ON SOME CATEGORIES OF SEQUENTIAL SPACES

Josef Šlapal

Abstract. The sequential spaces dealt with in the paper have arbitrary ordinals as domains of sequences and are subjected, besides the axiom of convergence of constant sequences, to only one other, quite special axiom. Categories of these sequential spaces are studied and the results are then applied to categories of associated closure spaces.

It is a usual way to define a topological structure on a set X by introducing a convergence on X, i.e. a map which assigns to any point x ∈ X a set of sequences (of points of X) converging to x and which is subjected to certain axioms. The topological structure on X is then given by a closure operation where the closure of a subset A ⊆ X is the set of all limits of sequences contained in A. The most known topological structures obtained by this way are the Fréchet S^{∗}-spaces - see e.g. [4]. Another kind of such topological structures is studied in this paper. But here, in contrast to the classical access, domains of sequences are considered to be arbitrary ordinals (and not only the least infinite ordinal). Also the axiom prescribed besides the usual axiom of convergence of constant sequences is quite special. We show that closure spaces induced by these sequential spaces have a useful categorical property: they form an exponential supercategory of the category of finitely generated topological spaces.

Let X be a set, α > 0 an ordinal, and π : X → exp(X^{α}) a map such that

(i) (x_{i}|i < α) ∈ π(x) whenever x_{i} = x for all ordinals i < α, (ii) (x_{i}|i < α) ∈ π(x) ⇔ for each ordinal j, 0 < j < α, there is a sequence (y_{i}^{j}|i < α) ∈ π(x_{j}) with {y_{i}^{j}; i < α} ⊆ {x_{i}; i < j}, and there is a sequence

(y_{i}|i < α) ∈ π(x) with {y_{i}; i < α} ⊆ {x_{i}; i < α}.

The pair (X, π) is called a sequential space of type α. Given a pair of sequential spaces of type α, (X, π) and (Y, ρ), by a homomorphism of (X, π) into (Y, ρ) we understand a map f : X → Y with (x_{i}|i < α) ∈ π(x) ⇒ (f(x_{i})|i < α) ∈ ρ(f(x)). We denote by Seq_{α} the (finitely productive) category of sequential spaces of type α with homomorphisms as morphisms. (All categories are considered to be concrete categories of structured sets and structure-compatible maps.)

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). 54A20.

Key words and phrases. Sequential space, diagonal sequential space, closure space, finitely generated topological space, exponential supercategory of a category.

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. .

By a closure space we mean a pair (X, u) where X is a set and u : expX → expX is a map fulfilling u∅ = ∅, A ⊆ X ⇒ A ⊆ uA, and A ⊆ B ⊆ X ⇒ uA ⊆ uB (such spaces are studied e.g. in [2]). The symbol Clo will denote the category of closure spaces and continuous maps (i.e. maps f : (X, u) → (Y, v) with f(uA) ⊆ vf(A) whenever A ⊆ X).

Let us assign to any sequential space (X, π) ∈ Seq_{α} the pair (X, u_{π}) where u_{π} : expX → expX is given as follows: A ⊆ X ⇒ u_{π}A = {x ∈ X; there is a sequence (x_{i}|i < α) ∈ π(x) such that x_{i} ∈ A for all i < α}.

Proposition 1. The assignation (X, π) ↦ (X, u_{π}) determines a full concrete embedding of Seq_{α} into Clo for any ordinal α > 0.

Proof. The proof is trivial when using the following, quite obvious lemma:

Lemma 1. Let (X, π) ∈ Seq_{α} (α > 0 an ordinal), x ∈ X and (x_{i}|i < α) ∈ X^{α}. Then (x_{i}|i < α) ∈ π(x) if and only if x ∈ u_{π}{x_{i}; i < α} and x_{j} ∈ u_{π}{x_{i}; i < j} for each ordinal j, 0 < j < α.

The full concrete embedding mentioned will be denoted by F_{α}. The objects of F_{α}(Seq_{α}) will be called the sequential closure spaces of type α. Clearly, objects of F_{1}(Seq_{1}) are precisely the quasi-discrete closure spaces [3], i.e. the closure spaces (X, u) fulfilling uA = ⋃_{x∈A} u{x} for each subset A ⊆ X.

