On a new convergence in topological spaces

Xiao-jun Ruan; Xiao-quan Xu

doi:10.1515/math-2019-0123

Artikel Open Access

On a new convergence in topological spaces

Xiao-jun Ruan und Xiao-quan Xu

Veröffentlicht/Copyright: 31. Dezember 2019

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Aus der Zeitschrift Open Mathematics Band 17 Heft 1

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 τ_SI₂(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.

Keywords: s2-continuous poset; weakly irreducible topology; SI2-continuous space; 𝓓-convergence

MSC 2010: 06B35; 06B75; 54F05

1 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 define 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 T₀ 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 T₀ topological spaces (see [5, 6]). In the absence of enough joins, Erné introduced the concept of s₂-continuous posets and the weak Scott topology by means of the cuts instead of joins (see [7]). The notion of s₂-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 defined a new way-below relation and a new topology constructed from any given topology on a set using irreducible sets in a T₀ 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 different 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 s₂-continuous posets to topology context. By the manner of Erné 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 τ_SI₂(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 T₀ topological spaces instead of restricting to posets.

2 Preliminaries

Let P be a partially ordered set (poset, for short). A nonempty set D ⊆ P is directed if for any d₁, d₂ in D there exists d in D above d₁ and d₂. 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 filter ↑ x and upper set ↑ A are defined dually. A^↑ and A^↓ denote the sets of all upper and lower bounds of A, respectively. A cut δ of A in P is defined 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 Alexandroff upper topology α(P). The weak Scott topology σ₂(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) ⊆ σ₂(P) ⊆ σ(P) ⊆ α(P).

On the other hand, for a topological space (X, τ), the specialization order ≤ on X is defined by y ≤ x if and only if y ∈ cl{x}. It is antisymmetric, hence a partial order, if and only if (X, τ) is T₀. 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 Alexandroff 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.

Remark 2.1

Let (X, τ) be a T₀ space.

If D ⊆ X is a directed set with respect to the specialization order, then D is irreducible;
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.

Definition 2.1

[7] Let P be a poset.

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;
P is called s₂-continuous if for all x ∈ P, x ∈ (⇓ x)^δ and ⇓ x is directed.

Indeed, since ⇓ x ⊆ ↓ x we have x = ⋁ ⇓ x if and only if x ∈ (⇓ x)^δ.

Definition 2.2

[5] Let (X, τ) be a T₀ space. For x, y ∈ X, define 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.

Definition 2.3

[5] Let (X, τ) be a T₀ space. X is called SI-continuous if the following conditions are satisfied:

↡_SI x is directed for all x ∈ X;
x = ⋁ ↡_SI x for all x ∈ X;
↟_SI x ∈ τ for all x ∈ X.

Definition 2.4

[5] Let (X, τ) be a T₀ space. A subset U ⊆ X is called SI-open if the following conditions are satisfied:

U ∈ τ;
For all F ∈ Irr_τ(X), ⋁ F ⋂ U ≠ ∅ implies F ⋂ U ≠ ∅ whenever ⋁ F exists.

The set of all SI-open sets of (X, τ) is denoted by τ_SI(X).

3 SI₂-continuous spaces

In this section, we define a SI₂-continuous space derived by the irreducible set of a topological space. Some properties of this derived SI₂-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).

Definition 3.1

Let (X, τ) be a T₀ space and x, y ∈ X. Define 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.

Remark 3.1

Let (X, τ) be a T₀ space and x, y, u, v ∈ X.

x ≪_r y implies x ≤ y;
u ≤ x ≪_r y ≤ v implies u ≪_r v.

Definition 3.2

Let (X, τ) be a T₀ space. A subset U ⊆ X is called weakly irreducibly open if the following conditions are satisfied:

U ∈ τ;
F^δ ⋂ U ≠ ∅ implies F ⋂ U ≠ ∅ for all F ∈ Irr_τ(X).

The set of all weakly irreducibly open sets of (X, τ) is denoted by τ_SI₂(X). Complements of all weakly irreducibly open sets are called weakly irreducibly closed sets.

Lemma 3.1

Let (X, τ) be a T₀ space. Then τ_SI₂(X) is a topology on X.

