Applied General Topology c© Universidad Politécnica de Valencia Volume 3, No. 1, 2002 pp. 55–64 Minimal TUD spaces A. E. McCluskey and W. S. Watson Abstract. A topological space is TUD if the derived set of each point is the union of disjoint closed sets. We show that there is a minimal TUD space which is not just the Alexandroff topology on a linear order. Indeed the structure of the underlying partial order of a minimal TUD space can be quite complex. This contrasts sharply with the known results on minimality for weak separation axioms. 2000 AMS Classification: 54D10, 06A10, 54A10, 54G20. Keywords: Minimal topologies, weak separation axioms. 1. Introduction Definition 1.1. [2] A topological space is said to be TUD if the derived set of each point is the (possibly empty) union of disjoint closed sets. In this introduction, we provide a complete brief survey and bibliography of minimality. The family LT(X) of all topologies definable for an infinite set X is a complete atomic and complemented lattice (under set inclusion). If T and S are two members of LT(X) with S ⊆ T , then S is said to be weaker than T . Given a topological invariant P, a member T of LT(X) is said to be minimal P if and only if T possesses property P but no weaker member of LT(X) possesses property P. The concept of minimal topologies was first introduced in 1939 by Parho- menko [27] when he showed that compact Hausdorff spaces are minimal Haus- dorff. Motivation for such an investigation is provided by realising that it is in seeking to identify those members of LT(X) which minimally satisfy an invari- ant that we are, in a very real sense, examining the topological essence of the invariant. Given a topological space (X,T ), (X,T ) is - minimal Hausdorff if and only if it is Hausdorff and every open filter- base which has a unique adherent point is convergent to this point (see [5], [9], [10], [27], [31], [32], and [36]) - minimal T1 if and only if T is the cofinite topology C on X 56 A. E. McCluskey and W. S. Watson - minimal regular if and only if it is regular and every regular filter-base which has a unique adherent point is convergent ([4], [8]) - minimal completely regular if and only if it is compact and Hausdorff ([4], [5]) - minimal normal if and only if it is compact and Hausdorff ([5]) - minimal Urysohn if and only if it is Urysohn and every filter with a unique adherence point converges to this point ([11], [34]) - minimal (locally compact and Hausdorff ) if and only if it is compact and Hausdorff ([5], [4]) - minimal paracompact if and only if it is compact and Hausdorff ([35]) - minimal metric only if it is compact and Hausdorff ([35]) - minimal completely normal only if it is compact and Hausdorff ([35]) - minimal completely Hausdorff only if it is compact and Hausdorff ([35]) - minimal T0 if and only if it is T0, nested and generated by the family {X \{x} : x ∈ X}∪{∅,X} ([1], [12], [19], [22], [26]) - minimal TD if and only if it is TD and nested ([1], [12], [19], [22], [26]) - minimal Tδ if and only if it is Tδ and nested ([1], [22]) - minimal Tξ if and only if it is Tξ and nested ([1]) - minimal TA if and only if it is TA and partially nested ([22]) - minimal TES if and only if either T = C or T = E(X \Y )∪(C∩I(Y )) for some non-empty proper subset Y of X ([21]) - minimal TEF if and only if T = C or T = I(x) or T = E(x) for some x ∈ X ([21]) - minimal TFF if and only if there exists x ∈ X such that either T = C∩ I(x) or T = C∩E(x) ([15]) - minimal TF if and only if either there exists x ∈ X such that T = C∩ I(x) or there exists a non-empty proper non-singleton subset Y of X such that T = D(Y ) ([15]) - minimal TY S if and only if T = W(P)∨(C∩I(K)) for some subset K of X and some partition P of X such that P is simply associated with K and is associated with X \K. ([16]) - minimal TDD if and only if T = WK(P) ∨ (C∩I(K)) for some subset K of X and partition P of X such that P is simply associated with K and associated with X \K ([16]) - minimal TY Y if and only if T = Mp(P) ∨ (C∩I(K)) for some p ∈ X, subset K of X \{p} such that P is simply associated with K ([23]) - minimal TY if and only if T = W(F)∨(C∩I(K)) for some degenerate K-cover F of X ([23]) - minimal TSA if and only if T = E(X\B)∨SK(P)∨(C∩I(K∪B)) for some disjoint subsets B and K of X such that K 6= ∅ and K ∪ B 6= X, and partition P of X \ B such that P is simply associated with X \ (K ∪B) and associated