10. are related to each other by the following implication scheme (cf. [20]).

[figure] [figure]
[figure] [figure] [figure] [figure]

diagonal

[figure] [figure] [figure] [figure]

[figure] [figure]

diagonal

precompact

locally precompact complete

t–regular ; and ; complete compact

locally compact

2.13 Theorem (cf. [19;3.12] and [20;3.15]). A semiuniform convergence space (X, J_{X}) is locally compact (resp. locally precompact) iff it is compactly generated (resp. precompactly generated), i.e. J_{X} is the final SUConv–structure with respect to the family (j_{i} : (K_{i}, J_{K_{i}}) → (X, J_{X})) of the inclusions of all compact (resp. precompact) subspaces of (X, J_{X}).

2.14 Definitions. 1) A topological category C, i.e. a concrete category over the category Set of sets (and maps) which is initially complete and small–fibred such that all constant maps between its objects are morphisms, is called hereditary provided that in C final (epi–) sinks are hereditary, i.e. if (f_{i} : A_{i} → A)_{i∈I} is a final (epi–) sink in C, B a subspace (= extremal subject) of A, B_{i} as subspace of A_{i} with underlying set f_{i}^{−1}[B] and g_{i} : B_{i} → B is the corresponding restriction of f_{i}, then (g_{i} : B_{i} → B)_{i∈I} is a final (epi–) sink in C too.

2) A cartesian closed topological category C is called a topological

universe provided that C is hereditary.

3) A topological universe C is called strong provided that in C prod-

ucts of quotients are quotients.

2.15 Theorem (cf. [17]). SUConv is a strong topological universe.

2.16 Corollaries. 1) SULim and Fil are strong topological universes (cf. [17]).

2) The category LPC–SUConv of locally precompact semiuniform

convergence spaces is a topological universe (cf. [20]).

3) The category Conv is a cartesian closed topological category in

which quotient maps are productive (cf. [18]).

4) ULim as well as the category LC–SUConv of locally compact

semiuniform convergence spaces are cartesian closed topological categories (cf. [2], [13] and [19]).

2.17 Remark. The power objects (= natural function spaces) in the category SUConv are defined as follows: Let X = (X, J_{X}) and Y = (Y, J_{X}) be semiuniform convergence spaces. Then the power–object Y^{X} is the set [X, Y] of all uniformly continuous maps from X into Y endowed with the SUConv–structure J_{X,Y} = {ϕ ∈ F ([X, Y] × [X, Y]) : ϕ(F) ∈ J_{Y} for each F ∈ J_{X}}, where ϕ(F) is the filter generated by {A(F) : A ∈ ϕ, F ∈ F} with A(F) = {(f(a), g(b)) : (f, g) ∈ A, (a, b) ∈ F }. J_{X,Y} is also called the uniformly continuous SUConv–structure.

2.18 Proposition. Let X = (X, J_{X}) be a semiuniform convergence space and Y = (Y, J_{Y}) a regular semiuniform limit space. Then ([X, Y], J_{X,Y}) is a complete semiuniform limit space provided that Y is complete.

Proof. By [2;1.5.1], ([X, Y], J_{X,Y}) is a semiuniform limit space provided that Y is a semiuniform limit space. Since 2.9 is valid, the proof of 2.18 parallels [13;theorem 3.9.]

2.19 Definition (cf. [16;2.5]). Let E ⊂ |SUConv|. Then a semiuniform convergence space (X, J_{X}) is called totally E–separated provided that for each x ∈ X the E–quasicomponent K_{x} of X containing x is a singleton, i.e. for any two distinct elements x, y ∈ X there exists an object (E, J_{E}) ∈ E and a uniformly continuous map f : (X, J_{X}) → (E, J_{E}) such that f(x) ≠ f(y). The class of all totally E–separated semiuniform convergence spaces is denoted by QE.