Proof

Clearly ∅, X ∈ τ_SI₂(X);
It claims that U ⋂ V ∈ τ_SI₂(X) for any U, V ∈ τ_SI₂(X). Indeed, for any F ∈ Irr_τ(X), if F^δ ⋂ (U ⋂ V) ≠ ∅, then F^δ ⋂ U ≠ ∅ and F^δ ⋂ V ≠ ∅. Note U, V ∈ τ_SI₂(X), we have F ⋂ U ≠ ∅ and F ⋂ V ≠ ∅. Since F ∈ Irr_τ(X), we have F ⋂ (U ⋂ V) ≠ ∅. Clearly U ∩ V ∈ τ, so U ⋂ V ∈ τ_SI₂(X).
Assume that {U_i : i ∈ I} ⊆ τ_SI₂(X). Firstly, ⋃_i∈I U_i ∈ τ. Secondly, for any F ∈ Irr_τ(X), if F^δ ⋂ (⋃_i∈I U_i) = ⋃_i∈I(F^δ ⋂ U_i) ≠ ∅, then there exists some i ∈ I such that F^δ ⋂ U_i) ≠ ∅. Note that F ∈ Irr_τ(X) and U ∈ τ_SI₂(X), then F ⋂ U_i ≠ ∅. Thus we have F ⋂ (⋃_i∈I U_i) ≠ ∅. Therefore ⋃_i∈I U_i ∈ τ_SI₂(X). □

Remark 3.2

Let (X, τ) be a T₀ space. Then τ_SI₂(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_SI₂(X)) is also simply denoted by SI₂(X).

The following example is due to Erné (see [7, Example 2.5]).

Example 3.1

Let P be poset delineated by Figure 1 and B = {b_n: n = 0, 1, 2, …}, C = {c_k : k = 1, 2, …}. The order ≤ on P = B ∪ C is defined as follows:

$Figure 1 A set which is SI-open is not weakly irreducibly open.$

Figure 1

A set which is SI-open is not weakly irreducibly open.

↓ a₀ = {a₀} ∪ B,

↓ a₁ = {a₁, b₀} ∪ C,

↓ a₂ = {a₂, b₀, b₁} ∪ C,

↓ a_n = {a_n} ∪ {b_m : m < n}(n = 3, 4, …),

↓ b_n = {b_n}(n = 0, 1, 2, …),

↓ c_n = {c_m : m ≤ n}(n = 1, 2, …).

x ≤ y ⇔ x ∈ ↓ y. Endow P with the Alexandroff upper topology.

Then it is easy to see that ↑ b₀ is open in τ_SI(P). C = {c_k : k = 1, 2, …} is an irreducible set with b₀ ∈ C^δ ∩ ↑ b₀ ≠ ∅ while C ∩ ↑ b₀ = ∅, and whence ↑ b₀ ∉ τ_SI₂(α(P)). Thus τ_SI₂(α(P)) is proper contained in τ_SI(α(P)).

Definition 3.3

Let (X, τ) be a T₀ space. X is called SI₂-continuous if the following conditions are satisfied:

↡_r x is directed for all x ∈ X;
x ∈ ( ↡_r x)^δ for all x ∈ X;
↟_r x ∈ τ for all x ∈ X.

Lemma 3.2

Let P be a poset. Then SI₂(P, α(P)) = (P, σ₂(P)).

Remark 3.3

Let P be a poset. Then P is an s₂-continuous poset if and only if it is an SI₂-continuous space with respect to the Alexandroff upper topology.
Let (X, τ) be a T₀ space. If X is an SI₂-continuous space, then it is also an s₂-continuous poset under the specialization order. But the converse may not be true.

Example 3.2

Let X be an infinite set with a cofinite topology τ. Then it is a T₁ space. Clearly it is an antichain under the specialization order, and hence it is an s₂-continuous poset. But ↟_r x = {x} ∉ τ for all x ∈ X, then (X, τ) is not an SI₂-continuous space.

Let us note that an SI₂-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 SI₂-continuous, since every lower half-plane

Ea={(x,y)∈R×R:y≤a}

is a directed set with Eaδ = ℝ × ℝ and ⋂ {E_a : a ∈ ℝ} = ∅, thus ≪_r is empty.

The following theorem shows that the SI₂-continuity of the topological space has the interpolation property.

Theorem 3.1

Let X be an SI₂-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 SI₂-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 definition of the SI₂-continuous space.

Lemma 3.3

Let X be an SI₂-continuous space. Then for any x ∈ X, ↟_r x ∈ τ_SI₂(X).

Proof

It follows from Definition 3.3 and Theorem 3.1. □

Lemma 3.4

Let (X, τ) be a T₀ space and y ∈ int_{τ_SI₂(X)} ↑ x. Then x ≪_r y, where int_{τ_SI₂(X)} ↑ x denotes the interior of ↑ x with respect to the topology τ_SI₂(X).