with K. ([24]) - minimal TSD if and only if T = SK(P) ∨ (C ∩I(K)) for some non- empty proper subset K of X and partition P of X such that P is simply associated with X \K and associated with K ([24]) Minimal TUD spaces 57 - minimal TFA if and only if either (X,T ) is minimal TES with at least one isolated point and at least two closed points or (X,T ) is minimal TSD or T = E(X \B)∨SK(P)∨D(B∪K)) for some non-empty, disjoint subsets B, K of X such that B ∪ K is a proper, infinite subset of X with |X \ (B ∪ K)| > 1, a subset G of X \ (B ∪ K) and a partition P of X \ (B ∪G) such that P is simply associated with X \K and associated with K. 2. Constructing the Partial Order We need an axiom for partial orders which implies the TUD axiom for topo- logical spaces. Definition 2.1. A partial order (X, �) is said to be T +UD if there is a family {Y (x) : x ∈ X} of subsets of X such that • (∀y ∈ Y (x))y � x∧y 6= x • (∀z � x)z 6= x ⇒ (∃y ∈ Y (x))z � y • (∀y,y′ ∈ Y (x))(∀z ∈ X)(z � y ∧z � y′) ⇒ y = y′ We shall write YX(x) if the underlying partial order is ambiguous. A few comments: We conjecture that the minimal TUD topologies must be the weak topologies on a minimal T +UD partial order. We conjecture that the minimal T +UD partial orders are just the T + UD and suitable partial orders. This would provide a characterization which requires for each pair of elements an infinite set which satisfies a first order formula. Maybe those weak separation axioms which have simpler minimality characterizations do so because of logical considerations, i.e., must all first order weak separation axioms have minimalities which are weak topologies for partial orders either without infinite chains or without infinite antichains? Next we describe a way in which two T +UD partial orders can be combined and yet preserve T +UD. Definition 2.2. If X0 ⊂ X1 are partial orders, where X0 has the order induced by X1, then we say that X0 is a simple subset of X1 if there are distinct x0,x1 ∈ X0 and w ∈ X1 −X0and A ⊂ X1 −X0 such that • x1 � w 6∈ A • x0 and x1 are incomparable in X0 • A is the set of all elements of X1 −X0 strictly below x0 • each element of A is minimal in X1 • (∀x ∈ X0)(∀y ∈ X1 −X0)(x � y ⇒ x � x1 � y = w) • (∀x ∈ X0)(∀y ∈ X1 −X0)(y � x ⇒ y � x0 � x) Proposition 2.3. If X0 is a simple subset of X1 and both X0 and X1 − X0 are T +UD, then X1 is also T + UD. Moreover, we can get YX1 (x) ∩X0 = YX0 (x) for each x ∈ X0. 58 A. E. McCluskey and W. S. Watson Proof. Let x0,x1 ∈ X0 and w ∈ X1 − X0 and A ⊂ X1 − X0 be as in defini- tion 2.2. Suppose x ∈ X1. We must define Y (x) as in definition 2.1. We do this by cases. (1) If x ∈ X0 and x 6= x0, then we let Y (x) = YX0 (x). (2) If x ∈ X1 −X0 and x 6= w, then we let Y (x) = YX1−X0 (x). (3) If x = x0, then we let Y (x) = YX0 (x) ∪A. (4) If x = w, then we let Y (x) = YX1−X0 (x) ∪{x1}. It suffices to show that definition 2.1 is satisfied by {Y (x) : x ∈ X1}. Verifying the first condition requires us to use only the facts (∀a ∈ A)a � x0 ∧a 6= x0 and x1 � w ∧x1 6= w. Verifying the second condition requires examination of the same four cases. (1) If y � x and y ∈ X1 −X0 then simplicity says that y � x0 � x. Now, since x 6= x0 ∈ X0, we know that (∃s ∈ YX0 (x))x0 � s and thus that y � s. (2) If y � x and y ∈ X0 then x = w which is impossible. (3) If x = x0 and y � x and y ∈ X1 −X0 then y ∈ A which suffices. (4) If x = w and y � x and y ∈ X0 then y � x1 which suffices. Verifying the third condition also requires the examination of these same four cases. (1) Suppose y0,y1 are distinct elements of Y (x) and z � y0,y1. Then z ∈ X1−X0 so that x0 �y0 and x0 �y1 by the sixth condition of simplicity— clearly a contradiction. (2) Suppose y0,y1 are distinct elements of Y (x) and z�y0,y1. Then z ∈ X0 so that, by the fifth condition of simplicity, y0 = w = y1! (3) The first case for YX0 (x) and the fact that A is a set of minimal points in X1 suffices. (4) The second case for YX1−X0 (x) leaves the possibility that there is z�x1 and z � y ∈ YX1−X0 (x). If z ∈ X1 − X0, then z � x0 � x1 which is impossible. If z ∈ X0, then z � x1 � y = w—yet y ∈ YX1−X0 (w)! The proof is complete. � Next, we describe when two incomparable elements of a T +UD partial order cannot be made comparable in a given “direction” without destroying T +UD. Moreover, since a TUD-topology may induce an order which is not T + UD, we stipulate a condition to ensure that the resulting