NET SPACES IN CATEGORICAL TOPOLOGY

Josef Šlapal

Abstract. In the paper we introduce the notion of 𝒮 -nets, 𝒮 a non-empty construct, as a generalization of the usual nets. 𝒮 -net spaces and continuous maps between them are then defined in a natural way. For 𝒮 -net spaces we introduce a number of axioms and study the categories obtained. We find some topological and cartesian closed categories of 𝒮 -net spaces and describe their relations to certain known topological categories.

0. Introduction

Cartesian closed topological categories belong to the most important objects studied in categorical topology. The usefulness of these categories for applications to many classical branches of mathematics is well known. And the interest in cartesian closed categories became even stronger after discovering the important role that they play in computer science. When looking for new cartesian closed topological categories categorical topologists usually start from the category of topological spaces and generalize or modify the considered axioms of topology. These axioms depend on the chosen access to topology (closure operations, neighborhood structures, systems of open sets, etc.). In this connection the most often used access is that one which views topologies as convergence structures and uses filters for describing the convergence. This filter access gave arise to a rich hierarchy of useful categories and an intensive study is being performed in the area at present.

In this paper we will also view topologies as convergence structures, but we shall use nets instead of filters for expressing the convergence. This access is closely related to that one using filters and some connections between certain types of categories of convergence filter spaces and those of convergence net spaces are described in [14]. Of course, from the classical point of view filters are more suitable for investigations in topology than nets which have, essentially, an order-theoretic nature. However, in our considerations we will not deal with the usual nets, but

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). AMS classification: 54A05, 54A20, 54B30, 18B30, 18D15.

Key words and phrases. Net, subnet, net space, convergence space, limit space, topological category, cartesian closed category, topological universe, diagonality axiom, condition of iterated limits, preclosure space, closure space, pseudo-topological space, topological space, R_{0}-space, T_{0}- space, T_{1}-space.

This paper was finished during the author’s stay at the Free University of Berlin as a fellow of the Konferenz der deutschen Akademien der Wissenschaften (Stipendienprogram gefördert durch die Volkswagenstiftung) from October 1994 to April 1995. .

with a certain type of generalized nets which have lost their order-theoretic nature in the benefit of a category-theoretic one. Our use of this generalized nets, called 𝒮 -nets (where 𝒮 is a construct), could be hardly duplicated by using filters.

For 𝒮 -net spaces introduced we will consider a number of axioms including, at the first place, the axioms that are extensions of the usual net axioms. By this way we obtain a number of full subcategories of the category of 𝒮 -net spaces (with extended continuous maps as morphisms) which are then studied. We will especially deal with the question of cartesian closedness of these subcategories and with their relations to some known topological categories (subcategories and supercategories of Top). We discover a number of cartesian closed topological categories of 𝒮 - net spaces some of which are trivial, some are isomorphic to certain well known categories, but many are new and useful for applications to various branches of mathematics. We also receive some known results (see e.g. [5] and [7]) as trivial consequences of certain more general facts.

1. Basic definitions

Throughout the paper, 𝒮 will denote a non-empty construct (i.e. concrete category over Set) whose objects have non-empty underlying sets. As usual, we will not distinguish notationally between 𝒮 -objects, resp 𝒮 -morphisms, and their underlying sets, resp. maps. If X, Y are sets and y ∈ Y is an element, then the constant map f : X → Y with f(x) = y for each x ∈ X will be called the y-constant map.

1.1. Definitions.

(1) Let X be a set. By an 𝒮 -net in X we understand any pair (S, f) where S ∈ 𝒮 is an object and f : S → X is a map. The class of all 𝒮 -nets in X will be denoted by ⟨X⟩_{𝒮} .

(2) If (S, f), (T, g) ∈ ⟨X⟩_{𝒮} , then (S, f) is called a subnet of (T, g) if there is an 𝒮 -morphism φ : S → T such that f = g ∘ φ.

1.2. Examples.

1. Let Dir be the category of directed sets whose morphisms are the cofinal maps, i.e. maps φ : (S_{1}, ≤_{1}) → (S_{2}, ≤_{2}) having the property that for any s_{2} ∈ S_{2} there is s_{1} ∈ S_{1} such that s_{2} ≤_{2} φ(s) whenever s ∈ S_{1}, s_{1} ≤_{1} s. If 𝒮 = Dir, then the notions of an 𝒮 -net and a subnet (of an 𝒮 -net) coincide with the usual notions of a net and a subnet (see e.g. [7]), respectively.

2. Let α > 0 be an ordinal and let \underline{α} be the category whose only object is α and whose morphisms are isotone injections of α into itself. Clearly, \underline{α} is a subcategory of Dir. For 𝒮 = \underline{α} the notions of an 𝒮 -net and a subnet coincide with the notions of a sequence of type α and a subsequence, respectively. Sequences of type ω are usually called briefly sequences.

3. Let Set^{+} be the category of non-empty sets. If 𝒮 = Set^{+} and (S, f), (T, g) ∈ ⟨X⟩_{𝒮} , then, clearly, (S, f) is a subnet of (T, g) iff f(S) ⊆ g(T).

1.3. Denotation. If X is a set and π : ⟨X⟩_{𝒮} → expX a map, then we will usually write (S, f) →^{π} x instead of x ∈ π(S, f), and (S, f) ↛^{π} x instead of x ∈ X − π(S, f).

The set π(S, f) can be viewed as the set of all limits of the net (S, f) w.r.t. π and therefore we will sometimes say that (S, f) converges to x (w.r.t. π) if (S, f) →^{π} x.

1.4. Definition.

(1) A pair (X, π) where X is a set and π : ⟨X⟩_{𝒮} → expX a map is called an 𝒮 -net space provided that (S, f) →^{π} x whenever f is the x-constant map.

(2) An 𝒮 -net space (X, π) is called a convergence 𝒮 -net space provided that from (S, f) →^{π} x it follows that (T, g) →^{π} x for each subnet (T, g) of (S, f).

(3) A convergence 𝒮 -net space (X, π) is called a limit 𝒮 -net space provided that, whenever (S, f) ↛^{π} x, there is a subnet (T, g) of (S, f) such that (U, h) ↛^{π} x for any subnet (U, h) of (T, g).

1.5. Examples.

1. Let 𝒮 = Dir or 𝒮 = \underline{α} where α > 0 is an ordinal. Let (X, 𝒪) be a topological space (given by the system 𝒪 of open sets). Let (X, π) be the 𝒮 -net space associated with (X, 𝒪), i.e., for any (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{π} x iff for each A ∈ 𝒪 with x ∈ A there exists s_{A} ∈ S such that f(s) ∈ A for every s ∈ S, s ≥ s_{A}. Then, obviously, (X, π) is a limit 𝒮 -net space.

2. The ℬ-convergence structures studied in [15] for special subcategories ℬ of Dir are nothing else than the maps π : ⟨X⟩_{ℬ} → expX for which (X, π) are convergence ℬ-net spaces.

3. Let 𝒮 = \underline{ω}. Then for convergence 𝒮 -net spaces (X, π) the maps π are precisely the multivalued convergences on X from [10]. The convergence or limit 𝒮 -net spaces (X, π) fulfilling card π(S, f) ≤ 1 for each (S, f) ∈ ⟨X⟩_{𝒮} are known as the (Fréchet) ℒ -spaces or ℒ^{∗}-spaces, respectively (see e.g. [4]).

1.6. Definition. Let (X, π), (Y, ρ) be a pair of 𝒮 -net spaces. A map F : X → Y is called a continuous map of (X, π) into (Y, ρ) provided that (S, F ∘ f) →^{ρ} F (x) whenever (S, f) →^{π} x. We denote by Net_{𝒮} the category of 𝒮 -net spaces and continuous maps, and by Conv_{𝒮} and Lim_{𝒮} its full subcategories of convergence 𝒮 -net spaces and limit 𝒮 -net spaces respectively. The set of all continuous maps of (X, π) into (Y, ρ) will be denoted by C((X, π), (Y, ρ)).

1.7. Examples.

1. Let (X, π) ∈ Net_{𝒮} be an object and let π′ : ⟨X⟩Set^{+} → expX be the map

given by (S, f) →^{π′} x iff f is the x-constant map or there exists (S, f) ∈ ⟨X⟩_{𝒮} such that (S, f) →^{π} x. Then the assignation (X, π) ↦ (X, π′) determines a concrete embedding of Net_{𝒮} into NetSet^{+}.

2. For 𝒮 = Dir the category Lim_{𝒮} is studied in [5] where its objects are called L^{∗}-spaces.

2. Properties of Net_{𝒮} , Conv_{𝒮} and Lim_{𝒮}

The category Net_{𝒮} is not fibre-small in general (it is fibre-small whenever 𝒮 is small). Therefore we do not suppose fibre-smallness when working with initially structured or topological categories. More precisely, by an initially structured category we understand a construct which is closed under formation of initial objects for all mono-sources and in which each constant map between its objects is a morphism. An initially structured category which is initially complete (i.e. closed under formation of initial objects for all sources) is said to be a topological category. Also the notion of a topological universe is meant here in the generalized sense obtained by omitting the requirement of fibre-smallness (cf. [1]). Thus, a topological universe is a cartesian closed topological category 𝒦 with the property that every 𝒦 -object C has a one-point extension, i.e. a 𝒦 -object \overline{C} = C ∪ \{∞_{C}\} where ∞_{C} ∉ C such that C is a subobject of \overline{C} and for any 𝒦 -object A, any subobject B of A and any 𝒦 -morphism φ : B → C the map \overline{φ} : A → C, given by \overline{φ}(a) = φ(a) whenever a ∈ B and \overline{φ}(a) = ∞_{C} whenever A ∉ B, is a 𝒦 -morphism. Let us remark here that by a subobject of an object we always understand an initial object with regard to an inclusion map. A topological universe is said to be strong provided that its quotient maps are productive (cf. [12]). Throughout the paper, products in 𝒮 are meant to be products over non-empty index sets.

