Topological Relations

1. Introduction

In [5] it is said that familiarity with categorical techniques can help those who are confronted with a new field to detect analogies and connections to familiar fields, to organize the new field properly, and to separate the general concepts, problems and results from the specials ones which deserve special investigation. Thus, categorical knowledge can help us to direct and organize our thoughts.

Closure operators have had multiple uses and applications. These applications have used finite set models [3, 4, 6], endorsing a pretopological structure to the relationships obtained from modelling the interactions between the elements (relationships that represent nearness, academic influence, domination, etc.).

A categorical approach to the definitions of previously cited works helps to achive the goals stated at the beginning of this section. Therefore, we propose structures that generalize the closure operator used in those works.

Moreover, Blass [2] says that a useful methodological principle in modern mathematics is that, when a kind of mathematics structures are defined, the corresponding morphism between two of such structures should be defined. Therefore, not only new structures are defined but also the morphisms among them.

The constructs proposed in this work have been defined in such a way that adjunctions to classical topological structures (Gconv and some of its full subconstructs) are obtained. We expect that analogous topological concepts can be defined in the future.

First, we determine the nature of the spaces to work as mathematical structures in the sense of [1]. Then our mathematical structures are laid inside a construct in the sense of [1, 5, 7]. As our main concern is to work with generalized forms of pretopological spaces, we endowed our constructs with the properties needed to be topological in the sense of [7]. We proceed to verify the “correct” behavior of our structures settling adjunctions with familiar topological constructs.

2. Preliminaries

Following [7], a construct is a category whose objects are structured sets, i.e. pairs (X, ε) where X is a set and ε is -structure on X, whose morphisms f : (X, ε) → (Y, η) are suitable maps between X and Y and whose composition law is the usual composition of maps.

We call X the underlying set of (X, ε) and f : X → Y the underlying map of f : (X, ε) → (Y, η). As an abuse of notation, we say a map f : (X, ε) → (Y, η) to refer to the underlying map f. An example of this is the following definition:

If (X, ε) and (X, η) are -constructs, we say that η is coarser than ε (or ε is finer than η) if the identity map 1_X : (X, ε) → (X, η) is a -morphism.

Again, according to [7], a construct is called topological if it satisfies the following conditions:

1. (1) Existence of initial structures: For any set X, any family (X_i , ε_i) _i∈I of - objects indexed by a class I and any family (f_i : X → X _i) i∈I of maps indexed by I there exists a unique -structure ε on X which is initial with respect to X, f_i ,(X_i , ε_i , I ) ; i.e., such that for any -object (Y, η) a map g : (Y, η) → (X, ε) is a -morphism iff for every i ∈ I the composite map f_{i O} g : (Y, η) → (X_i , ε_i) is a -morphis

2. (2) For any set X, the class {(Y, η) ∈ Ob( ) : X = Y } of all -objects with underlying set X is a se

3. (3) For any set X with cardinality at most one, there exists exactly one -object with underlying set X.

We mainly work with the construct Gconv and some of its full subcategories. Recall that Gconv denotes the category of generalized convergence spaces (and continuous maps), that is:

• For each set X let F(X) be the set of all filters on X. Then a generalized convergence space is a pair (X, q) where X is a set and q ⊆ F(X) × X such that the following axioms are satisfied:

  1. (, x) ∈ q for each x ∈ X, where = {A ⊆ X : x ∈ A};

  2. (G, x) ∈ q whenever (F, x) ∈ q and G ⊇ F.

• A map f : (X, q) → (Y, p) between generalized convergence spaces is continuous provided that (f(F), f(x)) ∈ p for each (F, x) ∈ q.

Now, we recall some important subcategories of Gconv: A generalized convergence space (X, q) is called

• a Kent convergence space provided that the following condition is satisfied:

  • (F ∩ x, ) ∈ q whenever (F, x) ∈ q,

• a limit space provided that the following condition is satisfied:

  • (F ∩ G, x) ∈ q whenever (F, x) and (G, x) ∈ q

• a pretopological space provided that the following condition is satisfied:

  • ( _q(x), x) ∈ q for all x ∈ X, where _q(x) = T {F ∈ F (X) : (F, x) ∈ q}.

