ON T_{D}-SPACES

Othman Echi

Abstract. This paper deals with some new properties of T_{D}-spaces. These properties are used in order to give an intrinsic topological characterization of the Goldman spectrum of a commutative ring.

1. Introduction. There are three famous separation axioms in topology namely, T_{0}, T_{1}, and T_{2}.

We denote by TOP the category of topological spaces with continuous maps as morphisms, and by TOP_{i} the full subcategory of TOP whose object are T_{i}-spaces. It is well-known that TOP_{i+1} is a reflective subcategory of TOP_{i}, for i = −1, 0, 1, with TOP_{−1} = TOP. Thus, TOP_{i} is reflective in TOP, for each i = 0, 1, 2 [10]. In other words, there is a universal T_{i}-space for every topological space X; we denote it by T_{i}(X). The assignment X ↦ T_{i}(X) defines a functor T_{i} from TOP onto TOP_{i}, which is a left adjoint functor of the inclusion functor TOP_{i} ↪ TOP.

The T_{D} separation axiom was introduced by Aull and Thron [1]. Recall that a topological space X is said to be a T_{D}-space if for each x ∈ X, {x} is locally closed.

For the separation axioms T_{0}, T_{1}, T_{2}, T_{D}, we have classically the following implications:

T_{2} ⟹ T_{1} ⟹ T_{D} ⟹ T_{0}.

We denote by TOP_{D} the full subcategory of TOP whose objects are T_{D}- spaces.

Unfortunately, the T_{D} property is not reflective in TOP. Indeed, in [2], Brümer has proved that the countable product of the Sierpinski space is not a T_{D}-space. On the other hand, according to Herrlich and Strecker [7], if a subcategory A is reflective in a category B, then for each category I, A is closed under the formation of I-limits in B [7]. [Taking I, a discrete category, one can see that in particular A is closed under products in B.] Therefore the full subcategory TOP_{D} of TOP whose objects are T_{D}-spaces is not reflective in TOP.

This paper deals with some new categorical properties of T_{D}-spaces (see Theorem 1.8).

Let R be a commutative ring with unit. We denote by Spec(R) the set of all prime ideals of R.

A topology T on a set X is defined to be spectral [8] (and (X, T ) is called a spectral space) if the following conditions hold:

(i) T is sober; (ii) the compact open subsets of X form a basis of T ; (iii) the family of compact open subsets of X is closed under finite inter-

sections.

In a remarkable paper, M. Hochster proved that a topological space is homeomorphic to the prime spectrum of some ring if and only if it is a spectral space [8]. In the same paper, Hochster characterized the space of maximal ideals of a ring. When a particular subset of the spectrum of a ring is given, a classical question, of whether we can give a topological characterization of that subspace, is asked.

A prime ideal 𝔭 of R is said to be a Goldman ideal (G-ideal, for short) if there exists a maximal ideal 𝔐 of the polynomial ring R[X] such that 𝔭 = 𝔐 ∩ R. Goldman ideals are important objects of investigation in algebra and algebraic geometry. Note, in particular, that G-ideals have been used by Goldman [5] and Krull [11] for a short inductive proof of the Nullstellensatz. It is a part of the folklore of algebra that 𝔭 is a G-ideal of R if and only if {𝔭} is locally closed in Spec(R) (endowed with the hull-kernel topology).

The subspace of Spec(R), whose elements are G-ideals is called the Goldman spectrum of R and it is denoted by Gold(R).

As in [4], by a goldspectral space we mean a topological space X which is homeomorphic to Gold(R) for some ring R.

A natural question is “give an intrinsic topological characterization of goldspectral spaces”.

The goal of this paper is to re-prove our characterization of goldspectral spaces [4] in a short elegant manner. More precisely, using our main result Theorem 1.5, we give an intrinsic topological characterization of the Goldman prime spectrum of a commutative ring (see Theorem 2.2). We prove that a topological space X is goldspectral if and only if X satisfies the following conditions:

(a) X is compact and has a basis of compact open subsets which is closed under finite intersections.

(b) X is a T_{D}-space.

2. T_{D}-spaces and Quasihomeomorphisms. Let us first recall some notions which were introduced by Grothendieck school, such as quasihomeomorphisms, strongly dense subsets and sober spaces.

If X is a topological space, we denote by 𝔒(X) the set of all open subsets of X. Recall that a continuous map q: Y ⟶ Z is called a quasihomeomorphism if U ⟼ q^{−1}(U) defines a bijection 𝔒(Z) ⟶ 𝔒(Y). A subset S of a topological space X is said to be strongly dense in X, if S meets every nonempty locally closed subset of X. Thus, a subset S of X is strongly dense if and only if the canonical embedding S ↪ X is a quasihomeomorphism. It is well-known that a continuous map q: X ⟶ Y is a quasihomeomorphism if and only if the topology of X is the inverse image by q of that of Y and the subset q(X) is strongly dense in Y [6].