2.1. Theorem. Net_{𝒮} is a strong topological universe.

Proof. Clearly, each constant map between objects of Net_{𝒮} is continuous. Let F_{i} : X → (X_{i}, π_{i}), i ∈ I, be a source in Net_{𝒮} and let π : ⟨X⟩_{𝒮} → expX be given as follows: (S, f) →^{π} x iff (S, F_{i} ∘ f) →^{π_{i}} F_{i}(x) for each i ∈ I. Clearly, (X, π) ∈ Net_{𝒮} . Let (Y, ρ) ∈ Net_{𝒮} be an object and G : Y → X a map such that F_{i}∘G : (Y, ρ) → (X_{i}, π_{i}) is continuous for each i ∈ I. Let (T, g) ∈ ⟨Y ⟩_{𝒮} , (T, g) →^{ρ} y. Then (T, F_{i} ∘ G ∘ g) →^{π_{i}} F_{i} ∘ G(y) for each i ∈ I. Hence (T, G ∘ g) →^{π} G(y), i.e. G : (Y, ρ) → (X, π) is a continuous map. Therefore (X, π) is the initial object of the source F_{i}, i ∈ I, and it is unique because Net_{𝒮} is transportable. Consequently, Net_{𝒮} is initially complete. We have shown that Net_{𝒮} is a topological category. Let (X, π), (Y, ρ) ∈ Net_{𝒮} be objects. Let Z = C((X, π), (Y, ρ)) and let σ : ⟨Z⟩_{𝒮} → expZ be given by z ∈ σ(S, g) iff from (S, f) →^{π} x it follows that (S, g^{f}) →^{ρ} z(x) where g^{f} : S → Y is defined by g^{f}(s) = g(s)(f(s)). Then (Z, σ) ∈ Net_{𝒮} and we will show that (Z, σ) is a function space of (X, π) and (Y, ρ) in Net_{𝒮} . To this end, let e : (X, π) × (Z, σ) → (Y, ρ) be the evaluation map, i.e. the map with e(x, z) = z(x) whenever (x, z) ∈ X × Z. Put (V, τ) = (X, π) × (Z, σ)

and let (S, f) ∈ ⟨V ⟩_{𝒮} , (S, f) →^{τ} (x, z). Then (S, pr_{X} ∘ f) →^{π} x and (S, pr_{Z} ∘

f) →^{σ} z. Hence (S, h^{p}) →^{ρ} z(x) where h = pr_{Z} ∘ f and p = pr_{X} ∘ f. We have h^{p}(s) = h(s)(p(s)) = e(p(s), h(s)) = e(pr_{X}(f(s)), pr_{Z}(f(s))) = e(f(s)) for every s ∈ S. Therefore h^{p} = e ∘ f which yields (S, e ∘ f) →^{ρ} e(x, z). Consequently, e : (X, π)×(Z, σ) → (Y, ρ) is continuous. Now, let F : (X, π)×(W, υ) → (Y, ρ) be a continuous map and let F^{∗} : W → Z be the map given by F^{∗}(w)(x) = F (x, w). Let

(S, g) ∈ ⟨W ⟩_{𝒮} , (S, g) →^{υ} w, and (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{π} x. Then (S, (f, g)) con-

verges to (x, w) in the space (X, π) × (W, υ). Hence (S, F ∘ (f, g)) →^{ρ} F (x, w). This results in (S, F^{∗} ∘ g) →^{σ} F^{∗}(w) because F ((f, g)(s)) = F^{∗}(g(s))(f(s)) = (F^{∗} ∘ g)^{f}(s) whenever s ∈ S. Consequently, F^{∗} : (W, υ) → (Z, σ) is a continuous map. Since clearly e ∘ (id_{X} × F^{∗}) = F, we have shown that (Z, σ) is a function space of (X, π) and (Y, ρ) in Net_{𝒮} . Therefore Net_{𝒮} is cartesian closed.

Let (X, π) ∈ Net_{𝒮} be an object and ∞_{X} a point with ∞_{X} ∉ X. Let (X ∪ {∞_{X}}, π′) ∈ Net_{𝒮} be defined as follows: if (S, f) ∈ ⟨X ∪ {∞_{X}}⟩_{𝒮} , then π′(S, f) = π(S, f) ∪ {∞_{X}} whenever ∞_{X} ∉ f(S) and π′(S, f) = X ∪ {∞_{X}} whenever ∞_{X} ∈ f(S). Clearly, (X, π) is a subspace (i.e. a subobject) of (X ∪ {∞_{X}}, π′). Let (Y, ρ) ∈ Net_{𝒮} , let (Z, σ) be a subspace of (Y, ρ) and let F : (Z, σ) → (X, π) be a continuous map. Let \overline{F} : Y → X ∪ \{∞_{X}\} be defined by \overline{F} (y) = F (y) whenever y ∈ Z and \overline{F} (y) = ∞_{X} whenever y ∈ Y − Z. It can easily be seen that \overline{F} : (Y, ρ) → (X ∪ \{∞_{X}\}, π′) is a continuous map. Consequently, Net_{𝒮} is a topological universe. We will show that this universe is strong. Let (X, π) ∈ Net_{𝒮} be an object, Y a set and F : X → Y a surjection. Evidently, an object (Y, ρ) ∈ Net_{𝒮} is the quotient object of (X, π) with regard to F if and only if the following condition is fulfilled: (S, g) →^{ρ} y iff there is (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{π} x, such that F ∘ f = g and F (x) = y. Let F_{i} : (X_{i}, π_{i}) → (Y_{i}, ρ_{i}), i ∈ I, be a family of quotient maps in Net_{𝒮} . Let (X, π) = ∏_{i∈I}(X_{i}, π_{i}), (Y, ρ) = ∏_{i∈I}(Y_{i}, ρ_{i}) be the

(cartesian) products and let p_{i} : (X, π) → (X_{i}, π_{i}), q_{i} : (Y, ρ) → (Y_{i}, ρ_{i}), i ∈ I, be the corresponding projections. Finally, let F : (X, π) → (Y, ρ) be the unique continuous map with q_{i} ∘ F = F_{i} ∘ p_{i} for all i ∈ I. Obviously, F is a surjection. Let (S, g) ∈ ⟨Y ⟩_{𝒮} , (S, g) →^{ρ} y. Then (S, q_{i} ∘ g) →^{ρ_{i}} q_{i}(y) for each i ∈ I. Thus, for

each i ∈ I there exists (S, f_{i}) ∈ ⟨X_{i}⟩_{𝒮} such that (S, f_{i}) →^{π_{i}} x_{i}, F_{i} ∘ f_{i} = q_{i} ∘ g

and F_{i}(x_{i}) = q_{i}(y). Let f : S → X be the map with p_{i} ∘ f = f_{i} for each i ∈ I, and let x ∈ X be the element with p_{i}(x) = x_{i}. Then (S, f) →^{π} x and we have F ∘ f = g because q_{i} ∘ F ∘ f = F_{i} ∘ p_{i} ∘ f = F_{i} ∘ f_{i} = q_{i} ∘ g for each i ∈ I. Next, we have F (x) = y because q_{i} ∘ F (x) = F_{i} ∘ p_{i}(x) = F_{i}(x_{i}) = q_{i}(y) for each i ∈ I. Conversely, let there is (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{π} x, F ∘ f = g and F (x) = y. Then

(S, p_{i} ∘ f) →^{π_{i}} p_{i}(x) and, for each i ∈ I, we have F_{i} ∘ p_{i} ∘ f = q_{i} ∘ F ∘ f = q_{i} ∘ g

and F_{i}(p_{i}(x)) = q_{i}(F (x)) = q_{i}(y). Consequently, (S, q_{i} ∘ g) →^{ρ_{i}} q_{i}(y) for each i ∈ I, i.e. (S, g) →^{ρ} y. We have shown that (Y, ρ) is a quotient object of (X, π) with regard to F. Hence quotient maps are productive in Net_{𝒮} . Thus Net_{𝒮} is a strong topological universe. The proof is complete.