A pretopological space (X, q) is called a topological space provided that the following condition is satisfied:

• For each U ∈ _q(x) there is some V ∈ q(x) such that U ∈ q(y) for all y ∈ V .

The corresponding full subcategories of Gconv are denoted by Kent, Lim, Prtop and Top, respectively.

3. Topological Relations

In this section the concept of topological relation is defined. Also, interesting properties of these relations are proved. Constructs are built with them, we show that those constructs are topological.

Let X be a set and R ⊆ X × P (X). We say that R is a R₁-relation on X if:

(RT₀) ∀x ∈ X [(x, ∅) ∈/ R ] ,

(RT₁) ∀x ∈ X [(x, {x}) ∈ R ] .

If R is a R ₁-relation on X, we call the pair (X, R) a R ₁-space. A function f : (X, R) → (Y, Q) between R ₁-spaces is R ₁-continuous if:

• ∀(x, U ) ∈ R ∃V (f(x), V ) ∈ Q ∧ f[U] ⊆ V .

The class of R₁-spaces, and R₁-continuous functions forms a construct. This construct is denoted by1. Observe that the identity map 1_X : (X, R ) → (X, R ₁) is an ₁- morphism between the ₁-objects (X, R ) and (X, R ₁) if R ⊆ R ₁. We denote by _xR the set {U ⊆ X | (x, U) ∈ R }. The union of this set, _xR is denoted by _x ^R or just _x when it is clear from context what relation we are referring to.

The following association can be made:

We associate to each (X, R ) ∈ ₁ an object (X, _qR) ∈ Gconv as follows:

We associate each R₁-continuous f : (X, R) → (Y, Q) the Gconv-morphism f : (X, _qR) → (Y, _qQ), with the same underlying function.

Remark 3.1. The above association is a functor:

Given that x R {x} for all x, and x˙ = {x}↑, we have that ( ˙x, x) ∈ _qR for all x ∈ X.

If G ⊇ F and (F, x) ∈ _qR, then ∃U [x R U ∧ F ⊇ U↑]. Thus we obtain that (G, x) ∈ _qR since G ⊇ U↑.

Suppose f : (X, R ) → (Y, Q) is R ₁-continuous and (F, x) ∈ _qR. It follows that ∃U [x R U ∧ F ⊇ U↑]. This implies that U ∈ F and f[U] ∈ f(F); by the R ₁ continuity we have that, for some V , f(x)QV where f[U] ⊆ V . All the aforementioned implies that f(F) ⊇ f[U]↑ ⊇ V ↑, and also f(F), f(x) , which proves that f : (X, _qR) → (Y, _qQ) is continuous.

The remaining properties to verify that the association is a functor are easily derived from the construct structure. We denote the previous functor by T ₁.

The following association can also be made:

We associate to each (X, q) ∈ Gconv an R₁-space, (X, R_q) as follows

We associate each Gconv-morphism f : (X, q) → (Y, p) the ₁-morphism f : (X, R _q) → (Y, R _p), with the same underlying function.

Remark 3.2. The previous association is also a functor:

By construction we have (RT₀). Given that (, x) ∈ q for all x, in any Gconv space, we have that (RT₁), therefore (X, R _q) is an 1-object.

Suppose that f : (X, q) → (Y, p) is a Gconv-morphism and that x R_q U. Then ∃ (F, x) ∈ q such that U =T F. By continuity of f we have that f(F), f(x) ∈ p. Also, construction, f[U] ⊆ V , for all V ∈ f(F). Thus f(F) ⊇ f[U]. Observe that f[U]↑ ⊇ f(F), this implies that f[U]↑, f(x) . By construction f[U] = f[U]↑, which implies that f(x) R_p f[U], which makes f an ₁-morphism.

Equally, the remaining properties are easily derived from properties of the construct.

We will denote the previous functor by W₁.

The following properties will be used to define different constructs.

Definition 3.3. Let X be a set and R ⊆ X × P(X):

(RT₂) ∀x ∈ X [x R U ⇒ x R U ∪ {x}].

(RT₃) ∀x ∈ X [x R U ∧ x R V ⇒ x R (U ∪ V )].

(RT₄) ∀x ∈ X [x R x]

(RT₅) ∀x ∈ X [x R U ∧ y ∈ U ⇒y ⊆x].