A subspace Y of X is called irreducible, if each nonempty open subset of Y is dense in Y (equivalently, if C_{1} and C_{2} are two closed subsets of X such that Y ⊆ C_{1} ∪ C_{2}, then Y ⊆ C_{1} or Y ⊆ C_{2}). Let C be a closed subset of a space X; we say that C has a generic point if there exists x ∈ C such that C = \overline{\{x\}}. Recall that a topological space X is said to be sober if any nonempty irreducible closed subset of X has a unique generic point.

The main result of this section is Theorem 1.5. Before stating it, we need a sequence of lemmas.

\underline{Lemma 1.1 ([3])}. Let X be a topological space. Then the following properties hold:

(1) if X is a T_{0}-space which has a basis of compact open subsets, then Gold(X) is strongly dense in X;

(2) if Gold(X) is strongly dense in X, then it is the smallest strongly dense subset of X.

\underline{Lemma 1.2 (Never two without three)}. Let p: X ⟶ Y and q: Y ⟶ Z be two continuous maps. If two among the three maps (p, q, q ∘ p) are quasihomeomorphisms, then so is the third one.

\underline{Lemma 1.3}. Let X be a topological space and A a strongly dense subset of X. Then A is strongly dense in each subspace of X containing A.

\underline{Proof}. The proof is straightforward; but I would like to check it in terms of quasihomeomorphisms. Clearly, B is strongly dense in X. Hence, the canonical embeddings i: A ↪ X, j: B ↪ X are quasihomeomorphisms. If we let t: A ↪ B be the canonical embedding, then i = j ∘ t. By Lemma 1.2, t: A ↪ B is a quasihomeomorphism; this means that A is strongly dense in B.

\underline{Lemma 1.4 ([3])}. Let q: X ⟶ Y be a quasihomeomorphism. Then the following properties hold:

(1) if X is a T_{0}-space, then q is injective;

(2) if X is sober and Y is a T_{0}-space, then q is a homeomorphism.

Now, we are in a position to state our main result.

\underline{Theorem 1.5}. Let X be a T_{0}-space and Y be a topological space such that Gold(X) is strongly dense in X and Gold(Y) is strongly dense in Y. Let q: X ⟶ Y be a quasihomeomorphism. Then the following properties hold:

(a) q(Gold(X)) = Gold(q(X)) = Gold(Y);

(b) the induced map q_{G}: Gold(X) ⟶ Gold(Y) which carries x to q(x) is a homeomorphism.

\underline{Proof}.

(a) Let us consider the map q_{1}: X ⟶ q(X) induced by q. Let j: q(X) ⟶ Y be the canonical embedding; then j is a quasihomeomorphism. Since q = j ∘ q_{1}, we get q_{1} a quasihomeomorphism (by “Never two without three”). Now, since X is a T_{0}-space, q_{1} is injective, by Lemma 1.4. Thus, q_{1} is a bijective quasihomeomorphism; so that it is a homeomorphism.

It follows that Gold(q(X)) = q_{1}(Gold(X)) = q(Gold(X)). Since X is homeomorphic to q(X), Gold(q(X)) is strongly dense in q(X). But q(X) is strongly dense in Y; this forces Gold(q(X)) to be strongly dense in Y.

On the other hand, Gold(Y) is the smallest strongly dense subset of Y (see Lemma 1.1); this yields

Gold(Y) ⊆ Gold(q(X)) = q(Gold(X)) ⊆ q(X).

By Lemma 1.3, Gold(Y) is strongly dense in q(X); but Lemma 1.1 says that Gold(q(X)) is the smallest strongly dense subset of q(X); consequently, Gold(q(X)) ⊆ Gold(Y). We conclude that q(Gold(X)) = Gold(q(X)) = Gold(Y).

(b) Since the induced map q_{1}: X ⟶ q(X) is a homeomorphism and q_{1}(Gold(X)) = Gold(Y), the mapping q_{G}: Gold(X) ⟶ Gold(Y) defined by x ⟼ q(x) is also a homeomorphism.

\underline{Proposition 1.6}. Every quasihomeomorphism between two T_{D}-spaces is a homeomorphism.

\underline{Proof}. It follows immediately from Theorem 1.5 (b).

Note also that one may give an easy direct proof. Indeed, let q: X ⟶ Y be a quasihomeomorphism between two T_{D}-spaces. Hence, q is injective by Lemma 1.4. On the other hand, q(X) is strongly dense in Y and every point set of Y is locally closed; so that q(X) = Y. Thus, q is a bijective quasihomeomorphism. Therefore, q is a homeomorphism.

The following concept, motivated by Proposition 1.6, proves to be useful.

\underline{Definition 1.7}. Let C be a category. By a categoroid of C we mean a full subcategory of C closed under isomorphisms in which all arrows are isomorphisms.

Theorem 1.4 and Definition 1.6 immediately give the following categorical properties of T_{D}-spaces.