Proposition 2. There exists a full concrete embedding of Seq_{α} into Seq_{β} whenever 0 < α < β.

Proof. Let (X, π) ∈ Seq_{α}, and let ρ : X → exp(X^{β}) be given by (x_{i}|i < β) ∈ ρ(x) ⇔ for each ordinal j, 0 < j < β, there is a sequence (y_{i}^{j}|j < α) ∈ π(x_{j}) with {y_{i}^{j}; i < α} ⊆ {x_{i}; i < j}, and there is a sequence (y_{i}|i < α) ∈ π(x) with {y_{i}; i < α} ⊆ {x_{i}; i < β}. For each j, 0 < j < β, let (z_{i}^{j}|i < β) ∈ X^{β} be the sequence defined by z_{i}^{j} = y_{i}^{j} for all i < α and z_{i}^{j} = y_{0}^{j} for all i with α ≤ i < β. Similarly, let (z_{i}|i < β) ∈ X^{β} be the sequence defined by z_{i} = y_{i} for all i < α and z_{i} = y_{0} for all i with α ≤ i < β. Then we have (z_{i}^{j}|i < β) ∈ ρ(x_{j}), {z_{i}^{j}; i < β} ⊆ {x_{i}; i < j} for each j, 0 < j < β, (z_{i}|i < β) ∈ ρ(x) and {z_{i}; i < β} ⊆ {x_{i}; i < β}. Conversely, let x ∈ X and (x_{i}|i < β) ∈ X^{β} be such that for each j, 0 < j < β, there is a sequence (z_{i}^{j}|i < β) ∈ ρ(x_{j}) with {z_{i}^{j}; i < β} ⊆ {x_{i}; i < j}, and there is a sequence (z_{i}|i < β) ∈ ρ(x) with {z_{i}; i < β} ⊆ {x_{i}; i < β}. Then for each j, 0 < j < β, there is a sequence (t_{i}^{j}|i < α) ∈ π(x_{j}) such that {t_{i}^{j}; i < α} ⊆ {z_{i}^{j}; i < β} and there is a sequence (t_{i}|i < α) ∈ π(x) such that {t_{i}; i < α} ⊆ {z_{i}; i < β}. Hence {t_{i}^{j}; i < α} ⊆ {x_{i}; i < j} for each j, 0 < j < β, and {t_{i}; i < α} ⊆ {x_{i}; i < β}. This results in (x_{i}|i < β) ∈ ρ(x). We have shown that ρ fulfils the axiom (ii). As the axiom (i) is fulfilled trivially, (X, ρ) ∈ Seq_{β}. The proof is complete.

Obviously, in the proof of Proposition 2 there holds the equality F_{α}(X, π) = F_{β}(X, ρ), which results in the following

Corollary 1. F_{α}(Seq_{α}) is a full subcategory of F_{β}(Seq_{β}) whenever 0 < α < β.

A sequential space (X, π) of type α will be called diagonal provided that the following condition is satisfied (cf. [8]): if ((x_{ij}|j < α)|i < α) ∈ (X^{α})^{α} is a sequence of sequences with (x_{ij}|j < α) ∈ π(y_{i}) for all i < α, (x_{ij}|i < α) ∈ π(z_{j}) for all j < α, (y_{i}|i < α) ∈ π(x) and (z_{j}|j < α) ∈ π(x), then (x_{ii}|i < α) ∈ π(x). The full subcategory of Seq_{α} given by the diagonal sequential spaces (of type α) will be denoted by S̃eq_{α}.

We denote by Fin the full subcategory of Clo given by the finitely generated topological spaces [5], i.e. closure spaces (X, u) satisfying uuA = uA = ⋃_{x∈A} u{x}

whenever A ⊆ X.

Definition (cf. [9]). If K is a category with finite products and L its full isomorphism closed subcategory, then K is said to be an exponential supercategory of L provided that for any pair of objects G ∈ L and H ∈ K there exists an object G^{H} ∈ L with |G^{H}| = Mor_{K}(H, G) such that the pair (G^{H}, e), where e : H × G^{H} → G is the evaluation map, is a couniversal map for G with respect to the functor H × − : K → K.