Conv_{𝒮} and Lim_{𝒮} are obviously closed under formation of initial objects in Net_{𝒮} and so they are topological. Consequently, Conv_{𝒮} is a bireflective subcategory of Net_{𝒮} and Lim_{𝒮} is a bireflective subcategory of Conv_{𝒮} . The bireflection of an object (X, π) ∈ Net_{𝒮} in Conv_{𝒮} is given by the identity map id_{X} : (X, π) → (X, π̂) where π̂ : ⟨X⟩_{𝒮} → expX is defined by (T, g) \underrightarrow{π̂} x iff there is (S, f) ∈ ⟨X⟩_{𝒮} with (S, f) →^{π} x such that (T, g) is a subnet of (S, f). The bireflection of an object (X, π) ∈ Conv_{𝒮} in Lim_{𝒮} is given by the identity map id_{X} : (X, π) → (X, π̂) where π̂ : ⟨X⟩_{𝒮} → expX is defined by (T, g) \underrightarrow{π̂} x iff for each subnet (S, f) of (T, g) there is a subnet (U, h) of (S, f) with (U, h) →^{π} x.

2.2. Theorem. Conv_{𝒮} is a bicoreflective subcategory of Net_{𝒮} .

Proof. Let (X, π) ∈ Net_{𝒮} and let π^{∗} : ⟨X⟩_{𝒮} → expX be the map given by (T, g) →^{π^{∗}} x iff, whenever (S, f) is a subnet of (T, g), (S, f) →^{π} x. Clearly, (X, π^{∗}) ∈ Conv_{𝒮} and id_{X} : (X, π^{∗}) → (X, π) is a bimorphism. Let (Y, ρ) ∈ Conv_{𝒮} be an object and F : (Y, ρ) → (X, π) a continuous map. Let (T, g) ∈ ⟨Y ⟩_{𝒮} , (T, g) →^{ρ} y. Then each subnet (U, h) of (T, g) has the property that (U, h) →^{ρ} y. Let (S, f) be a subnet of (T, F ∘ g). Then there is an 𝒮 -morphism φ : S → T with f = F ∘ g ∘ φ. Since (S, g ∘ φ) is a subnet of (T, g), we have (S, g ∘ φ) →^{ρ} y. Consequently, (S, f) =

(S, F ∘ g ∘ φ) →^{π} F (y). Hence (T, F ∘ g) →^{π^{∗}} F (y), i.e. F : (Y, ρ) → (X, π^{∗}) is

continuous. Hence id_{X} : (X, π^{∗}) → (X, π) is a bicoreflection of (X, π) in Conv_{𝒮} and the proof is complete.

2.3. Corollary. Conv_{𝒮} is a cartesian closed category.

2.4. Theorem. If 𝒮 has finite cartesian products, then Lim_{𝒮} is cartesian closed.

Proof. Let (X, π), (Y, ρ) ∈ Lim_{𝒮} . Let Z = C((X, π), (Y, ρ)) and let σ : ⟨Z⟩_{𝒮} → expZ be the map defined by (T, g) →^{σ} z iff from (S, f) →^{π} x it follows that (S × T, g^{f}) →^{ρ} z(x) where g^{f} : T × S → Y is the map given by g^{f}(s, t) = g(t)(f(s)). Let z ∈ Z and let (T, g) ∈ ⟨Z⟩_{𝒮} where g is the z-constant map. Let (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{π} x. Then g^{f}(s, t) = g(t)(f(s)) = z(f(s)). Hence g^{f} = z ∘ f ∘ pr_{S}. As z is continuous, (S, z ∘ f) →^{π} z(x). Therefore (S × T, g^{f}) →^{ρ} z(x), i.e. (T, g) →^{σ} z. Let (T, g) ∈ ⟨Z⟩_{𝒮} , (T, g) →^{σ} z, and let (S, f) →^{π} x. Let (U, h) be a subnet of (T, g). Then there is an 𝒮 -morphism φ : U → T such that h = g∘φ. We have (S×T, g^{f}) →^{ρ} z(x). As the map ψ : S ×U → S ×T given by ψ(s, u) = (s, φ(u)) is obviously an 𝒮 - morphism, (S ×U, g^{f} ∘ψ) is a subnet of (S ×T, g^{f}) and thus (S ×U, g^{f} ∘ψ) →^{σ} z(x). But g^{f}(ψ(s, u)) = h(s, φ(u)) = g(φ(u))(f(s)) = p(u)(f(s)) = h^{f}(s, u) for any (s, u) ∈ S × U and consequently (U, h) →^{σ} z. Let (T, g) ∈ ⟨Z⟩_{𝒮} , (T, g) ↛^{σ} z. Then there exists (S, f) ∈ ⟨X⟩_{𝒮} such that (S, f) →^{π} x but (S × T, g^{f}) ↛^{ρ} z(x). Thus, there exists a subnet (U, h) of (S × T, g^{f}) such that (V, p) ↛^{ρ} z(x) for any subnet (V, p) of (U, h). Let φ : U → S × T be the 𝒮 -morphism with h = g^{f} ∘ φ. As pr_{T} ∘ φ : U → T is an 𝒮 -morphism, (U, g ∘ pr_{T} ∘ φ) is a subnet of (T, g). Let (W, r) be a subnet of (U, g ∘ pr_{T} ∘ φ). Then there is an 𝒮 -morphism ψ : W → U such that r = g ∘ pr_{T} ∘ φ ∘ ψ. Let χ : W → S × W be the map given by χ(w) = (pr_{S}(φ(ψ(w))), w). Then χ is an 𝒮 -morphism and we have g^{f}(φ(ψ(w))) = g^{f}(pr_{S}(φ(ψ(w))), pr_{T} (φ(ψ(w)))) = g(pr_{T} (φ(ψ(w))))(f(pr_{S}(φ(ψ(w))))) = r(w)(f(pr_{S}(χ(w)))) = r^{f}(pr_{S}(χ(w)), w) = r^{f}(χ(w)) for every w ∈ W. Hence (W, g^{f} ∘ φ ∘ ψ) is a subnet of (S × W, r^{f}). Since (W, g^{f} ∘φ∘ψ) is a subnet of (U, h), we have (W, g^{f} ∘φ∘ψ) ↛^{ρ} z(x) and consequently (S×W, r^{f}) ↛^{ρ} z(x). Hence (W, r) ↛^{σ} z which means that no subnet of (U, g∘pr_{T} ∘φ) converges to z. We have shown that (Z, σ) ∈ Lim_{𝒮} . Let e : (X, π) × (Z, σ) → (Y, ρ) be the evaluation map. Denote (V, τ) = (X, π) ×

(Z, σ) and let (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{τ} (x, z). Then (S, pr_{X} ∘ f) →^{π} x and

(S, pr_{Z}∘f) →^{σ} z. Hence (S×S, h^{g}) →^{ρ} z(x) where G = pr_{x}∘f and h = pr_{Z}∘f. Let φ :

S → S ×S be the map given by φ(s) = (s, s). Then φ is an 𝒮 -morphism and consequently (S, h^{g} ∘φ) is a subnet of (S ×S, h^{g}). Thus (S, h^{g} ∘φ) →^{ρ} z(x). For any s ∈ S we have h^{g}(φ(s)) = h^{g}(s, s) = pr_{Z}(f(s))(pr_{X}(f(s))) = e(pr_{X}(f(s)), pr_{Z}(f(s))) = e(f(s)). Therefore (S, e ∘ f) →^{ρ} e(x, z) and we have shown that e is continuous. Let (W, υ) ∈ Lim_{𝒮} and let F : (X, π) × (W, υ) → (Y, ρ) be a continuous map. Let F^{∗} : W → Z be the map given by F^{∗}(w)(x) = F (x, w). Let (S, f) ∈ ⟨S⟩_{𝒮} , (S, f) →^{π}

x, and (T, g) ∈ ⟨W ⟩_{𝒮} , (T, g) →^{υ} w. Then the 𝒮 -net (S × T, (f ∘ pr_{S}, g ∘ pr_{T})) =

(S × T, f × g) converges to (x, w) in (X, π) × (W, υ) and thus (S × T, F ∘ (f × g)) →^{ρ} F (x, w). For any (s, t) ∈ S × T we have F ((f × g)(s, t)) = F (f(s), g(t)) = F^{∗}(g(t))(f(s)) = (F^{∗} ∘ g)^{f}(s, t). Hence (S × T, (F^{∗} ∘ g)^{f}) →^{ρ} F^{∗}(w)(x) which results in (S, F^{∗} ∘ g) →^{σ} F^{∗}(w). Thus, F^{∗} : (W, υ) → (Z, σ) is a continuous map. As the equality e ∘ (id_{X} × F^{∗}) = F is clearly valid, we have shown that (Z, σ) is the function space of (X, π) and (Y, σ) in Lim_{𝒮} . The proof is complete.

2.5. Example. The cartesian closedness of Lim_{𝒮} for 𝒮 = Dir, which is an immediate consequence of Theorem 2.4, is also proved in [5].

3. Diagonal 𝒮 -net spaces

In the previous paragraph it has been shown that Conv_{𝒮} is a cartesian closed topological subcategory of the topological preuniverse Net_{𝒮} but the function spaces in Conv_{𝒮} are not inherited from Net_{𝒮} . The same is valid also for Lim_{𝒮} provided that 𝒮 has finite cartesian products. In this paragraph we will find such cartesian closed topological subcategories of Conv_{𝒮} and Lim_{𝒮} whose function spaces are inherited from Net_{𝒮} . These subcategories will be received as intersection of Conv_{𝒮} and Lim_{𝒮} with certain full subcategory of Net_{𝒮} . This subcategory itself will be topological and cartesian closed with function spaces inherited from Net_{𝒮} and, moreover, its function spaces will have the point-wise structure.