2.20 Examples. 1) Let E be the class of all T_{0}–,T_{1}– or T_{2}–spaces respectively, then E = QE (Note: A semiuniform convergence space (X, J_{X}) is called a

a) T_{0}–space provided that for each pair (x, y) ∈ X × X, (ẋ, y) ∈ q_{γ_{J_{X}}} and (ẏ, x) ∈ q_{γ_{J_{X}}} imply x = y,

b) T_{1}–space provided that for each pair (x, y) ∈ X × X, (ẋ, y) ∈ q_{γ_{J_{X}}} implies x = y,

c) T_{2}–space provided that each filter on X converges to at most one element of X in (X, q_{γ_{J_{X}}})).

2) If D_{H}E denotes the class of all totally E–disconnected semiuniform

convergence spaces, then QD_{H}E = D_{H}E (cf. [18;6.30] and [15;1.3.13]).

2.21 Proposition. Let X = (X, J_{X}) and Y = (Y, J_{Y}) be semiuniform convergence spaces. Then ([X, Y], J_{X,Y}) is totally E–separated provided that Y is totally E–separated.

Proof. Let f_{0} ∈ [X, Y] and let Q_{f_{0}} be the E–quasicomponent of [X, Y] containing f_{0}. For each x ∈ X define

h_{x} : [X, Y] → X × [X, Y]

by h_{x}(f) = (x, f). Obviously, h_{x} is uniformly continuous (provided that X × [X, Y] is endowed with the product structure with respect to SUConv). Since Q_{f_{0}} is E–connected with respect to [X, Y], h_{x}[Q_{f_{0}}] = {x} × Q_{f_{0}} is E–connected with respect to X × [X, Y] (cf. [15;1.3.7(3)(b)]), and, since the evaluation map e_{X,Y} : X × Y^{X} → Y is uniformly continuous, e_{X,Y} [{x} × Q_{f_{0}}] = {f(x) : f ∈ Q_{f_{0}}} is E–connected with respect to Y. Since Y ∈ QE, it follows that f(x) = f_{0}(x) for each f ∈ Q_{f_{0}} and for each x ∈ X, i.e. Q_{f_{0}} is a singleton. Thus ([X, Y], J_{X,Y}) is totally E–separated.

2.22 Remark. Obviously, the above proposition is valid in every cartesian closed topological category.

3. Precompactness and compactness in the power objects of SUConv

3.1 Definition. A semiuniform convergence space (X, J_{X}) is called semi–pseudouniform provided the following is satisfied:

(Ps) F ∈ J_{X} whenever U ∈ J_{X} for each ultrafilter U ⊃ F.

3.2 Remarks. 1) Every semi–pseudouniform space (X, J_{X}) is a semiuniform limit space (cf. [2;1.1.5]).

2) The category SUnif of semiuniform spaces (and uniformly con-

tinuous maps) in the sense of Čech [5] can be embedded into the category SPsU of semi–pseudouniform spaces (and uniformly continuous maps) as a bireflective subcategory (cf. [2;1.4.10]).

3) SPsU is a bireflective subcategory of SULim (cf. [2;1.3.2. and

1.3.6.a)] and SULim is a bireflective subcategory of SUConv (cf. [2;1.3.4] or [17;2.2.2)b)]).

3.3 Proposition. Let (X, J_{X}) be a semi–pseudouniform space and A ⊂ X a precompact subset. Then the filter (Δ_{A}) on X × X generated by the diagonal Δ_{A} in A × A belongs to J_{X}.

Proof. Obviously, A is a precompact subset of X iff each ultrafilter U on X containing A is a J_{X}–Cauchy filter. Let W be an ultrafilter on X × X finer than (Δ_{A}). Then there is an ultrafilter U on X containing A such that Δ_{U} = W, where Δ_{U} is the filter on X × X generated by the filter Δ′_{U} = {Δ_{U} : U ∈ U} on Δ_{X}, namely U = {F ⊂ X : Δ_{F} ∈ W} (note: since U is an ultrafilter, Δ_{U} ⊂ W is an ultrafilter too, because Δ′_{U} is an ultrafilter). Since U × U ⊂ Δ_{U} and A is a precompact subset of X, Δ_{U} = W belongs to J_{X}. Thus, since (X, J_{X}) is semi–pseudouniform, (Δ_{A}) belongs to J_{X}.

