On a new convergence in topological spaces

Abstract In this paper, we introduce a new way-below relation in T0 topological spaces based on cuts and give the concepts of SI2-continuous spaces and weakly irreducible topologies. It is proved that a space is SI2-continuous if and only if its weakly irreducible topology is completely distributive under inclusion order. Finally, we introduce the concept of 𝓓-convergence and show that a space is SI2-continuous if and only if its 𝓓-convergence with respect to the topology τSI2(X) is topological. In general, a space is SI-continuous if and only if its 𝓓-convergence with respect to the topology τSI(X) is topological.


Introduction
Domain theory which arose from computer science and logic, started as an outgrowth of theories of order. Rapidly progress in this domain required many materials on topologies (see [1][2][3]). Conversely, it is well known that given a topological space one can also de ne order structures (see [3][4][5][6]). At the 6th International Symposium in Domain Theory, J.D. Lawson emphasized the need to develop the core of domain theory directly in T topological spaces instead of posets. Moreover, it was pointed out that several results in domain theory can be lifted from the context of posets to T topological spaces (see [5,6]). In the absence of enough joins, Erné introduced the concept of s -continuous posets and the weak Scott topology by means of the cuts instead of joins (see [7]). The notion of s -continuity admits to generalize most important characterizations of continuity from dcpos to general posets and has the advantage that not even the existence of directed joins has to be required. In [6], Erné further proved that the weak Scott topology is the weakest monotone determined topology with a given specialization order. In [5], Zhao and Ho de ned a new way-below relation and a new topology constructed from any given topology on a set using irreducible sets in a T topological space replacing directed subsets and investigated the properties of this derived topology and k-bounded spaces. It was proved that a space X is SI-continuous if and only if SI(X) is a C-space.
Many convergent classes in posets were studied in [3,[8][9][10][11][12]. By di erent convergences, not only many notions of continuity are characterized, but also they make order and topology across each other. In [3], the concept of S-convergence for dcpos was introduced by Scott to characterize continuous domains. It was proved that for a dcpo, the S-convergence is topological if and only if it is a continuous domain. The main purpose of this paper is to lift the notion of s -continuous posets to topology context. By the manner of Erné we introduce a new way-below relation in T topological spaces based on cuts and give the concepts of SIcontinuous spaces and weakly irreducible topologies. It is proved that a space is SI -continuous if and only if its weakly irreducible topology is completely distributive under inclusion order. Finally, we introduce the concept of D-convergence and show that a space is SI -continuous if and only if its D-convergence with respect to the topology τ SI (X) in a topological space is topological. Furthermore, a space is SI-continuous if and only if its D-convergence with respect to the topology τ SI (X) is topological. The work carry out here is another response to the call by J. D. Lawson to develop domain theory in the wider context of T topological spaces instead of restricting to posets.

Preliminaries
Let P be a partially ordered set (poset, for short). A nonempty set D ⊆ P is directed if for any d , d in D there exists d in D above d and d . The principal ideal generated by x ∈ P is ↓ x = {y ∈ P ∶ y ≤ x}. ↓ A = ⋃ a∈A ↓ a is the lower set or downset generalized by A ⊆ P; The principal lter ↑ x and upper set ↑ A are de ned dually. A ↑ and A ↓ denote the sets of all upper and lower bounds of A, respectively. A cut δ of A in P is de ned by A δ = (A ↑ ) ↓ for every A ⊆ P. Notice that x ∈ A δ means x ≤ ⋁ A whenever A has a join (supremum).
On the one hand, given a poset P, we can generate some intrinsic topologies. The upper sets form the Alexandro upper topology α(P). The weak Scott topology σ (P) consists of all upper sets U such that D δ ∩ U = ∅ implies D ∩ U = ∅ for all directed sets D of P. In case P is a dcpo, the weak Scott topology coincides with the usual Scott topology σ(P), which consists of all upper sets U such that ⋁ D ∈ U implies D ∩ U = ∅ for all directed sets D in P. The upper topology generating by the complements of the principal ideals is denoted by υ(P). Clearly, υ(P) ⊆ σ (P) ⊆ σ(P) ⊆ α(P).
On the other hand, for a topological space (X, τ), the specialization order ≤ on X is de ned by y ≤ x if and only if y ∈ cl{x}. It is antisymmetric, hence a partial order, if and only if (X, τ) is T . The specialization order on X is denoted by ≤ τ if there is need to emphasize the topology τ. Note that the specialization order of the Alexandro upper topology on a poset coincides with the underlying order and ↓ x = cl{x}.
If not otherwise stated, in topological contexts, lower sets, upper sets and related notions refer to the specialization order. (1) If D ⊆ X is a directed set with respect to the specialization order, then D is irreducible; (2) If U ⊆ X is an open set, then U is an upper set; Similarly, if F ⊆ X is a closed set, then F is a lower set.
(1) For any x, y ∈ P, we say that x is way below y, written x ≪ y if for all directed sets D ⊆ P with y ∈ D δ , there exists d ∈ D such that x ≤ d. The set {y ∈ P ∶ y ≪ x} will be denoted by ⇓ x and {y ∈ P ∶ x ≪ y} denoted by ⇑ x;

De nition 2.2. [5]
Let (X, τ) be a T space. For x, y ∈ X, de ne x ≪ SI y if for all irreducible sets F, y ≤ ⋁ F implies there exists e ∈ F such that x ≤ e whenever ⋁ F exists. The set {y ∈ X ∶ y ≪ SI x} is denoted by ↡ SI x and the set {y ∈ X ∶ x ≪ SI y} by ↟ SI x.