3.1. Definition. An 𝒮 -net space (X, π) is called diagonal provided that it satisfies the following condition (called the diagonality axiom): Let {(S, f_{i}); i ∈ S} and {(S, g_{j}); j ∈ S} be families of 𝒮 -nets in X such that f_{i}(j) = g_{j}(i) whenever i, j ∈ S and let (S, f_{i}) →^{π} x_{i} whenever i ∈ S and (S, g_{j}) →^{π} y_{j} whenever j ∈ S. Let f, g, h : S → X be the maps given by f(i) = x_{i}, g(j) = y_{j} and h(i) = f_{i}(i). Then from (S, f) →^{π} x and (S, g) →^{π} x it follows that (S, h) →^{π} x.

We denote by DNet_{𝒮} the full subcategory of Net_{𝒮} whose objects are precisely the diagonal 𝒮 -net spaces.

3.2. Examples.

1. If 𝒮 = \underline{ω}, then the diagonality of 𝒮 -net spaces coincides with the diagonality defined in [13].

2. Let 𝒮 = \underline{1}. Then DNet_{𝒮} is isomorphic with the category of quasi-discrete topological spaces [3] (i.e. topological spaces with completely additive closures).

3.3. Lemma. Let (X, π) ∈ Net_{𝒮} and (Y, σ) ∈ DNet_{𝒮} be objects, let Z = C((X, π), (Y, σ)), and let (Z, σ) be the function space of (X, π) and (Y, ρ) in Net_{𝒮} . Then the space (Z, σ) has point-wise structure, i.e., whenever (S, g) ∈ ⟨Z⟩_{𝒮} , (S, g) →^{σ} z iff (S, g^{x}) →^{ρ} z(x) for each x ∈ X where g^{x} : S → Y is the map given by g^{x}(i) = g(i)(x).

Proof. Recall that (S, g) →^{σ} z iff (S, f) →^{π} x implies (S, g^{f}) → z(x) where g^{f} : S → Y is the map given by g^{f}(i) = g(i)(f(i)) - see the proof of Theorem 2.1. Let τ : ⟨Z⟩_{𝒮} → expZ be the map given by (S, g) →^{τ} z iff (S, g^{x}) →^{ρ} z(x) for

each x ∈ X. Let (S, g) ∈ ⟨Z⟩_{𝒮} , (S, g) →^{τ} z, and let (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{π} x.

Then for each i ∈ S we have (S, g(i) ∘ f) →^{ρ} g(i)(x). Put u_{i} = g(i) ∘ f for each i ∈ S and v_{j} = g^{f(j)} for each j ∈ S. Then u_{i}(j) = g(i)(f(j)) = g^{f(j)}(i) = v_{j}(i) whenever i, j ∈ S. Obviously, (S, u_{i}) →^{ρ} g(i)(x) for each i ∈ S and (S, v_{j}) →^{ρ} z(f(j)) for each j ∈ S. Further, we have (S, g^{x}) →^{ρ} z(x) and (S, z ∘ f) →^{ρ} z(x). As g^{f}(i) = g(i)(f(i)) = u_{i}(i) for each i ∈ S, we have (S, g^{f}) →^{ρ} z(x) and consequently (S, g) →^{ρ} z. Thus the implication (s, g) →^{τ} z ⇒ (S, g) →^{σ} z is valid. The inverse implication is obvious: For any x ∈ X we have (S, f) →^{π} x whenever f : S → X is the x-constant map, thus from (S, g) →^{σ} z it follows that (S, g^{f}) →^{ρ} z(x) where g^{f}(i) = g(i)(f(i)) = g(i)(x) = g^{x}(i). Hence (S, g) →^{τ} z. Therefore σ = τ which proves the statement.

Let us denote DConv_{𝒮} = DNet_{𝒮} ∩ Conv_{𝒮} and DLim_{𝒮} = DNet_{𝒮} ∩ Lim_{𝒮} . Then we have

3.4. Theorem. DNet_{𝒮} , DConv_{𝒮} and DLim_{𝒮} are cartesian closed topological categories. Their function spaces are inherited from Net_{𝒮} and have point-wise structure.

Proof. Let F_{k} : X → (X_{k}, π_{k}), k ∈ K, be a source in DNet_{𝒮} and let (X, π) be its initial object in Net_{𝒮} , i.e. (S, f) →^{π} x iff (S, F_{k} ∘ f) →^{π_{k}} F_{k}(x) for each k ∈ K - see the proof of Theorem 2.1. Let {(S, f_{i}); i ∈ S} and {(S, g_{j}); j ∈ S} be families of 𝒮 -nets in X such that f_{i}(j) = g_{j}(i) whenever i, j ∈ S and let (S, f_{i}) →^{π} x_{i} whenever i ∈ S and (S, g_{j}) →^{π} y_{j} whenever j ∈ S. Further, let f, g, h : S → X be the maps given by f(i) = x_{i}, g(j) = y_{j} and h(i) = f_{i}(i), and suppose that (S, f) →^{π} x,

(S, g) →^{π} x. Then, for each k ∈ K, {(S, F_{k} ∘ f_{i}); i ∈ S} and {(S, F_{k} ∘ g_{j}), j ∈ S} are

families of 𝒮 -nets in X_{k} such that (F_{k} ∘ f_{i})(j) = (F_{k} ∘ g_{j})(i) whenever i, j ∈ S,

(S, F_{k} ∘ f_{i}) →^{π_{k}} F_{k}(x_{i}) whenever i ∈ S and (S, F_{k} ∘ g_{j}) →^{π_{k}} F_{k}(y_{j}) whenever j ∈ S.

Next, for each k ∈ K, the map F_{k} ∘ f : S → X_{k} fulfils (F_{k} ∘ f)(i) = F_{k}(x_{i}) for each i ∈ S, the map F_{k} ∘ g : S → X_{k} fulfils (F_{k} ∘ g)(j) = F_{k}(y_{j}) for each j ∈ S, and the map F_{k} ∘ h : S → X_{k} fulfils (F_{k} ∘ h)(i) = F_{k}(f_{i}(i)) for each i ∈ S. As

(S, F_{k} ∘ g) →^{π_{k}} F_{k}(x) and (S, F_{k} ∘ g) →^{π_{k}} F_{k}(x) for each k ∈ K, (S, F_{k} ∘ h) →^{π_{k}} F_{k}(x) for

each k ∈ K too. Consequently, (S, h) →^{π} x. We have shown that DNet_{𝒮} is closed in Net_{𝒮} under formation of initial objects. As Conv_{𝒮} and Lim_{𝒮} are closed in Net_{𝒮} under formation of initial objects, so are also DConv_{𝒮} and DLim_{𝒮} . Hence DNet_{𝒮} , DConv_{𝒮} and DLim_{𝒮} are topological. Let (X, π), (Y, ρ) ∈ DNet_{𝒮} be objects and let (Z, σ) be the function space of (X, π) and (Y, ρ) in Net_{𝒮} . By the Lemma 3.3 we have (S, p) →^{σ} z iff (S, p^{x}) →^{σ} z(x) for each x ∈ X where p^{x} : S → Y is the map defined by g^{x}(s) = g(s)(x). Let {(S, f_{i}); i ∈ S} and {(S, g_{j}); j ∈ S} be families of 𝒮 -nets in Z such that f_{i}(j) = g_{j}(i) whenever i, j ∈ S and let (S, f_{i}) →^{σ} y_{i} whenever i ∈ S and (S, g_{j}) →^{σ} z_{j} whenever j ∈ S. Let f, g, h : S → Z be the maps given by f(i) = y_{i}, g(j) = z_{j}, h(i) = f_{i}(i), and let

(S, f) →^{σ} z, (S, g) →^{σ} z. Then for any x ∈ X, {(S, f_{i}^{x}); i ∈ S} and {(S, g_{j}^{x}); j ∈ S}

are families of 𝒮 -nets in Y such that f_{i}^{x}(j) = f_{i}(j)(x) = g_{j}(i)(x) = g_{j}^{x}(i) whenever

i, j ∈ S, (S, f_{i}^{x}) →^{ρ} y_{i}(x) whenever i ∈ S and (S, g_{j}^{x}) →^{ρ} z_{j}(x) whenever j ∈ S.