3.4 Corollary. Every precompact semi-pseudouniform space is diagonal.

3.5 Proposition (cf. [2;1.5.1.d)] or [13;3.11]). Let X = (X, J_{X}) and Y = (Y, J_{Y}) be semiuniform convergence spaces. Then ([X, Y], J_{X,Y}) is a semi–pseudouniform space provided that Y is a semi-pseudouniform space. The inverse implication is valid provided that X ≠ ϕ.

3.6 Definiton. Let X = (X, J_{X}) and Y = (Y, J_{Y}) be semiuniform convergence spaces and M ⊂ [X, Y]. Then M is called uniformly equicontinuous provided that (Δ_{M}) ∈ J_{X,Y}, i.e. for each F ∈ J_{X} the filter (Δ_{M})(F) generated by {Δ_{M}(F) : F ∈ F} with Δ_{M}(F) = {(f(a), f(b)) : f ∈ M, (a, b) ∈ F } belongs to J_{Y}.

3.7 Remarks. 1) If X and Y are principal uniform limit spaces (= uniform spaces), then “uniformly equicontinuous“ in the sense of 3.6 means “uniformly equicontinuous“ in the usual sense.

2) Let (X, X) be a topological R_{0}–space (= symmetric topological space)

and X = (X, J_{γ_{q_{X}}}) its corresponding semiuniform convergence space, i.e. J_{γ_{q_{X}}} = {F ∈ F (X × X) : ∃x ∈ X such that F ⊃ U(x) × U(x)}, where U(x) denotes the neighborhood filter of x with respect to the topology X. Furthermore, let Y be a principal uniform limit space (=uniform space). Then “uniformly equicontinuous“ means “equicontinuous′′ in the usual sense.

3.8 Definition. Let Δ–SUConv be the category of diagonal semiuniform convergence spaces (and uniformly continuous maps). If X is a set and (Y, J_{Y}) ∈ |Δ−SUConv|, then the Δ–SUConv–structure

J_{Y^{X}}^{u} = {ϕ ∈ F (Y^{X} × Y^{X}) : ϕ((Δ_{X})) ∈ J_{Y} }

is called the structure of uniform convergence on Y^{X}.

3.9 Remark. If the set X in 3.8 is endowed with the discrete Δ–SUConv–structure [(Δ_{X})]={F ∈F (X×X) : F ⊃(Δ_{X})}, then [(X, [(Δ_{X})]), (Y, J_{Y})]_{Δ−SUConv} =[(X, [(Δ_{X})]), (Y, J_{Y})]_{SUConv} = Y^{X} and the uniformly continuous SUConv–structure J_{X,Y} (cf. 2.17) coincides with the structure of uniform convergence on Y^{X}.

3.10 Proposition. Let X = (X, J_{X}) and Y = (Y, J_{X}) be diagonal semiuniform convergence spaces, J_{X,Y} the uniformly continuous SUConv–structure on [X, Y]_{SUConv} and J_{X,Y}^{u} the structure induced on [X, Y]_{SUConv} by J_{Y^{X}}^{u} . Then

1_{[X,Y]} : ([X, Y], J_{X,Y}) → ([X, Y], J_{X,Y}^{u})

is uniformly continuous, i.e. J_{X,Y} is finer than J_{X,Y}^{u}.

Proof. Since X is diagonal, (Δ_{X}) ∈ J_{X} and, by definition of J_{X,Y} , ϕ((Δ_{X})) ∈ J_{Y} for each ϕ ∈ J_{X,Y}. If i : [X, Y] → Y^{X} denotes the inclusion map, then, since ϕ ⊂ (i × i)(ϕ), it follows that (i × i)(ϕ)((Δ_{X})) ∈ J_{Y}, i.e. (i × i)(ϕ) ∈ J_{Y^{X}}^{u}. Thus ϕ ∈ J_{X,Y}^{u} for each ϕ ∈ J_{X,Y}.