With these properties it is possible to define _n constructs as the full subconstructs of ₁ such that their objects satisfy the RT_i properties with i ≤ 5

Remark 3.4. Let (X, R) ∈ 1. We define as (x, U) ∈ ⇔ ∃(x, V ) ∈ R [U ⊆ V ].

Then

1. T₁((X, R)) = T₁((X,));

2. if R = the

This means that if we are interested in studying _n constructs through the functors T ₁ and W ₁, then we may assume R =. Furthermore, we can replace (RT₀) by

We will now see that the constructs _n, with 1 ≤ n ≤ 5, are topological. First, we will show this for ₁ and then for the others.

And

Theorem 3.5. Let {(X_i , R_i)}_i∈I and {fi : X → (X_i , R_i)}_i∈I be a family of 1-spaces and maps, respectively. The structure R over X defined as

is an initial structure.

Proof. Because fi(x) R_i {fi(x)} for all x ∈ X and all i ∈ I, we obtain that x R {x}. Therefore we obtain RT₁; the verification of RT₀ is straightforward. The previous shows that (X, R) is an ₁ structure. Next we are going to verify the initiality.

Let g : (Y, Q) → (X, R) a map such that f_i o g : (Y, Q) → (X, R _i) is an ₁-morphism. We have to prove that g is an ₁-morphism. Let y Q V . We will prove that ∃U ⊆ X g(y) R U ∧ g[V ] ⊆ U . Because f_i o g is an ₁ morphism for each i ∈ I, then

By the construction of R, we have that g(y) R g[V ]. Now we will show that R is the coarsest structure that makes each f_i R₁-continuous. Suppose that R´ does it too. Let 1_X : (X, R ´ ) → (X, R ) be the identity map and (x, U) ∈ R ´ . We will prove that

but this is equivalent to fi : (X, R ´ ) → (X, R_i) being ₁-continuous, which is true by hypothesis.

To show that _i is also topological for 1 < i ≤ 5, it is enough to prove that the structure R defined in Theorem 3.5 is an _i-structure when {fi : X → (X _i , R _i )}_i∈I is an _i-source.

Theorem 3.6. The constructs _i when 1 < i ≤ 5 are topological.

Proof. 1) ₄ is topological. Indeed, let {f_i : X → (X _i , R _i)}_i∈I with {(X _i , R _i)}_i∈I ⊆ ₄. Let (X, R) be defined as in indeed, Theorem 3.5. We shall prove that (x, _xR) ∈ R. ∀U ∈ _x R ∃V_U f_i(x) R_i V_U ∧ fi [U] ⊆ V_U . Because each (X _i , R _i) is an ₄-object, then (f_i(x), f_i(x)R _i) ∈ R _i . It follows that

which concludes that x R _xR.

The proof that ₂ and ₃ are topological constructs is similar.

2) ₅ is topological. To see this, let {f_i : X → (X _i , R _i)}_i∈I with {(X _i , R _i)}_i∈I ⊆ ₅; (X, R) as defined in Theorem 3.5; (x, U) ∈ R and y ∈ U. We shall prove that _y ⊆ _x. Since (x, U ) ∈ R, we have that ∀i ∈ I ∃V_i f_i(x) R _i V _i ∧ f_i [U] ⊆ V_i . It follows that f_i(y) ∈ f_i [U] ⊆ V _i for each i and, since each (X _i , R _i) is ₅, we obtain _fi(y) ⊆ _fi(x) . Let z ∈ _y then ∃A [y R A ∧ z ∈ A]. Observe that z ∈ _x is equivalent to ∃U ´ x R U ´ ∧ z ∈ U ´ . By construction we have that ∀i ∈ I ∃W_i f_i(y) R_i W_i ∧ f_i [A] ⊆ W_i . From this it follows that fi(z) ∈ f_i [A] ⊆ WA ⊆ _fi(y) ⊆ _fi(x) . So, we obtain that x R {z}.

From an ₁-object, (X, R), we can construct _n-objects as follows:

By letting R _k = R ∪ R ^∗ , with R ^∗ = {(x, U ∪ {x}) | x R U }, we obtain an 2-object. Defining R _l = R ∪R ^∗ , with R ^∗ = {(x, ∪A) | A ⊆ xR ∧|A| < ℵ₀}, we obtain an ₃-object. And R_C = R ∪ R ^∗ , with R ^∗ = {(x, ∪A) | A ⊆ xR }, is an ₄-object.