Further, for any x ∈ X we have f^{x}(i) = y_{i}(x) whenever i ∈ S, g^{x}(j) = z_{j}(x) whenever j ∈ S, h^{x}(i) = f_{i}^{x}(i) whenever i ∈ S, (S, f^{x}) →^{ρ} z(x) and (S, g^{x}) →^{ρ} z(x). Therefore (S, h^{x}) →^{ρ} z(x) for each x ∈ X. This results in (S, h) →^{σ} z, i.e. (Z, σ) ∈ DNet_{𝒮} . Consequently, DNet_{𝒮} is cartesian closed, and its function spaces are inherited from Net_{𝒮} and have point-wise structure. Let (X, π), (Y, σ) ∈ DConv_{𝒮} be objects and let (Z, σ) be their function space in Net_{𝒮} . Let (S, p) →^{σ} z and let x ∈ X. Let (T, q) be a subnet of (S, p). Then there is an 𝒮 -morphism φ : T → S with q = p ∘ φ, and (T, p^{x} ∘ φ) is a subnet of (S, p^{x}). Therefore (T, p^{x} ∘ φ) →^{ρ} z(x). But for every t ∈ T we have p^{x}(φ(t)) = p(φ(t))(x) = q(t)(x) = q^{x}(t), i.e. p^{x} ∘ φ = q^{x}. Hence (T, q^{x}) →^{ρ} z(x). We have shown that (T, q) →^{σ} z. Consequently, (Z, σ) ∈ Conv_{𝒮} and regarding the previous part of the proof (Z, σ) ∈ DConv_{𝒮} . Hence DConv_{𝒮} is cartesian closed, and its function spaces are inherited from Net_{𝒮} and have point-wise structure. Finally, let (X, π), (Y, σ) ∈ DLim_{𝒮} be objects and let (Z, σ) be their function space in Net_{𝒮} . Let (S, f) ∈ ⟨Z⟩_{𝒮} has the property that each its subnet has a subnet converging to z (w.r.t. σ). Let x ∈ X and let (T, g) be a subnet of (S, f^{x}). Then there is an 𝒮 -morphism φ : T → S with g = f^{x} ∘ φ and we have (f^{x} ∘ φ)(t) = f(φ(t))(x) = (f ∘ φ)^{x}(t) for each t ∈ T, i.e. g = (f ∘ φ)^{x}. As (T, f ∘ φ) is a subnet of (S, f), there is a subnet (U, g) of (T, f ∘ φ) such that (U, g) →^{σ} z. Hence (U, g^{x}) →^{ρ} z(x). Let ψ : U → T be the 𝒮 -morphism with g = f ∘ φ ∘ ψ. As g^{x}(u) = g(u)(x) = f(φ(ψ(u)))(x) = (f ∘ φ)^{x}ψ(u) = g(ψ(u)) for each u ∈ U, (U, g^{x}) is a subnet of (T, g). Consequently, (S, f^{x}) →^{ρ} z(x) and hence (S, f) →^{σ} z. Thus, with regard to the previous part of the proof, (Z, σ) ∈ DLim_{𝒮} . Therefore DLim_{𝒮} is cartesian closed, and its function spaces are inherited from Net_{𝒮} and have point-wise structure. The statement is proved.

4. Relations of Net_{𝒮} and Lim_{𝒮} to some supercategories of Top

We denote by Pcl the category of preclosure spaces (i.e. pairs (X, u) where X is a set and u : expX → expX is a map fulfilling uϕ = ϕ, A ⊆ X ⇒ A ⊆ uA and A ⊆ B ⊆ X ⇒ uA ⊆ uB) and continuous maps (i.e. maps F : (X, u) → (Y, v) with f(uA) ⊆ vf(A) whenever A ⊆ X). Further, we denote by Clo and Wet the full subcategories of Pcl whose objects are precisely the closure spaces (i.e. the preclosure spaces (X, u) fulfilling u(A ∪ B) = uA ∪ uB whenever A, B ⊆ X) and the weakly topological spaces (i.e. the preclosure spaces (X, u) fulfilling uuA = uA whenever A ⊆ X) respectively. Finally, we denote by Top the category of topological spaces, i.e. Top = Clo ∩ Wet.

It is well known that for 𝒮 = Dir the category Top can be fully concretely embedded into Lim_{𝒮} - see [7]. It is also known that for 𝒮 = \underline{ω} the full subcategory of Lim_{𝒮} given by the limit 𝒮 -net spaces in which each sequence converges to at most one point can be fully concretely embedded into Top - see [4]. We will find some next subcategories of Net_{𝒮} and Lim_{𝒮} that can be, for special categories 𝒮, fully concretely embedded into Pcl, Clo, Wet or Top.

For any 𝒮 -net space (X, π) let u_{π} : expX → expX be the map given by A ⊆ X ⇒ u_{π}A = {x ∈ X; the exists (S, f) ∈ ⟨X⟩_{𝒮} such that f(S) ⊆ A and (S, f) →^{π} x}. The following assertion is evident:

4.1. Proposition. The assignation π ↦ u_{π} determines a concrete functor from Net_{𝒮} into P cl.

The concrete functor from 4.1, and also every its restriction, will be denoted by ℱ_{𝒮} .

4.2. Example. For 𝒮 = \underline{1} the functor ℱ_{𝒮} is a full concrete embedding of Net_{𝒮} into Pcl. Namely, in this case, ℱ_{𝒮} is an isomorphism of Net_{𝒮} onto the full subcategory of Clo whose objects are precisely the quasi-discrete closure spaces [3], i.e. the closure spaces with completely additive closures.

4.3. Definition. The category 𝒮 is called

(1) an additively hereditary category if for any 𝒮 -object S and any pair of sets T, U with S = T ∪ U there is a subobject of S whose underlying set is T or U,

(2) a fork category if for any pair of 𝒮 -objects T, U and any pair of elements t ∈ T, u ∈ U there exists an 𝒮 -object S such that both the t-constant map of S into T and the u-constant map of S into U are 𝒮 -morphisms.

4.4. Example.

1. If 𝒮 = Dir, then 𝒮 is additively hereditary.

2. If in 𝒮 all constant maps are 𝒮 -morphisms, then 𝒮 is a fork category.

3. If 𝒮 is initially complete, then 𝒮 is both additively hereditary and fork category.

4.5. Proposition. If 𝒮 is additively hereditary, then ℱ_{𝒮} is a concrete functor from Net_{𝒮} into Clo.

Proof. Let (X, π) ∈ Net_{𝒮} , A, B ⊆ X and x ∈ u_{π}(A ∪ B). Then there is (S, f) ∈ ⟨X⟩_{𝒮} with (S, f) →^{π} x such that f(S) ⊆ A∪B. Put C = f(S)∩A and D = f(S)∩B. Then f(S) = C ∪ D, i.e. S = f^{−1}(C) ∪ f^{−1}(D). Thus, there is a subobject T of S whose underlying set is f^{−1}(C) or f^{−1}(D). Without loss of generality we can suppose that f^{−1}(C) is the underlying set of T. Then (f^{−1}(C), f ↾ f^{−1}(C)) is a subnet of (S, f). Hence (f^{−1}, f ↾ f^{−1}(C)) →^{π} x and, since (f ↾ f^{−1}(C))(f^{−1}(C)) = C ⊆ A, we have x ∈ u_{π}A ⊆ u_{π}A ∪ u_{π}B. Therefore u_{π}(A ∪ B) ⊆ u_{π}A ∪ u_{π}B. As the converse inclusion is obvious, (X, u_{π}) ∈ Clo. Now the statement follows from 4.1.

4.6. Definition. We denote by Ñet_{𝒮} and L̃im_{𝒮} the full subcategories of Net_{𝒮} and Lim_{𝒮} , respectively, given by those their objects (X, π) for which the following condition is satisfied: If (S, f) →^{π} x and (T_{s}, g_{s}) →^{π} f(s) for each s ∈ S, then there is (U, h) ∈ ⟨X⟩_{𝒮} such

that (U, h) →^{π} x and h(U) ⊆ ⋃_{s∈S} g_{s}(T_{s}).

4.7. Proposition. ℱ_{𝒮} is a concrete functor from Ñet_{𝒮} into Wet.

Proof. Let (X, π) ∈ Ñet_{𝒮} , A ⊆ X and x ∈ u_{π}u_{π}A. Then there is a net (S, f) ∈ ⟨X⟩_{𝒮} with (S, f) →^{π} x such that f(S) ⊆ u_{π}A. Further, for each s ∈ S there is a net (T_{s}, g_{s}) ∈ ⟨X⟩_{𝒮} with (T_{s}, g_{s}) →^{π} f(s) such that g_{s}(T_{s}) ⊆ A. Thus, there is a net

(U, h) ∈ ⟨X⟩_{𝒮} with (U, h) →^{π} x and h(U) ⊆ ⋃_{s∈S} g_{s}(T_{s}) ⊆ A. Therefore x ∈ u_{π}A and

we have proved the inclusion u_{π}u_{π}A ⊆ u_{π}A. As the converse inclusion is obvious, (X, u_{π}) ∈ Wet and the statement follows from 4.1.

4.8. Corollary. If 𝒮 is additively hereditary, then ℱ_{𝒮} is a concrete functor from Ñet_{𝒮} into Top.

4.9. Lemma. Let 𝒮 be a fork category, X a set and (S, f), (T, g) ∈ ⟨X⟩_{𝒮} . If f(S) ⊆ g(T), then there exists a subnet of (S, f) which is also a subnet of (T, g).

Proof. Let x ∈ f(S) and let s ∈ S, t ∈ T be arbitrary elements with f(s) = g(t) = x. Then there is an 𝒮 -object U such that both the s-constant map φ : U → S and the t-constant map ψ : U → T are 𝒮 -morphisms. Let h : U → X be the x-constant map. Then clearly (U, h) is a subnet of both (S, f) and (T, g).

4.10. Lemma. Let 𝒮 be a fork category and (X, π) ∈ Lim_{𝒮} an object. Then for any (S, f) ∈ ⟨X⟩_{𝒮} and any x ∈ X we have (S, f) →^{π} x iff x ∈ u_{π}g(T) for each subnet (T, g) of (S, f).