Proof

Let y ∈ int_{τ_SI₂(X)} ↑ x. For every irreducible set E with y ∈ E^δ, we have y ∈ E^δ ∩ int_{τ_SI₂(X)} ↑ x ≠ ∅, and hence int_{τ_SI₂(X)} ↑ x ∩ E ≠ ∅. Thus there exists e ∈ int_{τ_SI₂(X)} ↑ x ∩ E. Thus we have x ≤ e and e ∈ E. This shows x ≪_r y. □

Theorem 3.2

Let (X, τ) be a T₀ space. Then the following statements are equivalent:

X is an SI₂-continuous space;
For all U ∈ τ_SI₂(X) and x ∈ U, there exists y ∈ U such that x ∈ int_{τ_SI₂(X)} ↑ y ⊆ ↑ y ⊆ U;
(τ_SI₂(X), ⊆) is a completely distributive lattice.

Proof

(1) ⇒ (2) Let X be an SI₂-continuous space and U ∈ τ_SI₂(X) with x ∈ U. Since x ∈ (↡_r x)^δ ⋂ U ≠ ∅, note that ↡_r x is directed(hence irreducible), then we have ↡_r x ⋂ U ≠ ∅. Thus there exists y ≪_r x such that y ∈ U. By Lemma 3.3, ↟_r y ∈ τ_SI₂(X), we have that x ∈ ↟_r y ⊆ ↑ y ⊆ U, that is, x ∈ int_{τ_SI₂(X)} ↑ y ⊆ ↑ y ⊆ U.

(2) ⇒ (1) For x ∈ X, consider the set E = {y ∈ X : x ∈ int_{τ_SI₂(X)} ↑ y}. Let y₁, y₂ ∈ E. Then x ∈ int_{τ_SI₂(X)} ↑ y₁ ⋂ int_{τ_SI₂(X)} ↑ y₂. By (2), there exists y ∈ int_{τ_SI₂(X)} ↑ y₁ ⋂ int_{τ_SI₂(X)} ↑ y₂ such that x ∈ int_{τ_SI₂(X)} ↑ y ⊆ ↑ y ⊆ int_{τ_SI₂(X)} ↑ y₁ ⋂ int_{τ_SI₂(X)} ↑ y₂, so y ∈ E and y₁, y₂ ≤ 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_SI₂(X)} ↑ y, which is open in τ. Hence X is an SI₂-continuous space.

(2) ⇔ (3) See [4]. □

Corollary 3.1

([7]) Let P be a poset. Then the following conditions are equivalent:

P is s₂-continuous;
For all U ∈ σ₂(P) and x ∈ U, there exists y ∈ U such that x ∈ int_σ₂(P) ↑ y ⊆ ↑ y ⊆ U;
(σ₂(P), ⊆) is a completely distributive lattice.

Corollary 3.2

([5]) Let (X, τ) be a T₀ space. Then the following statements are equivalent:

X is an SI-continuous space;
For all U ∈ τ_SI(X) and x ∈ U, there exists y ∈ U such that x ∈ int_{τ_SI(X)} ↑ y ⊆ ↑ y ⊆ U;
(τ_SI(X), ⊆) is a completely distributive lattice.

4 𝓓-convergence in SI₂-continuous spaces

In this section, the concept of 𝓓-convergence in a topological space is introduced. It is proved that a space X is SI₂-continuous if and only if the 𝓓-convergence with respect to the topology τ_SI₂(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.

Definition 4.1

Let (X, τ) be a T₀ 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

x ∈ D^δ;
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_j)_j∈J, x) with x ≡_𝓓 lim x_j. Then 𝓞(𝓓) = {U ⊆ X : whenever ((x_j)_j∈J, x) ∈ 𝓓 and x ∈ U, then eventually x_j ∈ U} is a topology.

Example 4.1

Let P = {a_j : j ∈ ℕ} ∪ {a, b}, where ℕ denotes the set of all natural numbers. The order on P is defined by a₁ ≤ a₂ ≤ ⋯ ≤ a, b ≤ a. Endow P with the Alexandroff upper topology. If x_j = a_j for all j ∈ ℕ, then (x_j)_j∈ℕ is a net. Take D = {a_j : j ∈ ℕ}, and then D is a directed subset of P with b ∈ D^δ. Moreover, for all d ∈ D, d ≤ x_j holds eventually, hence the net (x_j)_j∈ℕ converges to b.

Proposition 4.1

Let X be an SI₂-continuous space. Then x ≡_𝓓 lim x_j if and only if the net (x_j)_j∈J converges to the element x with respect to the topology τ_SI₂(X). That is, the 𝓓-convergence is topological.