The object G^{H} from the definition is called a power of G and H and it is uniquely determined by G and H whenever K is transportable. Obviously, if K is an exponential supercategory of L, then the powers fulfil the first exponential law, i.e. (G^{H})^{M} ≃ G^{H×M} whenever G ∈ L and H, M ∈ K. If K is an exponential supercategory of itself, then K is cartesian closed (see [6]).

Lemma 2. Let α > 1 be an ordinal and (X, π) ∈ S̃eq_{α} an object. Let x ∈ X, (x_{i}|i < α) ∈ π(x), and let (y_{j}|j < α) ∈ X^{α} be the constant sequence with y_{j} = x_{0} for each j < α. Then (y_{j}|j < α) ∈ π(x) and (y_{j}|j < α) ∈ π(x_{i}) for each i < α.

Proof. Obviously, (y_{j}|j < α) ∈ π(x_{0}). Let i_{0} be an ordinal, 0 < i_{0} < α, and suppose that (y_{j}|j < α) ∈ π(x_{i}) for each j < i_{0}. Put x_{ij} = x_{0} for each i, j < α and let (z_{i}|i < α) be the sequence given by z_{i} = x_{i} for all i < i_{0} and z_{i} = x_{0} for all i with i_{0} ⩽ i < α. Then clearly (x_{ij}|j < α) ∈ π(z_{i}) for all i < α and (x_{ij}|i < α) ∈ π(z_{j}) for all j < α. But from (x_{i}|i < α) ∈ π(x) it also follows that (z_{i}|i < α) ∈ π(x_{i_{0}}). Hence (x_{ii}|i < α) ∈ π(x_{i_{0}}). Because (x_{ii}|i < α) = (y_{j}|j < α), according to the principle of transfinite induction we have (y_{j}|j < α) ∈ π(x_{i}) for all i < α. Consequently, (x_{ij}|j < α) ∈ π(x_{i}) for each i < α and (x_{ij}|i < α) ∈ π(x_{j}) for each j < α. As (x_{i}|i < α) ∈ π(x), we have (x_{ii}|i < α) = (y_{j}|j < α) ∈ π(x). The lemma is proved.

Theorem 1. Seq_{α} is an exponential supercategory of S̃eq_{α} for each ordinal α > 0.

Proof. Let G = (X, π) ∈ S̃eq_{α} and H = (Y, ρ) ∈ Seq_{α} be objects and put G^{H} = (Z, σ) where Z =Mor(H, G) and σ : Z →exp(Z^{α}) is given by (z_{i}|i < α) ∈ σ(z) iff (z_{i}(y)|i < α) ∈ π(z(y)) for each y ∈ Y. In virtue of the Lemma 2 it can easily be seen that G^{H} ∈ Seq_{α}. But G^{H} is evidently diagonal, so that G^{H} ∈ S̃eq_{α}. Let e : H × G^{H} → G be the evaluation map. Denote (W, τ) = H × G^{H} and let (w_{i}|i < α) ∈ τ(w). Then there are (y_{i}|i < α) ∈ ρ(y) and (z_{i}|i < α) ∈ σ(z) such that (y_{i}, z_{i}) = w_{i} for each i < α and (y, z) = w. Since z_{i} ∈Mor(H, G) for each i < α, we have (z_{i}(y_{j})|j < α) ∈ π(z_{i}(y)) for each i < α. But we have also (z_{i}(y_{j})|i < α) ∈ π(z(y_{j})) for each j < α. Further, (z_{i}(y)|i < α) ∈ π(z(y)) and (z(y_{j})|j < α) ∈ π(z(y)) is evidently true. Thus, because of the diagonality of G, (z_{i}(y_{i})|i < α) ∈ π(z(y)). But z_{i}(y_{i}) = e(y_{i}, z_{i}) = e(w_{i}) for each i < α and z(y) = e(w). Therefore e : H × G^{H} → G is a homomorphism. Now, if K = (V, ν) ∈ Seq_{α} and h : H × K → G is a homomorphism, then the map ĥ : K → Z given by ĥ(v)(y) = h(y, v) is clearly a unique homomorphism of K into G^{H} with the property h = e ∘ (id_{Y} × ĥ). Hence the pair (G^{H}, e) is a co-universal map for G with regard to the functor H × − : Seq_{α} → Seq_{α}. This completes the proof.