3.11 Remark. It is easily verified that Δ–SUConv is a bireflective subcategory of SUConv and therefore initial structures in Δ–SUConv are formed as in SUConv. Consequently, J_{X,Y}^{u} is also a Δ–SUConv–structure. On the other hand, ([X, Y], J_{X,Y}) in 3.10 is generally not diagonal, e.g. if IR_{u} denotes the set IR of real numbers endowed with the usual uniformity, then ([IR_{u}, IR_{u}], J_{IR,IR}) is a uniform limit space, which is not diagonal (cf. [10;2.2.]). Thus, J_{X,Y} ≠ J_{X,Y}^{u} in general.

3.12 Theorem (Necessary conditions for precompactness in ([X, Y], J_{X,Y})). Let X = (X, J_{X}) be a semiuniform convergence space, Y = (Y, J_{Y}) a semi-pseudouniform space and M a precompact subset of ([X, Y], J_{X,Y}). Then the following are satisfied:

(1) M is uniformly equicontinuous,

(2) M(x) = {f(x) : f ∈ M} ⊂ Y is precompact for each x ∈ X.

Proof. Since Y is semi-pseudouniform, it follows from 3.5 that ([X, Y], J_{X,Y}) is semipseudouniform. Thus, by 3.3, M is uniformly equicontinuous. For each x ∈ X, the restriction map e_{X,Y} |_{{x}×M} : {x} × M → Y of the evaluation map e_{X,Y} : X × [X, Y] → Y is uniformly continuous. Since {x} and M are precompact, M ×{x} is precompact (cf. [18;4.2]). Consequently (e_{X,Y} |_{{x}×M})[{x} × M] = M(x) ⊂ Y is precompact (cf.[20;2.11]).

3.13 Definition. A semi-pseudouniform space Y = (Y, J_{Y}) is called a semi-Ascoli space provided that for every precompact uniform limit space X = (X, J_{X}) and every uniformly equicontinuous set M ⊂ [X, Y] the subspace structure of M with respect to the product structure of Y^{X} coincides with the subspace structure of M in ([X, Y], J_{X,Y}).

3.14 Remark. Obviously, a semi-pseudouniform space Y = (Y, J_{Y}) is a semi-Ascoli space iff for every precompact uniform limit space X = (X, J_{X}) and every uniformly equicontinuous set M ⊂ [X, Y] the subspace structure of M in ([X, Y], J_{X,Y}) contains the subspace structure of M with respect to the product struture of Y^{X} (i.e. the structure of simple convergence on M).

3.15 Theorem (Sufficient conditions for precompactness in ([X, Y], J_{X,Y}).

Let X = (X, J_{X}) be a locally precompact uniform limit space and Y = (Y, J_{Y}) a semi-Ascoli space. Then M ⊂ [X, Y] is a precompact subset of ([X, Y], J_{X,Y}) provided that the following are satisfied:

(1) M is uniformly equicontinuous,

(2) M(x) = {f(x) : f ∈ M} ⊂ Y is precompact for each x ∈ X.

Proof. Let N be the space (M(J_{Y^{X}})_{M}), where (J_{Y^{X}})_{M} denotes the subspace structure of M in the product space (Y^{X}, J_{Y^{X}}). Since M ⊂ ∏_{x∈X} M(x) ⊂ Y^{X}, N is precompact