Proof. Suppose that (S, f) ↛^{π} x. Then there is a subnet (T, g) of (S, f) such that for any subnet (U, g) of (T, g) there holds (U, h) ↛^{π} x. Let (V, p) ∈ ⟨X⟩_{𝒮} , p(V) ⊆ g(T). By Lemma 4.9, there exists a subnet (W, q) of (V, p) which is also a subnet of (T, g). But then (W, q) ↛^{π} x, hence (V, p) ↛^{π} x. Therefore x ∉ u_{π}g(T). We have proved that from x ∈ u_{π}g(T) for each subnet (T, g) of (S, f) it follows that x ∈ u_{π}g(T). As the converse implication is obvious, the proof is complete.

4.11. Theorem. If 𝒮 is a fork category, then ℱ_{𝒮} is a full concrete embedding of Lim_{𝒮} into Pcl.

Proof. Let (X, π), (X, ρ) ∈ Lim_{𝒮} be objects, π ≠ ρ. Without loss of generality we can suppose that there are (S, f) ∈ ⟨X⟩_{𝒮} and x ∈ X such that (S, f) →^{π} x but (S, f) ↛^{ρ} x. Thus, by Lemma 4.10, x ∈ u_{π}g(T) for each subnet (T, g) of (S, f) and there is a subnet (T_{0}, g_{0}) of (S, f) such that x ∉ u_{ρ}g_{0}(T_{0}). Hence u_{π} ≠ u_{ρ}, i.e. ℱ_{𝒮} is injective on objects. Let (X, π), (Y, ρ) ∈ Lim_{𝒮} and let F : (X, u_{π}) → (Y, u_{ρ}) be a continuous map. Let (S, f) →^{π} x. Then, by Lemma 4.9, for each subnet (T, g) of (S, f) we have x ∈ u_{π}g(T) and consequently F (x) ∈ u_{ρ}F (g(T)). But every subnet of (S, F ∘ f) has the form (T, F ∘ g) where (T, g) is a subnet of (S, f). Thus, by Lemma 4.10, (S, F ∘ f) →^{ρ} F (x). Therefore ℱ_{𝒮} is full. The assertion now results from 4.1.

4.12. Corollary. If 𝒮 is an additively hereditary and fork category, then ℱ_{𝒮} is a full concrete embedding of Lim_{𝒮} into Clo.

Proof. It follows from 4.5 and 4.11.

4.13. Corollary.

(a) If 𝒮 is a fork category, then ℱ_{𝒮} is a full concrete embedding of L̃im_{𝒮} into Wet.

(b) If 𝒮 is an additively hereditary and fork category, then ℱ_{𝒮} is a full concrete embedding of L̃im_{𝒮} into Top.

Proof. It is a consequence of 4.7, 4.8 and 4.11.

4.14. Example. Let 𝒮 = \underline{ω}. It is well known that the full subcategory of Lim_{𝒮} whose objects are precisely the ℒ^{∗}-spaces (see 1.5, example 3) is embeddable by ℱ_{𝒮} into Clo. The ℒ^{∗}-spaces that are objects of L̃im_{𝒮} are called 𝒮^{∗}-spaces (see [6]). It is known that ℱ_{𝒮} is a full concrete embedding of the category of 𝒮^{∗}-spaces into Top (more precisely, ℱ_{𝒮} is an isomorphism of the category of 𝒮^{∗}-spaces onto the category of those Fréchet topological spaces in which every sequence has at most one limit).

5. The condition of iterated limits

The condition introduced in 4.6 has several lacks. It is known that for 𝒮 = Dir the functor ℱ_{𝒮} : L̃im_{𝒮} → Top is surjective on objects but it is not an isomorphism. Next, Ñet_{𝒮} and L̃im_{𝒮} are evidently not closed under formation of finite products in Net_{𝒮} , and the same is valid for the full subcategories of Ñet_{𝒮} and L̃im_{𝒮} given by their diagonal objects. Consequently, these subcategories are not closed under formation of function spaces in Net_{𝒮} . We will therefore replace the condition from 4.6 with a more convenient condition, the condition of iterated limits, which is an extension of the known condition introduced by J. L. Kelley for the case 𝒮 = Dir (see [7]).

5.1. Definition. Let 𝒮 have cartesian products and let (X, π) be a limit 𝒮 -net space. We say that (X, π) fulfils the condition of iterated limits if the following is valid: Let S ∈ 𝒮 be an object and let (T_{s}, g_{s}) ∈ ⟨X⟩_{𝒮} for each s ∈ S. Let U = S × ∏_{s∈S} T_{s} and let h : U → X be the map given by h(s, t) = g_{s}(t(s)). Then

from (T_{s}, g_{s}) →^{π} x_{s} for each s ∈ S and (S, f) →^{π} x where f : S → X is given by f(s) = x_{s} it follows that (U, h) →^{π} x.

We denote by Cil_{𝒮} the full subcategory of Lim_{𝒮} whose objects are precisely the limit 𝒮 -net spaces fulfilling the condition of iterated limits. Clearly, Cil_{𝒮} is a full subcategory of L̃im_{𝒮} and thus ℱ_{𝒮} is a concrete functor from Cil_{𝒮} into Wet.

5.2. Definition. Let 𝒮 have cartesian products. Then 𝒮 is called fine if for each 𝒮 -object S there is a map i ↦ S^{(i)} of S into the class of all subobjects of S such that

(1) φ : S → T is an 𝒮 -morphism iff for any j ∈ T there exists i ∈ S such that φ(S^{(i)}) ⊆ T^{(j)},

(2) j ∈ S^{(i)} ⇒ S^{(j)} ⊆ S^{(i)},

(3) ( ∏_{k∈K} S_{k})^{(v)} ⊆ ∏_{k∈K} S_{k}^{(v(k))}.

5.3. Example.

1. The category Dir is fine. For any directed set S = (S, ≤) and any i ∈ S the directed subset S^{(i)} of S is the set S^{(i)} = {s ∈ S; s ≥ i} directed by ≤.

2. The category Set^{+} is fine. For any non-empty set S and any i ∈ S the subset S^{(i)} of S is given by S^{(i)} = S.

5.4. Lemma. Let 𝒮 be fine and let (X, π) ∈ Cil_{𝒮} be an object. Then for any (S, f) ∈ ⟨X⟩_{𝒮} and any x ∈ X we have (S, f) →^{π} x iff x ∈ u_{π}g(T) for each subnet (T, g) of (S, f).

Proof. Let x ∈ u_{π}g(T) for each subnet (T, g) of (S, f). Let (T, g) be a subnet of (S, f). Then, for each i ∈ T, (T^{(i)}, g ↾ T^{(i)}) is a subnet of (S, f) and hence x ∈ u_{π}g(T^{(i)}). Consequently, for each i ∈ T there is a net (U_{i}, h_{i}) ∈ ⟨X⟩_{𝒮} with

(U_{i}, h_{i}) →^{π} x and h(U_{i}) ⊆ g(T^{(i)}). Put W = T × ∏_{i∈T} U_{i} and let p : W → X be

the map given by p(i, v) = h_{i}(v(i)). Now, since (T, q) →^{π} x whenever q : T → X is the x-constant map, we have (W, p) →^{π} x. We will show that (W, p) is a subnet of (T, g). For each i ∈ T let φ_{i} : U_{i} → T^{(i)} be an arbitrary map with the property h_{i} = g ∘φ_{i}. Now we can define a map φ : W → T by putting φ(i, v) = φ_{i}(v(i)). Let j ∈ T be an arbitrary element. Then for any v ∈ ∏_{i∈T} U_{i} and any (k, w) ∈ W^{(j,v)}

we have (k, w) ∈ (T × ∏_{i∈T} U_{i})^{(j,v)} ⊆ T^{(j)} × ∏_{i∈T} U_{i}^{(v(i))}. Hence k ∈ T^{(j)} and w(i) ∈

U_{i}^{(v(i))} for each i ∈ T. Consequently, φ(k, w) = φ_{k}(w(k)) ∈ T^{(k)} ⊆ T^{(j)}. Thus φ(k, w) ∈ T^{(j)} and therefore φ : W → T is an 𝒮 -morphism. For any (i, v) ∈ W we have g(φ(i, v)) = g(φ_{i}(v(i))) = h_{i}(v(i)) = p(i, v), hence g ∘ φ = p. We have shown that (W, p) is a subnet of (T, g). But (T, g) has been chosen arbitrarily and so it follows that (S, f) →^{π} x. As the converse inclusion is obvious, the proof is complete.

5.5. Theorem. If 𝒮 is fine, then ℱ_{𝒮} is a full concrete embedding of Cil_{𝒮} into Wet.

Proof. Using Lemma 5.4 we can prove Theorem 5.5 in an analogous way as we proved Theorem 4.11 by using Lemma 4.10.

¿From 4.5 and 5.5 we immediately get

5.6. Corollary. If 𝒮 is fine and additively hereditary, then ℱ_{𝒮} is a full concrete embedding of Cil_{𝒮} into Top.

5.7. Example. It is well known that for 𝒮 = Dir the functor ℱ_{𝒮} is a concrete isomorphism of Cil_{𝒮} onto Top - see [7]. Thus, if 𝒮 is fine and additively hereditary, then Cil_{𝒮} can be embedded into CilDir.