De nition 2.3. [5]
Let (X, τ) be a T space. X is called SI-continuous if the following conditions are satis ed: De nition 2.4. [5] Let (X, τ) be a T space. A subset U ⊆ X is called SI-open if the following conditions are satis ed: The set of all SI-open sets of (X, τ) is denoted by τ SI (X).

SI -continuous spaces
In this section, we de ne a SI -continuous space derived by the irreducible set of a topological space. Some properties of this derived SI -continity are investigated.
Let (X, τ) be a topological space. A nonempty subset F ⊆ X is called irreducible if for every closed sets B and C, whenever F ⊆ B ⋃ C, one has either F ⊆ B or F ⊆ C. The set of all irreducible sets of the topological space (X, τ) will be denoted by Irr τ (X) or Irr(X).
De nition 3.1. Let (X, τ) be a T space and x, y ∈ X. De ne x ≪ r y if for ervery irreducible set E, y ∈ E δ implies there exists e ∈ E such that x ≤ e. We denote the set {y ∈ X ∶ y ≪ r x} by ↡ r x and the set {y ∈ X ∶ x ≪ r y} by ↟ r x.

De nition 3.2. Let (X, τ) be a T space. A subset U ⊆ X is called weakly irreducibly open if the following conditions are satis ed:
The set of all weakly irreducibly open sets of (X, τ) is denoted by τ SI (X). Complements of all weakly irreducibly open sets are called weakly irreducibly closed sets. Lemma 3.1. Let (X, τ) be a T space. Then τ SI (X) is a topology on X. Proof.

Remark 3.2.
Let (X, τ) be a T space. Then τ SI (X) is always coarser than τ SI (X), and if any irreducible set in X has a supremum, then both topologies coincide. In the following, the space (X, τ SI (X)) is also simply denoted by SI (X).

Then it is easy to see that ↑ b is open in τ SI
. Thus τ SI (α(P)) is proper contained in τ SI (α(P)).

De nition 3.3. Let (X, τ) be a T space. X is called SI -continuous if the following conditions are satis ed:
Lemma 3.2. Let P be a poset. Then SI (P, α(P)) = (P, σ (P)).

Remark 3.3.
(1) Let P be a poset. Then P is an s -continuous poset if and only if it is an SI -continuous space with respect to the Alexandro upper topology.
(2) Let (X, τ) be a T space. If X is an SI -continuous space, then it is also an s -continuous poset under the specialization order. But the converse may not be true.

Example 3.2. Let X be an in nite set with a co nite topology τ. Then it is a T space. Clearly it is an antichain
under the specialization order, and hence it is an s -continuous poset. But ↟ r x = {x} ∈ τ for all x ∈ X, then (X, τ) is not an SI -continuous space.
Let us note that an SI -continuous space is SI-continuous space, but the converse may not be true:

Example 3.3. Consider the Euclidean plane R × R under the usual topology. It is an SI-continuous space, but it is not SI -continuous, since every lower half-plane
is a directed set with E δ a = R × R and ⋂{Ea ∶ a ∈ R} = ∅, thus ≪ r is empty.
The following theorem shows that the SI -continuity of the topological space has the interpolation property.
Theorem 3.1. Let X be an SI -continuous space and x, y ∈ X. If x ≪ r y, then there exists z ∈ X such that x ≪ r z ≪ r y.
Proof. Let X be an SI -continuous space and x ≪ r y. Then we have ↡ r y is directed and y ∈ (↡ r y) δ . Since the union of a directed family of directed sets, E = ⋃{↡r z ∶ z ∈↡ r y} is still a directed set( hence an irreducible set) and y ∈ E δ . So there exists z ∈↡ r y such that x ≤ u ≪ r z for some u ∈ X. Thus x ≪ r z ≪ r y.

Remark 3.4. In Theorem 3.1, when we prove the interpolation property, we do not need the third condition in the de nition of the SI -continuous space.
Lemma 3.3. Let X be an SI -continuous space. Then for any x ∈ X, ↟ r x ∈ τ SI (X).
Proof. It follows from De nition 3.3 and Theorem 3.1.

Lemma 3.4.
Let (X, τ) be a T space and y ∈ int τ SI (X) ↑ x. Then x ≪ r y, where int τ SI (X) ↑ x denotes the interior of ↑ x with respect to the topology τ SI (X).
Proof. Let y ∈ int τ SI (X) ↑ x. For every irreducible set E with y ∈ E δ , we have y ∈ E δ ∩ int τ SI (X) ↑ x = ∅, and hence int τ SI (X) ↑ x ∩ E = ∅. Thus there exists e ∈ int τ SI (X) ↑ x ∩ E. Thus we have x ≤ e and e ∈ E. This shows x ≪ r y.