(note (2) and cf. [18;4.2]). Consider first the case where X is precompact. Since Y is a semi–Ascoli space and M is uniformly equicontinuous, N = (M, (J_{X,Y})_{M}) where (J_{X,Y})_{M} denotes the subspace structure of M with respect to J_{X,Y}. Then the restriction h of the evaluation map e_{X,Y} : X × ([X, Y], J_{X,Y}) → Y to X × N is uniformly continuous. Since ([X, Y], J_{X,Y}) is a power object in SUConv, there is a unique uniformly continuous map \overline{h} : N → ([X, Y], J_{X,Y}) such that e_{X,Y} ∘(1_{X} ×\overline{h}) = h. Then \overline{h}(f) = f for every f ∈ M and consequently \overline{h}[N] = (M, (J_{X,Y})_{M}). Since N is precompact, it follows that (M, (J_{X,Y})_{M}) is precompact (cf. [20;2.11]). Consider next any locally precompact uniform limit space X. By 2.13, the family (j_{i} : K_{i} → X)_{i∈I} of inclusions of all precompact subspaces K_{i} = (K_{i}, J_{K_{i}}) of X is a final epi–sink in SUConv. Then the source (h_{i} : ([X, Y], J_{X,Y}) → ([K_{i}, Y], J_{K_{i},Y})_{i∈I} with h_{i}(f) = f ∘ j_{i} = f|K_{i} is an initial mono–source in SUConv. For each i ∈ I, M_{i} = h_{i}[M] is uniformly equicontinuous (namely, since h_{i} is uniformly continuous, h_{i} × h_{i}((Δ_{M})) = (Δ_{M_{i}}) ∈ J_{K_{i},Y} for each i ∈ I). Furthermore, for each x_{i} ∈ K_{i}, M_{i}(x_{i}) = M(x_{i}) is precompact by assumption. Thus, using the first case of this proof, M_{i} is precompact for each i ∈ I. Since (h_{i}|M : (M, (J_{X,Y})_{M}) → (M_{i}, (J_{K_{i},Y})_{M_{i}})_{i∈I} is an initial mono–source in SUConv, (M, (J_{X,Y})_{M}) is precompact (cf. [18;4.2]).

3.16 Remark. 1) Every uniform space (=principal uniform limit space) (X, J_{X}) is a semi–Ascoli space (cf. [23;4.7] and note that every Ascoli–space in the sense of Wyler is a semi–Ascoli space).

2) The function space ([IR_{u}, IR_{u}], J_{IR,IR}) considered under 3.11 is a semi–

Ascoli space (since it is an Ascoli–space in the sense of Wyler by [23;4.4]), which is not uniform (otherwise it would be diagonal).

3.17 Corollary 1. (Necessary conditions for compactness in ([X, Y], J_{X,Y})). Let X = (X, J_{X}) be a semiuniform convergence space and Y = (Y, J_{Y}) is a semi– pseudouniform T_{2}–space. If M is a compact subset of ([X, Y], J_{X,Y}), then the following are satisfied:

(1) M is closed in ([X, Y], J_{X,Y}), i.e. M = \overline{M}, where \overline{M} denotes the closure

of M with respect to the underlying Kent convergence space of ([X, Y], J_{X,Y}), (2) M is uniformly equicontinuous,

(3) M(x) ⊂ Y is compact for each x ∈ X.

Proof. (1) By 2.20 and 2.21, ([X, Y], J_{X,Y}) is a T_{2}–space. Since M is compact, M is closed (cf. [9;3.16.14.]).

(2) is obvious, since M is precompact and 3.12 is valid.

(3) cf. the corresponding proof under 3.12 and replace “precompact“ by “compact“.

3.18 Corollary 2 (Sufficient conditions for compactness in ([X, Y], J_{X,Y})). Let X = (X, J_{X}) be a locally precompact uniform limit space and Y = (Y, J_{Y}) a regular complete semi–Ascoli space (e.g. a complete uniform space). Then M ⊂ [X, Y] is a compact subset of ([X, Y], J_{X,Y}) provided that the following are satisfied:

(1) M is closed in ([X, Y], J_{X,Y}),

(2) M is uniformly equicontinuous,

(3) M(x) ⊂ Y is compact for each x ∈ X.

Proof. By 2.18 and 3.2.1), ([X, Y], J_{X,Y}) is complete. By 3.15, M is precompact. Since M is closed, M is complete too (cf. [18;3.12]). Thus, M is compact.

3.19 Corollary 3. Let (X, X) be a compact topological T_{2}–space and (Y, W) a separated uniform space (= uniform T_{2}–space). Further, let C(X, Y) be the set of all continuous maps from (X, X) into (Y, Y_{W}), where Y_{W} denotes the topology on Y induced by the uniformity W for Y. If C_{u}(X, Y) is the set C(X, Y) endowed with the uniform struture of uniform convergence (cf. also [18;7.6.2)]), then M ⊂ C_{u}(X, Y) is relatively compact, i.e. the closure \overline{M} of M with respect to the topology of uniform convergence is compact, if and only if M is equicontinuous and for each x ∈ X, the set M(x) is relatively compact in (Y, W).

