ANALISIS DE RIESGOS LABORALES

implying that τ₁ is closed under arbitrary unions. It is easy to see that

∅, X ∈ τ₁ as ∅, X_α ∈ τ_r1. Hence τ₁ is a topology on X, corresponding to τ_r1. Similarly τ₂ is a topology on X corresponding to τ_r2. It is easy to see that both τ₁ and τ₂ are weaker than τ.

It is obvious thatτ₁ is weaker thanτ₂ and (by Lemma 4.1) that τ₁ is only one set less than τ₂. That is, τ₁ is a strong reduction of τ₂.

As τ_r1 and τ_r2 in C_Γ are arbitrary it follows that there corresponds to C_Γ a chain C of topologies on X of pair-wise comparable topologies which can be arranged in such a way that each one is strictly weaker than the next by only one set. If we let the elements of C to represent the (hypothetical) range space (X_α, τ_α)—one after the other—in the collection of sub-base for weak topologies on X while leaving the other range spaces unchanged, the required chain of weaker weak topologies on X will emerge. The proof is complete.

Theorem 4.2 indicates that a fixed family of functions can generate a family of pairwise comparable weak topologies. Further research may now embark on finding more considerations for this result. This is part of the developments in the sections ahead.

NOTE

So far, all the chains of strong reduction of topologies given in this section are countable. The question then arises as to whether there can be an un-countable chain of strong reductions of some topology. For example, can an uncountable chain of strong reductions be obtained for the ususal topology of R? Further, if a range topology for a weak topology has an uncountable chain of strong reductions, what is the implication of this on the weak topology?

That is, does the weak topology in this case inherit this property? Can we characterize the weak topologies for which there exist families of other weak topologies which are chains of strong reductions of the given weak topologies?

Answers to these questions are as yet unknown.

3. R is anti-symmetric; in that xRy and yRx implies x=y.

Definition 4.17 A set X on which a partial order is defined is called a partially ordered set; in brief, a poset.

Definition 4.18 If X is a poset, with partial order R, and xRy, then we say that x precedes y, written x ≺ y. We then analogously also say that y dominates x. If x precedes y and x6=y, we say that x properly precedes y or y properly dominates x.

Definition 4.19 Let X be a poset with R. Then x is called a lower bound of y if x≺y; and then y is called an upper bound of x.

Definition 4.20 Let X be a poset with R. An element x₀ of X is called the first or the least element of X if x₀ precedes every other element of X. The last or greatest element of X is that which dominates every other element of X.

Definition 4.21 Let X be a poset. An element x₀ of X is called a minimal element if no element of X properly precedes x₀.

NOTE

Ifx₀ is a minimal element of a poset X andx≺x₀, then x=x₀. Also, every first element is a minimal element but a minimal element may not be a first element.

Definition 4.22 Let X be a poset. An element y0 of X is called a maximal element if no element of X properly dominates y₀.

Definition 4.23 LetX be a poset. LetT be a subset ofX. A lower bound of T is an element of X which precedes every element ofT. The greatest lower bound (g.l.b.) of T is the lower bound which dominates every other lower bound of T. The g.l.b. of T is also called the infimum of T, and denoted inf(T).

Definition 4.24 Let X be a poset and let T be a subset of X. An upper bound of T is an element of X which dominates every element of T. The least upper bound (l.u.b.) of T is the upper bound which precedes every other upper bound of T. The l.u.b. of T is also called the supremum of T, and denoted sup(T).

Definition 4.25 Two elements x, y of a poset X are said to be comparable if either x≺y or y ≺x.

Definition 4.26 A lattice is a poset in which every two elements have a g.l.b and an l.u.b.

DEVELOPMENTS

Let C = {τα : α ∈ ∆} be a chain of (weak or strong) reductions of a topology τ on a set X. Then C, with the relation of set inclusion ⊆ is a poset. We also see that C is totally ordered (in that any two elements of C are comparable). If τα1 and τα2 are two topologies inC such that, say,τα1 is weaker than τ_α2, then the g.l.b. of the sub-family T = {τ_α1, τ_α2} of C, that is, inf(T), is τ_α1. Also sup(T) = τ_α2. Hence C is a lattice of topologies by set inclusion.

Let R be another relation on the chainC, whereτ_αRτ_r ifτ_α ≤τ_r. That is, the relation R(≤) on C, now, is that of comparison of topologies as topolo-gies. With this relation on C, we see again thatC is a lattice of topologies.

Corollary 4.1 Every chain C of reductions of a topology on a set X is a lattice in at least two ways.

OBSERVATIONS

Every set (on which a partial order is defined) is not a lattice. In particu-lar, every family of topologies is not a lattice. For example, if the topologies in a family F are not comparable, then the family F would not be a lattice in either of the ways; but F would still be a poset in the two ways (of set inclusion and comparison of topologies).

If a family of subsets of a set X is pairwise comparable by set inclusion (i.e. totally ordered by set inclusion), then it generates a topology (on X) which has a chain of reductions. This indeed is a theorem which marks the end and climax of this section.

Theorem 4.3 Any (set inclusion) pairwise comparable family F of subsets of a set X generates a reducible topology τ onX. And the chain C of reduc-tions of τ can be constructed in such a way that card(F) = card(C).

Proof:

Let F = {A_α : A_α ⊂ X}_α∈∆ be a family of (set inclusion) pairwise com-parable subsets of X. Let A_α1 and A_α2 be two elements of F such that, say, A_α1 ⊂A_α2. Let γ₁ =A_α1-induced topology onX and γ₂ =A_α2-induced topology onX. Ifγ₁ andγ₂ are not comparable, letτ₁ =γ₁ andτ₂ =γ₁5γ₂, the join of γ₁ and γ₂ (defined as the weakest topology, onX, finer than both γ₁ and γ₂). Thenτ₁ and τ₂ are two comparable topologies on X. Precisely, τ₁ is strictly weaker than τ₂.

Since F is pairwise comparable, the sets in F can be arranged such that A_α ⊂A_r ⊂ · · ·.

It follows from the construction above that these sets inF have, correspond-ing to them, a family C = {τα}α∈∆ of topologies on X, which is pairwise comparable in that

τ_α ≤τ_r ≤ · · ·.

It is easy to see that C is equivalent to F; that is, card(C) =card(F).

NOTE

It is easier to see the existence of the chainC, constructed in the proof of the theorem, if we remember that the construction can actually be done through inducement by the discrete topologies of A_α1 andA_α2; or, by what is similar, first getting a topology on A_α2 and then using this to induce a topology on A_α1; and then finally using these two topologies to construct subset-induced topologies on X.

4.1.5 Generalization of Hyperplane-Open Topologies