We define recursively over ω the following sets for each x ∈ X: _xR⁰ = _xR, _xRⁿ⁺¹ = {V ∈ _yRⁿ | ∃U ∈ _xRⁿ [y ∈ U]} ∪ _xRⁿ, _xR^ω = S {_xRⁿ | n ∈ ω}. Hence, (X,(R^ω)_C ) is an ₅-object.

Theorem 3.7. ₄ is a reflective subcategory of ₁.

Proof. Let (X, R) be an ₁-object and let R_C be as previously defined. Let us see, in fact, that (X, R_C ) is an ₄-object. It is enough to show that _xR_C = _x R. One inclusion follows by definition. Let x ∈ _x R^C , if x ∈ U ∈ R we have finished. Suppose that x ∈ U ∈ R ^∗ . This implies that ∃V ∈ R [x ∈ V ∈ _xR], so x ∈ _xR, which shows that (X, R_C ) is an ₄-object.

We will see that the identity map 1_x : (X, R ) → (X, R_C ) is a morphism and serves as a reflector; for which we shall prove that the following diagram commutes if has f as an underlying map.

Let f : (X, R) → (Y, Q) and (x, U ) ∈ R_C . If U ∈ R, we have finished. Suppose that U ∈ R ^∗ ; then U = A. By the ₁-continuity of f, we have that ∀W ∈ A∃V_W f(x)QV_W ∧ f[W] ⊆ V_W ; all this implies that

The proofs that ₂ and ₃ are reflective are similar to that for ₄.

Theorem 3.8. Let (X, R ) ∈ ₁ and R^ω as previously defined. Then:

• (X, R^ω) satisfies RT₅,

• for any ₁-morphism f : (X, R) → (Y, Q) with (Y, Q) an ₅-object, the diagram

commutes if and f have the same underlying map.

Proof. a) Let (x, U) ∈ R^ω and y ∈ U. We shall prove that _y ⊆ _x. Let z ∈ _y = _yR^ω , then ∃V ⊆ X ∃n ∈ ω z ∈ V ∧ V ∈ _yRⁿ ; let be the least n ∈ ω which satisfies the property. Since U ∈ _xR^ω we have that ∃m ∈ ω [U ∈ _xR^ω ]; let be the least m ∈ ω which satisfies the property. Let l = max(,). With this we obtain that z ∈ V ∈ _xR^l+1 ⊆ _xR^ω ⊆ _x .

b) Let (x, U) ∈ R^ω . Let n ∈ ω the least natural number such that (x, U) ∈ R ⁿ . We shall prove by induction over n that, if 0 < n, there are N ∈ ω, {A_i}_i∈N+1 ⊆ P(X) and {xi}_i∈N+1 ⊆ X such that:

Base case, n = 1. By definition we have that

Letting N = 0, A₁ = V and x₀ = y we obtain what is required.

Inductive step. Let (x, U) ∈ R^M+1 (remark our hypothesis, M + 1 is the least natural number which satisfies that membership). By definition we have that ∃V ⊆ X ∃ y ∈ X (y, U) ∈ R^M ∧ y ∈ V ∧ (x, V ) ∈ R^M . Using the induction hypothesis with (y, U) ∈ R^M and (x, V ) ∈ R^M we obtain that there are N ∈ ω, {Ai}_i∈N+1 ⊆ P(X), {x_i}_i∈N+1 ⊆ X and, N ´ ∈ ω, {A´_i }_i∈N0+1 ⊆ P(X) y {x´_i }_{i∈N +1} ⊆ X, respectively. For j ∈ N + N ´ + 2 we define

This construction satisfies what is required: B₀ = A₀ = U, _zN+N0+1 = _{x ´ N´} = x and that _zN = y ∈ V = A´₀ = B_N+1.

Let M = N + N ´ + 1. By construction and the ₁-continuity of f we have that there is a family {V _j}_j∈M+1 ⊆ P (Y ) which satisfies that f(z_j ) Q V _j and f[B _j ] ⊆ V _j for each j ∈ M + 1. The previous and that (Y, Q) is an ₅-object implies that:

Since f(x)QV_M and from RT₄, we have that f(x)Q _f(x) , which ends the proof.