If 𝒮 has cartesian products, then we denote by DCil_{𝒮} the full subcategory of Cil_{𝒮} determined by those its objects that are diagonal.

5.8. Theorem. Let 𝒮 have cartesian products. Then DCil_{𝒮} is a cartesian closed topological category with function spaces inherited from Net_{𝒮} .

Proof. Let F_{k} : X → (X_{k}, π_{k}), k ∈ K, be a source in DCil_{𝒮} and let (X, π) be its initial object in Net_{𝒮} , i.e. let (S, f) →^{π} x iff (S, F_{k} ∘f) →^{π_{k}} F_{k}(x) for each k ∈ K - see the proof of 2.1. Let S ∈ 𝒮 be an object, let (T_{s}, g_{s}) ∈ ⟨X⟩_{𝒮} , (T_{s}, g_{s}) →^{π} x_{s} for each s ∈ S, and let (S, f) →^{π} x where f : S → X is the map given by f(s) = x_{s}. Further, let U = S × ∏_{s∈S} T_{s} and let h : U → X be the map given by h(s, t) = g_{s}(t(s)). Then

for each k ∈ K we have (T_{s}, F_{k} ∘g_{s}) ∈ ⟨X_{k}⟩_{𝒮} , (T_{s}, F_{k} ∘g_{s}) →^{π_{k}} F_{k}(x_{s}) for each s ∈ S,

and (S, F_{k} ∘ f) →^{π_{k}} F_{k}(x) where F_{k} ∘ f(s) = F_{k}(x_{s}). Next, we have F_{k} ∘ h : U → X_{k}, (F_{k} ∘ h)(s, t) = (F_{k} ∘ g_{s})(t(s)). Therefore (U, F_{k} ∘ h) →^{π_{k}} F_{k}(x) and consequently (U, h) →^{π} x. Hence the initial object (X, π) fulfils the condition of iterated limits. Now, because DLim_{𝒮} is closed under formation of initial objects in Net_{𝒮} , so is also DCil_{𝒮} . Therefore DCil_{𝒮} is a topological category. Let (X, π), (Y, ρ) ∈ DCil_{𝒮} be objects and let (Z, σ) be their function space in Net_{𝒮} , i.e. let Z = C((X, π), (Y, ρ)) and let (T, g) →^{σ} z iff (V, g^{x}) →^{ρ} z(x) for each x ∈ X where g^{x} : V → Y is the map defined by g^{x}(s) = g(s)(x) - see 3.3. Let S ∈ 𝒮 be an object, let (T_{s}, g_{s}) ∈ ⟨Z⟩_{𝒮} , (T_{s}, g_{s}) →^{σ} z_{s} for each s ∈ S, and let (S, f) →^{π} z where f : S → Z is the map given by f(s) = z_{s}. Further, let U = S × ∏_{s∈S} T_{s}

and let h : U → Z be the map given by h(s, t) = g_{s}(t(s)). Then for each x ∈ X

we have (T_{s}, g_{s}^{x}) ∈ ⟨Y ⟩_{𝒮} , (T_{s}, g_{s}^{x}) →^{ρ} z_{s}(x) for each s ∈ S, and (S, f^{x}) →^{ρ} z(x)

where f^{x} : S → Y fulfils f^{x}(s) = f(s)(x) = z_{s}(x) for each s ∈ S. Next, we have h^{x} : U → Y and h^{x}(s, t) = h(s, t)(x) = g_{s}(t(s))(x) = g_{s}^{x}(t(s)). Therefore (U, h^{x}) →^{ρ} z(x) and consequently (U, h) →^{σ} z. We have shown that the function space (Z, σ) fulfils the condition of iterated limits. Because DLim_{𝒮} has function spaces inherited from Net_{𝒮} , so has DCil_{𝒮} .

5.9. Example. If 𝒮 = Dir, then from 5.6 and 5.8 it follows that ℱ_{𝒮} (DCil_{𝒮} ) is a cartesian closed topological subcategory of Top. But this subcategory is precisely the category of indiscrete topological spaces.

5.10. Remark. Obviously, the condition of iterated limits can be considered also for 𝒮 -net spaces or convergence 𝒮 -net spaces, respectively. Then, by 3.4 and the proof of 4.8, the full subcategory of DNet_{𝒮} or DConv_{𝒮} , respectively, given by those its objects that fulfil the condition of iterated limits is a cartesian closed topological category with function spaces inherited from Net_{𝒮} .

6. 𝒮 -net spaces and separation axioms

In what follows we will deal with the question under which conditions on an 𝒮 - net space (X, π) the preclosure space (X, u_{π}) fulfils the lower separation axioms R_{0}, T_{0} or T_{1}, respectively. For preclosure spaces these axioms are defined analogically as for topological spaces. It means that a preclosure space (X, u) is said to be an R_{0}-space if x ∈ u{y} ⇒ y ∈ u{x} whenever x, y ∈ X, a T_{0}-space if x ∈ u{y}, y ∈ u{x} ⇒ x = y whenever x, y ∈ X, a T_{1}-space if u{x} = {x} whenever x ∈ X. In [8] R_{0}-spaces are called B^{∗}-spaces, and in [2] T_{0}-spaces and T_{1}-spaces are called K-spaces and B-spaces, respectively. We denote by R_{0}-Pcl, T_{0}-Pcl and T_{1}-Pcl the full subcategories of Pcl whose objects are precisely the R_{0}-spaces, T_{0}-spaces and T_{1}-spaces, respectively. Clearly, a preclosure space is a T_{1}-space iff it is both R_{0}-space and T_{0}-space, i.e. T_{1}-Pcl = R_{0}-Pcl ∩ T_{0}-Pcl.

The following assertion is obvious:

6.1. Proposition. If (X, π) is an 𝒮 -net space, then

(1) (X, u_{π}) ∈ R_{0}-Pcl iff for any pair x, y ∈ X and any (S, f) ∈ ⟨X⟩_{𝒮} such that f is the x-constant map and (S, f) →^{π} y there exists (T, g) ∈ ⟨X⟩_{𝒮} such that g is the y-constant map and (T, g) →^{π} x,

(2) (X, u_{π}) ∈ T_{0}-Pcl iff for any x, y ∈ X and any pair (S, f), (T, g) ∈ ⟨X⟩_{𝒮} such that f is the x-constant map and g is the y-constant map from (S, f) →^{π} y and (T, g) →^{π} x it follows that x = y,

(3) (X, u_{π}) ∈ T_{1}-Pcl iff cardπ(S, f) = 1 whenever (S, f) ∈ ⟨X⟩_{𝒮} and f is a constant map.

The full subcategory of Net_{𝒮} whose objects are the 𝒮 -net spaces (X, π) fulfilling (X, u_{π}) ∈ R_{0}-Pcl is a bicoreflective subcategory of Net_{𝒮} but, in general, it is not closed under formation of finite products in Net_{𝒮} and hence is not closed under formation of function spaces in Net_{𝒮} . Neither the full subcategory of Net_{𝒮} whose objects are the 𝒮 -net spaces (X, π) fulfilling (X, u_{π}) ∈ T_{0}-Pcl, which is closed under formation of initial objects for monosources in Net_{𝒮} (and hence initially structured), is closed under formation of function spaces in Net_{𝒮} in general. In each of these two categories we will therefore find a full cartesian closed subcategory with function spaces inherited from Net_{𝒮} .

6.2. Definition. We denote by R_{0}-Net_{𝒮} the full subcategory of Net_{𝒮} whose objects are precisely the 𝒮 -net spaces (X, π) fulfilling the following condition: Let {(S, f_{i}); i ∈ S} be a family of 𝒮 -nets in X such that f_{i}(i) = x (x ∈ X an element) for each i ∈ S. Let (S, f_{i}) →^{π} x_{i} for each i ∈ I and let (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{π} x. For every j ∈ S let g_{j} : S → X be the map given by g_{j}(i) = f_{i}(j) and let g : S → X be the map given by g(i) = x_{i}. Then (S, g_{j}) →^{π} f(j) for each j ∈ S

and (S, g) →^{π} x.

R_{0}-Net_{𝒮} is clearly closed under formation of initial objects in Net_{𝒮} , i.e. R_{0}- Net_{𝒮} is a topological category.

As an immediate consequence of 4.1 and 6.2 we get

6.3. Corollary. ℱ_{𝒮} is a concrete functor from R_{0}-Net_{𝒮} into R_{0}-Pcl.

6.4. Lemma. Let (X, π) ∈ R_{0}-Net_{𝒮} , (S, f) ∈ ⟨X⟩_{𝒮} and (S, f) →^{π} x.

(1) If j, k ∈ S and f_{j,k} : S → X is the map given by f_{j,k}(i) = f(j) whenever

i ≠ j and f_{j,k}(j) = f(k), then (S, f_{j,k}) →^{π} f(j).

(2) If j ∈ S and g_{j} : S → X is the map given by g_{j}(i) = f(j) whenever i ≠ j

and g_{j}(j) = x, then (S, g_{j}) →^{π} f(j).