\underline{Theorem 1.8}. Let C be the category where objects are topological spaces X such that Gold(X) is strongly dense in X and arrows are quasihomeomorphisms. Let C_{1} be the full subcategory of C whose objects are T_{D}-spaces. Then C_{1} is a coreflective categoroide of C. The coreflector is μ_{X}: Gold(X) ↪ X.

\underline{Remark 1.9}. C_{1} is strictly contained in C. Let Y be an infinite set equipped with the cofinite topology and ω ∉ Y. Set X = Y ∪ {ω} and equip it with the topology whose closed sets are X and the closed sets of Y. Clearly, Gold(X) = Y is strongly dense in X. However, X is not a T_{D}-space, since {ω} is not locally closed.

3. The Goldman Spectrum of a Ring. Our next investigation of Theorem 1.5 is a new proof of our main result of [4], which gives an intrinsic topological characterization of the Goldman spectrum of a commutative ring.

We need to recall the the notion of soberification of a topological space which proved to play an important part in the next theorem.

Let X be a topological space and 𝕊(X) the set of all nonempty irreducible closed subsets of X [6]. Let U be an open subset of X and set

Ũ = {C ∈ 𝕊(X) | U ∩ C ≠ ∅}.

Then the collection (Ũ, U is an open subset of X) provides a topology on 𝕊(X) and the following properties hold [6]:

(i) the map μ_{X}: X ⟶ 𝕊(X) which carries x to \overline{\{x\}} is a quasihomeomorphism;

(ii) 𝕊(X) is a sober space.

The topological space 𝕊(X) is called the soberification of X.

Before stating the result which characterizes goldspectral spaces, let us give a straightforward remark.

\underline{Remark 2.1}. If q: X ⟶ Y is a quasihomeomorphism, then the following properties hold:

(a) Let U be an open subset of Y, then U is compact if and only if q^{−1}(U) is compact.

(b) X has a basis of compact open subsets closed under finite intersections if and only if so is Y.

\underline{Theorem 2.2}. Let X be a topological space. Then X is goldspectral if and only if X satisfies the following properties:

(a) X is compact;

(b) X has a basis of compact open subsets;

(c) the intersection of two compact open subsets is compact;

(d) X is a T_{D}-space.

\underline{Proof}.

• For each ring R, Gold(R) = Gold(Spec(R)) is a T_{D}−space. Since the canonical embedding Gold(R) ↪ Spec(R) is a quasihomeomorphism and Spec(R) satisfies properties (a), (b), and (c), then so is Gold(R), by Remark 2.1.

• Conversely, let X be a space satisfying properties (a), (b), (c), and (d). Let 𝕊(X) be the soberification of X and μ_{X}: X ⟶ 𝕊(X) defined by μ_{X}(x) = \overline{\{x\}} the canonical embedding of X into 𝕊(X).

According to Theorem 1.5, X is homeomorphic to Gold(𝕊(X)). Thus, Remark 2.1 implies immediately that 𝕊(X) is a spectral space, completing the proof.

\underline{Acknowledgement}. I wish to thank the referee for his valuable suggestions. I also thank Professors H. P. Kunzi and G. Brümer from the University of Cape Town (South Africa) for their comments on some categorical properties about T_{D}-spaces.

\underline{References}

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2. G. Brümmer, “Initial Quasi-Uniformities,” Indag. Math., 31 (1969), 403–409.

3. E. Bouacida, O. Echi, G. Picavet, and E. Salhi, “An Extension Theorem for Sober Spaces and the Goldman Topology,” Int. J. Math. Math. Sc., 51 (2003), 3217–3239.

4. O. Echi, “A Topological Characterization of the Goldman Prime Spectrum of a Commutative Ring,” Comm. Algebra, 28 (2000), 2329–2337.

5. O. Goldman, “Hilbert Rings and the Hilbert Nullstallensatz,” Math. Z., 54 (1951), 136–140.

6. A. Grothendieck and J. Dieudonné, Eléments de Géométrie Algébrique, Die Grundlehren der mathematischen Wissenschaften, Vol. 166, Springer-Verlag, New York, 1971.

7. H. Herrlich, G. E. Strecker, Category Theory. An Introduction, second edition, Sigma Series in Pure Mathematics 1, Heldermann Verlag, Berlin, 1979.

8. M. Hochster, “Prime Ideal Structure in Commutative Rings,” Trans. Amer. Math. Soc., 142 (1969), 43–60.

9. M. Hochster, “The Minimal Prime Spectrum of a Commutative Ring,” Canad. J. Math., 23 (1971), 749–758.

10. S. MacLane, Categories for the Working Mathematician, Springer, New York, 1971.

11. W. Krull, “Jacobsonsche Ringe, Hilbertscher Nullstellensatz Dimensionenstheorie,” Math. Z., 54 (1951), 354–387.

Mathematics Subject Classification (2000): 54B30, 54D10, 54F65

Othman Echi Department of Mathematics Faculty of Sciences of Tunis University of Tunis El-Manar “Campus Universitaire” 2092 El-Manar II Tunis, TUNISIA email: othechi@yahoo.com email: othechi@math.com