Remark 3.9. To prove that : (X,(R^ω)_C ) → (Y, Q) is continuous, the case when U = A with A ⊆ _xR^ω remains. In this case we proceed in the same way as Theorem 3.7, where the existence for each V_W is obtained in the same way as it done in Theorem 3.8.

To prove that (X,(R^ω)_C ) satisfies the property RT₅, observe that _x ^{R ω} = _x ^Rωc .

From the previous observation we can deduce the following:

Theorem 3.10. ₅ is reflexive in ₁.

Example 3.11. Let X = ω.

1. Let _nR = {{n}, {n + 1}, {n + 2}}. (X, R) is an ₁-space but not an ₂-space.

2. Let _nR = {{n}, {n, n + 1}, {n, n + 2}}. (X, R) is an ₂-space but not an ₃-space.

3. Let _nR = {[n, N] ⊆ ω | N ∈ ω ∧ n ≤ N}. (X, R) is an ₃-space but not an ₄-space.

4. For n ‡ 0, let _nR = {[n − 1, ω), {n}}. For n = 0, let ₀R = {{0}, ω}. (X, R ) is an ₄-space but not an ₅-space.

5. Let _nR = {[n, ω), {n}}. (X, R) is an ₅-space.

Remark 3.12. We said that these constructs generalize the closure operators generated by a reflexive poset. Indeed: If ≤ is a reflexive partial order over X, define an ₄-object, (X, R), as x R U ↔ U = {y ∈ X | x ≤ z}. Then, the canonical closure constructed from (X, ≤) coincides with the canonical closure constructed from T ₁((X, R)).

4. Adjunction

We will show that the functors T ₁ y W ₁ are adjoints; since the functors assign the same underlying map to the morphism, it will be enough to show that the unity and co-unity must have the identity map as their underlying maps.

Having proved that the identity maps works as unity and co-unity, we get that T₁(η(_X,R)) : (X, _qR ) → (X, q_RqR ) has an identity as their underlying map. Since the identity map ε(X,_qR ) : (X, _qRqR ) → (X, _qR ) is a morphism and by definition of the Gconv continuity we can conclude that qR = qRqR . Those structures being equal, ε(X,qR) and T₁(η(X,R)) are the identity morphism over (X, qR). This trivializes the triangular identity ε_T1 o T₁η = 1_T1 , since both maps from the left side are 1_T1 . Analogously this can be shown for W_1ε o η_W1 = 1_W1 .

Theorem 4.1. The identity map ε_q : (X, q_Rq ) → (X, q) is Gconv continuous.

Proof. Let (F, x) ∈ q_Rq . By construction we have the following

and ∃(G, x) ∈ q [U = G] . Then U↑ ⊇ G, which implies that (U↑, x) ∈ q. Since F ⊇ U↑ ⊇ G, we conclude (F, x) ∈ q.

Theorem 4.2. The identity map η_R : (X, R) → (X, R_qR ) is ₁-continuous.

Proof. Let x RU. By definition, (U↑, x) ∈ _qR. Since U = U↑, we conclude that x R_qR U.

Example 4.3. Let (X, R) be an ₁-object.

If (X, R) is an ₁-object such that X ∈ _x R, then (F, x) ∈ _qR for all F ∈ F (X). In this case we get the proper inclusion R R_qR .

Let X = (0, 1] ⊆ . Define V as the filter generated by the family {(0, δ] | 0 < δ ≤ 1} and let q(x) denote the set {F ∈ F (X) | (F, x) ∈ q}. Let (X, q) be such that q (1) has 1˙, V and all the superfilters of V as members. Then q_Rq (1) = {1˙}. In this case we get the proper inclusion q_Rq q.

The following observation will be useful to prove Theorem 4.5 below.

Remark 4.4. Let I be a non empty set, M = {Fi}i∈I ⊆ F(X) and N = { S {g(i)}i∈I | g ∈ Q i∈I Fi}. Therefore,

1. M = N.

2. If L = { F_i | i ∈ I} then M = L.

1) For M ⊆ N, if U ∈ M, it is enough to choose g such that g(i) = U for all i ∈ I. For N ⊆ M it is enough to see that for each ∈ I, g() ⊆ {g(i)}_i∈I .