Lemma 3. Let α > 0 be an ordinal, (X, π) ∈ Seq_{α} an object, and let u_{π}u_{π}A = u_{π}A whenever A ⊆ X. Then (x_{i}|i < α) ∈ π(x) implies x ∈ u_{π}{x_{0}} and x_{i} ∈ u_{π}{x_{0}} for each ordinal i < α.

Proof. Clearly, x_{0} ∈ u_{π}{x_{0}} is valid. Let i_{0} < α be an ordinal and suppose that x_{i} ∈ u_{π}{x_{0}} is valid for each ordinal i < i_{0}. Then, by Lemma 1, x_{i_{0}} ∈ u_{π}{x_{i}; i < i_{0}} ⊆ u_{π}u_{π}{x_{0}} = u_{π}{x_{0}}. Thus, according to the principle of transfinite induction, x_{i} ∈ u_{π}{x_{0}} for each i < α. But, as x ∈ u_{π}{x_{i} : i < α}, we have also x ∈ u_{π}u_{π}{x_{0}} = u_{π}{x_{0}}.

Remark 1. From Lemma 3 it immediately follows that for any object (X, π) ∈ Seq_{α} the condition (X, u_{π}) ∈ Fin is equivalent to u_{π}u_{π}A = u_{π}A for each A ⊆ X.

Theorem 2. F_{α}(S̃eq_{α}) = Fin for each ordinal α > 0.

Proof. Let (X, u_{π}) ∈ Fin. Let ((x_{ij}|j < α)|i < α) ∈ (X^{α})^{α} be a sequence of sequences such that (x_{ij}|j < α) ∈ π(y_{i}) for all i < α, (x_{ij}|i < α) ∈ π(z_{j}) for all j < α, (y_{i}|i < α) ∈ π(x) and (z_{j}|j < α) ∈ π(x). Then, by Lemma 3, for each i < α we have y_{i} ∈ u_{π}{x_{i0}} and x_{ij} ∈ u_{π}{x_{i0}} whenever j < α, and for each j < α we have x_{ij} ∈ u_{π}{x_{0j}} whenever i < α. We also have x ∈ u_{π}{y_{0}}. Thus, x ∈ u_{π}{y_{0}} ⊆ u_{π}u_{π}{x_{00}} and x_{i_{0}i_{0}} ∈ u_{π}{x_{i_{0}0}} ⊆ u_{π}u_{π}{x_{00}} = u_{π}{x_{00}}. Consequently, x ∈ u_{π}{x_{ii}; i < α} and x_{i_{0}i_{0}} ∈ u_{π}{x_{ii}; i < i_{0}} for each i_{0}, 0 < i_{0} < α. Therefore, according to Lemma 1, (x_{ii}|i < α) ∈ π(x). Hence (X, π) ∈ S̃eq_{α}. Conversely, let (X, π) ∈ S̃eq_{α}. Let A ⊆ X and x ∈ u_{π}u_{π}A. Then there is a sequence (x_{i}|i < α) ∈ (u_{π}A)^{α} such that (x_{i}|i < α) ∈ π(x). By Lemma 2, x ∈ u_{π}{x_{0}}. As x_{0} ∈ u_{π}A, there exists a sequence (y_{i}|i < α) ∈ A^{α} such that (y_{i}|i < α) ∈ π(x_{0}). Therefore, applying again Lemma 2, ve have x_{0} ∈ u_{π}{y_{0}}. Consequently, x ∈ u_{π}u_{π}{y_{0}} = u_{π}{y_{0}} ⊆ u_{π}A. We have proved the inclusion u_{π}u_{π}A ⊆ u_{π}A. As the converse inclusion is trivially fulfilled, the equality u_{π}u_{π}A = u_{π}A is valid. Thus, by Remark 1, (X, u_{π}) ∈ Fin and the statement is proved.

Corollary 2. F_{α}(Seq_{α}) is an exponential supercategory of Fin for each ordinal α > 0.