Theorem 3.2. Let (X, τ) be a T space. Then the following statements are equivalent:
(1) X is an SI -continuous space; (2) For all U ∈ τ SI (X) and x ∈ U, there exists y ∈ U such that x ∈ int τ SI (X) ↑ y ⊆↑ y ⊆ U; Let y , y ∈ E. Then x ∈ int τ SI (X) ↑ y ⋂ int τ SI (X) ↑ y . By (2), there exists y ∈ int τ SI (X) ↑ y ⋂ int τ SI (X) ↑ y such that x ∈ int τ SI (X) ↑ y ⊆↑ y ⊆ int τ SI (X) ↑ y ⋂ int τ SI (X) ↑ y , so y ∈ E and y , y ≤ y. This shows that E is a directed set. It is not hard to show x ∈ E δ . By Lemma 3.4, we have E ⊆↡ r x. Thus we have that ↡ r x is directed(hence irreducible) and x ∈ (↡ r x) δ , so we also have ↡ r x ⊆↓ E. Thus ↡ r x =↓ E. From the above discussion we can derive that ↟ r x = ⋃ y∈↑x int τ SI (X) ↑ y, which is open in τ. Hence X is an SI -continuous space.

D-convergence in SI -continuous spaces
In this section, the concept of D-convergence in a topological space is introduced. It is proved that a space X is SI -continuous if and only if the D-convergence with respect to the topology τ SI (X) in X is topological. In general, a space X is SI-continuous if and only if its D-convergence with respect to the topology τ SI (X) in X is topological.

De nition 4.1. Let (X, τ) be a T space. A net (x j ) j∈J in X is said to converge to x ∈ X if there exists a directed set D ⊆ X with respect to the specialization order such that
(1) x ∈ D δ ; (2) For all d ∈ D, d ≤ x j holds eventually.
In this case we write x ≡ D lim x j .
Let D denote the class of those pairs (( x) ∈ D and x ∈ U, then eventually x j ∈ U} is a topology. Proof. Firstly, suppose that a net (x j ) j∈J in X converges to x ∈ X and x ∈ U ∈ τ SI (X). Then there exists a directed set D ⊆ X (hence an irreducible set) such that d ≤ x j holds eventually for all d ∈ D and x ∈ D δ , and hence U ⋂ D = ∅, that is, there exists d ∈ U ⋂ D. Clearly, the net x j ∈ U holds eventually as U is an upper set. Conversely, assume that the net (x j ) j∈J converges to an element x with respect to the topology τ SI (X). For all y ∈↡ r x, then one has x ∈↟ r y ∈ τ SI (X) by Lemma 3.3. Thus there exists k ∈ J such that x j ∈↟ r y for any j ≥ k.
By SI -continuity of X, we have that x ∈ (↡ r x) δ and ↡ r x is directed.
It is easy to show that ↡ r x is directed. Finally, it follows that ↟ r x ∈ τ for all x ∈ X. Indeed, if y ∈↟ r x, then there exists z ∈ X such that x ≪ r z ≪ r y by Remark 3.4. From the above argument, as long as we replace (1) X is SI -continuous; (2) The D-convergence with respect to the topology τ SI (X) is topological.
Similarly, we also have: Let (X, τ) be a T space. Then the following statements are equivalent: (1) X is SI-continuous; (2) The D-convergence with respect to the topology τ SI (X) is topological. (1) P is s -continuous; (2) The S-convergence in P is topological for the weak Scott topology, that is, for all x ∈ P and all nets (x j ) j∈J in P, x ≡ S lim x j if and only if (x j ) j∈J converges to the element x with respect to the weak Scott topology.

Corollary 4.2. ([3])
Let P be a dcpo. Then the following statements are equivalent: (1) P is a domain; (2) The S-convergence in P is topological for the Scott topology, that is, for all x ∈ P and all nets (x j ) j∈J in P, x ≡ S lim x j if and only if (x j ) j∈J converges to the element x with respect to the Scott topology.

Conclusion
At the Sixth International Symposium on Domain Theory, J.D. Lawson encouraged the domain theory community to consider the scienti c program of developing domain theory in the wider context of T spaces instead of restricting to posets. In this paper, we introduce a new way-below relation in T topological spaces based on the cuts and give the concepts of SI -continuous spaces and weakly irreducible topologies. It is proved that a space is SI -continuous if and only if its weakly irreducible topology is completely distributive under inclusion order. Finally, we introduce the concept of D-convergence and show that a space is SIcontinuous if and only if its D-convergence with respect to the topology τ SI (X) is topological. In general, a space is SI-continuous if and only if its D-convergence with respect to the topology τ SI (X) is topological. The present paper can be seen as one of the some works towards the new direction, which may deserve further investigation. Indeed there are some questions to which we possess no answers. The following is such one.
In the rst condition of the de nition of the D-convergence, whether we can change directed set into irreducible set?