(3) If g : S → X is the x-constant map, then (S, g) →^{π} f(j) for each j ∈ S.

Proof. Let the assumption of (1) be fulfilled and for each i ∈ S let f_{i} : S → X be the map given by f_{i}(m) = f(j) whenever i ≠ j and f_{j}(m) = f(m). Then (S, f_{i}) ∈ ⟨X⟩_{𝒮} and f_{i}(i) = f(j) for each i ∈ S. Next, we have (S, f_{i}) →^{π} f(j) whenever i ≠ j and (S, f_{j}) →^{π} x. Let h : S → X be the f(j)-constant map. Then (S, h) →^{π} f(j). Finally, let g_{k} : S → X be the map defined by g_{k}(i) = f_{i}(k). Then, by the definition of R_{0}-Net_{𝒮} , (S, g_{k}) →^{π} h(k) = f(j). But it is evident that g_{k} = f_{j,k}. Let the assumption of (2) be fulfilled and let f_{i}, i ∈ I, and h be defined in the same way as in the previous part of the proof. Let g : S → X be the map given by g(i) = f(j) whenever i ≠ j and g(j) = x. Then, by the definition of R_{0}-Net_{𝒮} , (S, g) →^{π} f(j). But it is evident that g = g_{j}. Finally, let the assumption of (3) be fulfilled and for each i ∈ S let f_{i} : S → X be the x-constant map. Then, for each i ∈ S, f_{i}(i) = x and (S, f_{i}) →^{π} x. Let j ∈ S and let g_{j} : S → X be the x-constant map. Then g_{j}(i) = f_{i}(j) for each i ∈ S and, because (S, f) →^{π} x, by the definition of R_{0}-Net_{𝒮} we get (S, g_{j}) →^{π} f(j). But g_{j} = g. The proof is complete.

6.5. Theorem. R_{0}-Net_{𝒮} is a cartesian closed topological category with function spaces inherited from Net_{𝒮} .

Proof. Clearly, R_{0}-Net_{𝒮} is closed under formation of initial objects in Net_{𝒮} . Let (X, π), (Y, ρ) ∈ R_{0}-Net_{𝒮} and let (Z, σ) be the function space of (X, π) and (Y, ρ) in Net_{𝒮} . Let {(S, h_{i}); i ∈ S} be a family of 𝒮 -nets in Z such that h_{i}(i) = z (z ∈ Z) for each i ∈ I. Let (S, h_{i}) →^{σ} z_{i} for each i ∈ I and let (S, h) ∈ ⟨Z⟩_{𝒮} , (S, h) →^{σ} z. Let j ∈ S and let q_{j} : S → Z be the map given by q_{j}(i) = h_{i}(j). Let (S, f) ∈ ⟨X⟩_{𝒮} , (S, f) →^{π} x. For any k ∈ S let f_{j,k} : S → X be the map defined in Lemma 6.4. Then (S, f_{j,k}) →^{π} f(j) and, obviously, f_{j,k}(k) = f(j) for each k ∈ K. Let g_{j} : S → X be the map defined in Lemma 6.4. Then(S, g_{j}) →^{π} f(j). Thus, for each i ∈ S we have (S, h_{i}^{p}) →^{ρ} z_{i}(f(j)) where p = f_{j,i}, and h_{i}^{p}(i) = z(f(j)) for each i ∈ S. Next, we have (S, h^{g_{j}}) →^{ρ} z(f(j)). Consequently, if r : S → Y is the map given by r(i) = h_{i}^{p}(j), then (S, r) →^{ρ} h^{g_{j}}(j). But r(i) = h_{i}^{p}(j) = h_{i}(j)(p(j)) = h_{i}(j)(f_{j,i}(j)) = h_{i}(j)(f(i)) = q_{j}(i)(f(i)) = q_{j}^{f}(i) for each i ∈ S and h^{g_{j}}(j) = h(j)(g_{j}(j)) = h(j)(x). Therefore (S, q_{j}) →^{σ} h(j). Now, let q : S → Z be the map given by q(i) = z_{i}. For any i ∈ S let f_{i} : S → X be the x-constant map. Then f_{i}(i) = x and by Lemma 6.4 we have (S, f_{i}) →^{π} f(i). Consequently, (S, h_{i}^{f_{i}}) →^{ρ} z_{i}(f(i)) and h_{i}^{f_{i}}(i) = z(x). Next, we also have (S, h^{f}) →^{ρ} z(x). Thus, if g : S → Y is the map given by g(i) = z_{i}(f(i)), then (S, g) →^{ρ} z(x). But g(i) = z_{i}(f(i)) = q(i)(f(i)) = q^{f}(i) and therefore (S, q) →^{σ} z. We have proved that (Z, σ) ∈ R_{0}-Net_{𝒮} . This completes the proof.

6.6. Example. If 𝒮 = \underline{1}, then ℱ_{𝒮} is an isomorphism of R_{0}-Net_{𝒮} onto the category of quasi-discrete closure R_{0}-spaces. In [3] closure R_{0}-spaces are called semi-uniformizable spaces.

6.7. Definition. We denote by T_{0}-Net_{𝒮} the full subcategory of Net_{𝒮} whose objects are precisely the 𝒮 -net spaces (X, π) fulfilling the following condition: Let {(S, f_{i}); i ∈ S} and {(S, g_{j}); j ∈ S} be families of 𝒮 -nets in X such that f_{i}(j) = g_{j}(i) for each i, j ∈ S and f_{i}(i) = x (x ∈ X an element) for each i ∈ S. Let

(S, f_{i}) →^{π} x_{i} for each i ∈ S and (S, g_{j}) →^{π} y_{j} for each j ∈ S. Let f, g : S → X be

the maps given by f(i) = x_{i}, g(j) = y_{j}, and let (S, f) →^{π} x and (S, g) →^{π} x. Then f_{i} is x-constant for each i ∈ S and also both f and g are x-constant.

Clearly, T_{0}-Net_{𝒮} is closed under formation of initial objects for monosources in Net_{𝒮} , hence T_{0}-Net_{𝒮} is initially structured. The following statement is an immediate consequence of 4.1 and 6.7:

6.8. Corollary. ℱ_{𝒮} is a concrete functor from T_{0}-Net_{𝒮} into T_{0}-Pcl.

6.9 Theorem. T_{0}-Net_{𝒮} is a cartesian closed initially structured category with function spaces inherited from Net_{𝒮} .

Proof. Clearly, T_{0}-Net_{𝒮} is closed under formation of initial objects for monosources in Net_{𝒮} . Let (X, π), (Y, ρ) ∈ T_{0}-Net_{𝒮} and let (Z, σ) be the function space of (X, π) and (Y, ρ) in Net_{𝒮} . Let {(S, p_{i}); i ∈ S} and {(S, q_{j}); j ∈ S} be families of 𝒮 -nets in Z such that p_{i}(j) = q_{j}(i) for each i, j ∈ S and p_{i}(i) = z for each i ∈ S. Let (S, p_{i}) →^{σ} z_{i} for each i ∈ S and (S, q_{j}) →^{σ} t_{j} for each j ∈ S. Let p, q : S → Z be the maps given by p(i) = z_{i}, q(j) = t_{j} and let (S, p) → z, (S, q) → z. Let x ∈ X be an element and f : S → X the x-constant map. Then p_{i}^{f}(j) = p_{i}(j)(f(j)) = q_{j}(i)(f(i)) = q_{j}^{f}(i) for each i, j ∈ S and p_{i}^{f}(i) = z(x) for each i ∈ S. Next, as (S, f) →^{π} x, we have (S, p_{i}^{f}) →^{ρ} z_{i}(x) and (S, q_{j}^{f}) →^{ρ} t_{j}(x). Let g, h : S → Y be the maps given by g(i) = z_{i}(x) and h(j) = t_{j}(x). Then (S, g) →^{ρ} z(x) and (S, h) →^{ρ} z(x) because g(i) = z_{i}(x) = p(i)(f(i)) = p^{f}(i) for each i ∈ S and h(j) = t_{j}(x) = q(j)(f(j)) = q^{f}(j) for each j ∈ S. Consequently, p_{i}^{f} is the z(x)-constant map and both g and h are the z(x)-constant maps. Thus, for each i ∈ S we have p_{i}(i)(x) = p_{i}^{f}(i) = z(x), hence p_{i} is the z-constant map. Similarly, we have p(i)(x) = z_{i}(x) = g(i) = z(x) for each i ∈ S and q(j)(x) = t_{j}(x) = h(j) = z(x) for each j ∈ S. Hence also p and q are the z-constant maps. This completes the proof.

6.10. Example. Let 𝒮 = \underline{1}. Then ℱ_{𝒮} is an isomorphism of T_{0}-Net_{𝒮} onto the category of quasi-discrete closure T_{0}-spaces. In [3] closure T_{0}-spaces are called feebly semi-separated spaces (and closure T_{1}-spaces are called semi-separated spaces).

6.11. Remark. R_{0}-Net_{𝒮} ∩ T_{0}-Net_{𝒮} is precisely the category of discrete 𝒮 -net spaces.

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Department of Mathematics, Technical University of Brno, Technická 2, 616 69 Brno, Czech Republic E-mail: slapal@kinf.fme.vutbr.cz