Proof. “ ⇐ “. Let U be the unique uniformity inducing X. It follows from the assumptions on (X, X) and (Y, W) that C_{u}(X, Y) = C_{c}(X, Y), where C_{c}(X, Y) denotes the set C(X, Y) endowed with the uniformity of compact convergence (cf. [21;II 4.5]). Furthermore, equicontinuity of M means uniform equicontinuity (cf. [4;chap. 10,§2, no 1, corollaire 2]). Thus, the closure \overline{M} of M in C_{u}(X, Y) coincides with the closure of M in Y^{X} endowed with the product uniformity [= structure of simple convergence] (cf. [4;chap. 10,§3, no 4, corollaire]). Since p_{x}[\overline{M}] = \overline{M}(x) ⊂ \overline{p_{x}[M]} = \overline{M(x)}, where p_{x} : Y^{X} → Y denotes the x–th projection for each x ∈ X, \overline{M}(x) is relatively compact and consequently precompact for each x ∈ X. Thus, by 3.15, \overline{M} is precompact in ([X, Y], J_{X,Y}), where X = (X, J_{X}) and Y = (Y, J_{Y}) with J_{X} = [U] and J_{Y} = [W] respectively (note: According to [4;chap. 10,§2, no 3, proposition 6], \overline{M} is uniformly equicontinuous). Hence, by 3.10, \overline{M} is precompact in ([X, Y], J_{X,Y}^{u}) = C_{u}(X, Y). Since \overline{M} ⊂ ∏_{x∈X} \overline{M}(x) ⊂ Y^{X} and \overline{M}(x) is compact and consequently complete, \overline{M} is complete with respect to the structure of simple convergence [as a closed subspace of a complete product space] and thus, \overline{M} is complete with respect to the structure of compact convergence (cf. [4;chap. 10, §1, no 5, corollaire 2]), i.e. as a subspace of C_{c}(X, Y) = C_{u}(X, Y). Hence, \overline{M} is precompact and complete, i.e. \overline{M} is compact.

“ ⇒ “. Let N = \overline{M}. By assumption, N is compact in C_{u}(X, Y) = ([X, Y], J_{X,Y}^{u}), where X and Y are defined as under “ ⇐ “. Then N is compact in ([X, Y], J_{X,Y}) too, since filter convergence with respect to J_{X,Y}^{u} coincides with filter convergence with respect to J_{X,Y} (use [13;3.5] and note that X and Y are diagonal uniform limit spaces). Thus, by 3.17, N is uniformly equicontinuous and p_{x}[N] = N(x) is compact in Y for each x ∈ X. It follows from M ⊂ N that M is uniformly equicontinuous (= equicontinuous) and, since Y is a T_{2}–space, p_{x}[M] = M(x) is relatively compact in Y, i.e. in (Y, W) (cf. [18;5.8]), for each x ∈ X.

3.20 Remarks. 1) Note, that for the first part of the proof of 3.19 (“ ⇐ “) the uniform space (Y, W) need not be separated.

2) If in 3.19 the uniform space (Y, W) is metrizable, then C_{u}(X, Y) is

metrizable (cf. [4;chap. 10, §3, no 1, proposition 1]) and from 3.19 results the classical Ascoli theorem: If (X, X) is a compact topological T_{2}–space and (Y, W) a metrizable uniform space, then the following are equivalent for each M ⊂ C(X, Y):

(1) Each sequence (f_{n}) in M has a subsequence converging uniformly to some f ∈ C(X, Y),

(2) M is equicontinuous and M(x) is relatively compact in (Y, W) for each x ∈ X. If (Y, W) = IR_{u}, then the above conditions are equivalent to

(3) M is equicontinuous and M(x) is bounded in IR for each x ∈ X.

3.21 Problem. Does there exist a semi–Ascoli space which is not an Ascoli space in the sense of Wyler [23]?

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