Denote by Clo^{∗} the full subcategory of Clo whose objects are precisely the closure spaces (X, u) ∈ Clo with the property that for each A ⊆ X and each x ∈ uA there exists an ordinal α > 0 and a sequence (x_{i}|i < α) ∈ A^{α} such that x_{j} ∈ u{x_{i}; i < j} for each j, 0 < j < α, and x ∈ u{x_{i}; i < α}.

Corollary 3 (cf. [10]). Clo^{∗} is an exponential supercategory of Fin.

Proof. In the quasicategory of all categories and functors let us consider the diagram D whose objects are the categories Seq_{α}, α > 0, and whose arrows are the full concrete embeddings F_{β}^{−1} ∘ F_{α} : Seq_{α} → Seq_{β}, 1 < α ⩽ β. Then D is a directed diagram and Clo^{∗} is a direct limit of D. The corresponding natural sink is formed by the embeddings F_{α}, α > 0. Now Corollary 3 follows from Corollary 2.

Remark 2. a) The product of a pair of objects (X, u), (Y, v) in Clo^{∗} is the space (X ×Y, w), where D ⊆ X × Y ⇒ wD = ⋃{(x, y) ∈ X × Y; there is an ordinal α > 0 and a sequence ((x_{i}, y_{i})|i < α) ∈ D^{α} such that x_{j} ∈ u{x_{i}; i < j} and y_{j} ∈ v{y_{i}; i < j} whenever 0 < j < α, x ∈ u{x_{i}; i < α} and y ∈ v{y_{i}; i < α}. The power G^{H} of objects G = (X, u) ∈ Fin and H = (Y, v) ∈ Clo^{∗} is given pointwise, i.e. by G^{H} = (Mor(H, G), p) where p(C) = {g ∈ Mor(H, G); there exists f ∈ C such that g(y) ∈ u{f(y)} for each y ∈ Y } whenever C ⊆ Mor(H, G).

b) It is well known that Fin is an exponential supercategory of itself, i.e. a cartesian closed category. The Corollary can be considered as a generalization of this fact.

c) If a space (X, u) ∈ Clo^{∗} is idempotent, i.e. fulfils A ⊆ X ⇒ uuA = uA, then (X, u) ∈ Fin (see Remark 1). On the other hand, the full subcategory of Clo^{∗} given by the additive closure spaces, i.e. closure spaces (X, u) ∈ Clo^{∗} with A, B ⊆ X ⇒ u(A ∪ B) = uA ∪ uB, is not finitely productive in Clo^{∗}. Each of these two facts means no new results could be obtained by restricting our considerations to topological spaces.

d) It is obvious that Clo^{∗} is a bicoreflective subcategory of Clo. For any object (X, u) ∈ Clo its bicoreflection in Clo^{∗} is the identity map id_{X} : (X, u^{∗}) → (X, u) where u^{∗} is given by A ⊆ X ⇒ u^{∗}A = ⋃{x ∈ X; there is an ordinal α > 0 and a sequence (x_{i}|i < α) ∈ A^{α} such that x_{j} ∈ u{x_{i}; i < j} for each j, 0 < j < α, and x ∈ u{x_{i}; i < α}}.

Examples. 1. Let (X, π) ∈ Seq_{2} be the sequential space with X = {a, b, c} and π(a) = {(a, a), (a, b)}, π(b) = {(b, b), (a, a), (a, b)}, π(c) = {(c, c), (a, b)}. Clearly, the closure space (X, u) with u = u_{π}, i.e. with (X, u) = F_{2}(X, π), is given by u∅ = ∅, u{a} = {a, b}, u{b} = {b}, u{c} = {c}, u{a, b} = u{a, c} = uX = X, u{b, c} = {b, c}. Obviously, (X, u) ∉ F_{1}(Seq_{1}).

2. Let (ω +1, u) be the closure space defined by u∅ = ∅, uA = ω whenever A ⊆ ω and 0 < cardA < ω, and uA = ω + 1 otherwise. Then (ω + 1, u) ∈ F_{ω}(Seq_{ω}) but (ω + 1, u) ∉ F_{α}(Seq_{α}) for any α < ω. Clearly, the closure space (X, u) is additive.

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Department of Mathematics, Technical University of Brno, Technická 2, 616 69 Brno, Czech Republic