order has no compatible TUD- topology. Definition 2.4. If X0 ⊂ X1 are partial orders, where X0 has the order induced by X1, and x0,x1 ∈ X0 are incomparable, then we say that X0 is a suitable subset in X1 with respect to (x0,x1), if there are, in X1, elements w, {yi : i ∈ ω} and {zi : i ∈ ω} all distinct from each other and from x0 and x1 such that • (∀i ∈ ω)zi � yi � w • (∀i ∈ ω)zi � x0 • x1 � w Minimal TUD spaces 59 • (∀F ∈ [X1]<ω)(((∀f ∈ F)w 6�f) ⇒ ((∃i ∈ ω)(∀f ∈ F)yi 6�f)) Note that this definition applies also when X0 = X1. Indeed, we can “make” a T +UD partial order suitable for two incomparable elements in a “simple” way. Proposition 2.5. If X is any T +UD partial order and x0,x1 ∈ X are incompa- rable, then there is a partial order Y ⊃ X such that • X is a simple subset of Y • X is a suitable subset of Y with respect to (x0,x1) • Y −X is countable and T +UD Proof. We let Y = X ∪ {yi,zi : i ∈ ω} ∪ {w} where all these elements are distinct and not in X. We declare • (∀i ∈ ω)zi � x0 • (∀i ∈ ω)zi � yi � w • x1 � w and close off under transitivity. To check that X is a simple subset of Y , define A = {zi : i ∈ ω}. Since nothing is defined to be below any zi, we know that each zi is minimal in Y . Thus we have conditions 1, 2 and 4 in definition 2.2. Further, clearly w cannot be below x0 nor can any yi be below x0, so that condition 3 is satisfied. If x ∈ X, y ∈ Y −X and x�y, then x1 �w must be a step in the calculation. Since nothing is defined to be above w, w is maximal in Y and so w = y as required. Thus condition 5 is satisfied. If x ∈ X, y ∈ Y −X and y �x, then zi �x0 must be a step in the calculation as required. Thus condition 6 is satisfied. To check that X is a suitable subset of Y with respect to (x0,x1), suppose that there exists finite F ⊂ Y such that (∀f ∈ F)w 6= f and (∀i ∈ ω)(∃f ∈ F)yi �f. If yi �f and yi 6= f, then some step in the calculation must be yi �w. Since w is maximal in Y , we must have f = w which is impossible. Thus we know that (∀i ∈ ω)(∃f ∈ F)yi = f and thus F ⊃{yi : i ∈ ω}! To check that Y − X is T +UD, let Y (w) = {yi : i ∈ ω}, Y (yi) = {zi} and Y (zi) = ∅. � Definition 2.6. A partial order (X, �) is said to be suitable if, for each x0,x1 ∈ X which are incomparable, X is suitable in itself with respect to (x0,x1). Suitability can be obtained in a “simple” increasing sequence if suitability with respect to each incomparable pair is accomplished along the way. Proposition 2.7. If {Xi : i ∈ ω} is an increasing sequence of (partially or- dered) subsets of a partial order X such that • each Xi is a simple subset of Xi+1 • ⋃ {Xi : i ∈ ω} = X • (∀ incomparable x0,x1 ∈ X)(∃i ∈ ω)Xi is a suitable subset of Xi+1 with respect to (x0,x1) 60 A. E. McCluskey and W. S. Watson then X is suitable. Proof. Given any incomparable elements x0, x1 in X, we must check that X is suitable in itself with respect to (x0,x1). Now suppose that Xi is a suitable subset of Xi+1 with respect to (x0,x1). This gives us distinct w, {yi : i ∈ ω} and {zi : i ∈ ω} as in definition 2.4. Thus the first three conditions of definition 2.4 are satisfied. We need to check the fourth condition. Suppose F ∈ [X]<ω and (∀f ∈ F)w 6�f and (∀i ∈ ω)(∃f ∈ F)yi � f. We shall argue that no such F can exist by mathematical induction. Find such an F with j∗ = minimum{max[j ∈ ω : F ∩ (Xj+1 − Xj) 6= ∅]} and furthermore such that F ∩ (Xj∗+1 −Xj∗) has minimum cardinality. Let F ′ = {f ∈ F : (∃i ∈ ω)yi � f}. We shall prove that j∗ ≤ i. Suppose j∗ > i and choose f ∈ F ′ ∩ (Xj∗+1 − Xj∗)—such a choice is possible because of the minimum nature of j∗. We know that for certain i ∈ ω, yi � f. Each such yi is an element of Xi+1 ⊂ Xj∗. Let the fact that Xj∗ is a simple subset of Xj∗+1 be witnessed by x∗0,x ∗ 1,w ∗. The fifth condition of simplicity gives us that yi �x∗1 �f = w ∗ so that F ′∩(Xj∗+1−Xj∗) = {w∗} = {f}. Thus f can be replaced by x∗1 ∈ Xj∗, giving a subset F∗ = (F −{f})∪{x∗1} of F which again contradicts the minimum nature of j∗. Thus j∗ ≤ i. It follows that F ⊂ Xi+1, contradicting the suitability of Xi in Xi+1 with respect to (x0,x1). � Finally we can accomplish our aim of making a T +UD partial order suitable without destroying T +UD. Proposition 2.8. Any countable T +UD partial order can be embedded in a suit- able T +UD partial order. Proof. First, we define a partition {Pi : i ∈ ω} of ω. Given Pi ⊂ ω for i < n, define Pn to be any infinite, co-infinite subset of ω− ⋃ i