2) For M ⊇ L, observe that M ⊆ F_i for each i ∈ I. For M ⊆ L, let us see M as N = { {g(i)}_i∈I | g ∈ _i∈I F_i}. Let z ∈ M. This means that for each g ∈ _i∈I F_i , there is an i such that z ∈ g(i). Suppose that z /∈ L. This implies that, for each i ∈ I, we can select an A_i ∈ F_i such that z /∈ A_i . If we define (i) as A_i we have that z /∈ { (i)}_i∈I , a contradiction.

Restricting the functor T₁ over _i and W ₁ over Kent,Lim, Prtop and Top (which will be denoted by T_n and W_n , respectively) we obtain the following:

Theorem 4.5. The following constructs are adjoints:

• ₂ and Kent,

• ₃ and Lim,

• ₄ and Prtop,

• ₅ and Top.

Proof. (a) T₂ is well defined: Let (X, R ) ∈ ₂, (F, x) ∈ _qR. By construction ∃U[x R U ∧ F ⊇ U↑]. By the (RT₂) property we have x R U ∪ {x}. Since F ∩ x˙ ⊇ (U ∪ {x})↑, we conclude that (F ∩ x, ) ∈ _qR.

W₂ is well defined: Let (X, q) ∈ Kent, x R_q U. By construction ∃(F, x) ∈ q [U = F ‡ ∅] . By the Kent property, we have that (F ∩ , x ) ∈ q. By Remark 4.4 (choosing M = {F,}) we have (F ∩) = U ∪ {x}, and, by the functor definition, we have that x R_q U ∪ {x}.

(b) T₃ is well defined: Similar to (a), it can be easily proven that if F,G ∈ _qR(x), we get that there are U, V ⊆ P (X) such that F ⊇ U↑,G ⊇ V ↑, x R U and x R V . Finally, by F ∩ G ⊇ (U ∪ V ) ↑ (Remark 4.4) we conclude that F ∩ G ∈ _qR(x).

W₃ is well defined: Again, using the technique from (a) and by Remark 4.4, we have that if x R_q U and xR_q V , then there are F, G ∈ q(x) such that F = U ‡ ∅ and G = V = ∅. But F ∩ G = U ∪ V , which implies that xR_qU ∪ V .

(c) T₄ is well defined: Let (X, R) ∈ ₄. By Remark 4.4 we have _F∈qR(x) F Э _x.

W₄ is well defined: Let (X, q) ∈ Prtop. By Remark 4.4 we have that _x = q(x) = { F_i | i ∈ I} = _x.

(d) T₅ is well defined: Let (X, R) ∈ ₅. Let U ∈ _x and take V as _x. If y ∈ V , then

From this and (RT₅), we have that _y ⊆ _x. Thus Ux ∈ _y and U ∈ _y. This shows that (X, _qR) is a topological space; observe that _x = _x↑ for all x, then (X, _qR ) is always an Alexandroff space.

W₅ is well defined: Let (X, q) ∈ Top, xR_qU and y ∈ U. By (c) it is known that _x = _x. y ∈ U, then there exists (F, x) ∈ q such that y ∈ F = U. Thus ∀V ∈ _x[y ∈ V ]. Let V ∈ _x. Since (X, q) ∈ Top, we have that ∃V ´ ∈ _x [∀z ∈ V ´ [V ∈ z]] . In particular V ∈ _y. Therefore _y ⊆ V . Since V was arbitrary, _y ⊆ _x = _x.

It remains to determine which properties the functor W₁ preserve, and how they will be stated in an ₁ construct.

Acknowledgments

The authors wish to thank the referees for their valuable observations and comments, which helped improve greatly this paper.

References

Author notes

eapmat@yandex.com

Additional information

To cite this article: E. Angulo-Perkins and J. Angoa-Amador, Topological Relations, Rev. Integr. temas mat. 39 (2021), No. 1, 79-89. doi: 10.18273/revint.v39n1-2021006

Revista Integración
ISSN: 0120-419X
Vol. 39
Num. 1
Año. 2021

Topological Relations

EmilioJuan Angulo-PerkinsAngoa-Amador
Benemérita Universidad Autónoma de PueblaBenemérita Universidad Autónoma de Puebla,MéxicoMéxico