Proof

Firstly, suppose that a net (x_j)_j∈J in X converges to x ∈ X and x ∈ U ∈ τ_SI₂(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_SI₂(X). For all y ∈ ↡_r x, then one has x ∈ ↟_r y ∈ τ_SI₂(X) by Lemma 3.3. Thus there exists k ∈ J such that x_j ∈ ↟_r y for any j ≥ k. By SI₂-continuity of X, we have that x ∈ (↡_r x)^δ and ↡_r x is directed. Thus ((x_j)_j∈J, x) ∈ 𝓓, that is, x ≡_𝓓 lim x_j. □

Proposition 4.2

Let (X, τ) be a T₀ space. If the 𝓓-convergence with respect to the topology τ_SI₂(X) is topological, then X is SI₂-continuous.

Proof

Suppose that the 𝓓-convergence with respect to the topology τ_SI₂(X) is topological. For all x ∈ X, take J = {(U, n, a) ∈ N(x) × ℕ × X : a ∈ U}, where N(x) consists of all weakly irreducibly open sets which contain x, and define an order on J as follows: (U, m, a) ≤ (V, n, b) if and only if V is proper subset of U or U = V and m ≤ n. Obviously, J is directed. Let x_j = a for all j = (U, m, a) ∈ J. Then it is not hard to show that the net (x_j)_j∈J converges to x with respect to the weakly irreducible topology τ_SI₂(X). By the condition, one has x ≡_𝓓 lim x_j. Thus there exists a directed set D with respect to the specialization order such that x ∈ D^δ and for all d ∈ D, d ≤ x_j holds eventually. Then there exists k = (U, m, a) ∈ J such that (V, n, b) = j ≥ k implies d ≤ x_j = b for all d ∈ D. Especially one has (U, m + 1, b) ≥ (U, m, a) = k for all b ∈ U. So x ∈ U ⊆ ↑ d. It follows that D ⊆ ↓ x and x ∈ int_{τ_SI₂(X)} ↑ d. By Lemma 3.4, d ≪_r x, and then D ⊆ ↡_r x. Thus x ∈ D^δ ⊆ (↡_r x)^δ. 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 J = {(U, n, a) ∈ N(x) × ℕ × X : a ∈ U} with I = {(U, n, a) ∈ N(y) × ℕ × X : a ∈ U}, where N(y) consists of all weakly irreducibly open sets containing y, similarly we have that (x_i)_i∈I converges to y with respect to the topology τ_SI₂(X), and then there exists a directed set D₁ of eventual lower bounds of the net (x_i)_i∈I such that y ∈ D1δ . Note z ≪_r y. Thus there exists d₁ ∈ D₁ such that z ≤ d₁. For d₁ ∈ D₁, there exists i₀ = (U₁, m, a) such that d₁ ≤ x_i = b for all i ≥ i₀, where U₁ ∈ τ_SI₂(X). In particular, we have (U₁, m + 1, u) ≥ (U₁, m, a) = i₀ for all u ∈ U₁. We conclude that d₁ ≤ u, and hence y ∈ U₁ ⊆ ↑ d₁. Since d₁ ≤ u for u ∈ U₁ and x ≪_r z with z ≤ d₁, we have U₁ ⊆ ↟_r x. This shows that ↟_r x ∈ τ. Hence X is SI₂-continuous. □

From Propositions 4.1 and 4.2, we immediately have:

Theorem 4.1

Let (X, τ) be a T₀ space. Then the following statements are equivalent:

X is SI₂-continuous;
The 𝓓-convergence with respect to the topology τ_SI₂(X) is topological.

Similarly, we also have:

Theorem 4.2

Let (X, τ) be a T₀ space. Then the following statements are equivalent:

X is SI-continuous;
The 𝓓-convergence with respect to the topology τ_SI(X) is topological.

Corollary 4.1

([17]) Let P be a poset. Then the following statements are equivalent:

P is s₂-continuous;
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:

P is a domain;
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.

5 Conclusion

At the Sixth International Symposium on Domain Theory, J.D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of T₀ spaces instead of restricting to posets. In this paper, we introduce a new way-below relation in T₀ topological spaces based on the 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 τ_SI₂(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 first condition of the definition of the 𝓓-convergence, whether we can change directed set into irreducible set?

Acknowledgement

The research was supported by the National Natural Science Foundation of China (11661057, 11661055, 61866023, 11561046), GAN PO555 Program of Jiangxi Province and the Natural Science Foundation of Jiangxi Province (20192ACBL20045, 20151BAB201020).

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Received: 2019-03-05

Accepted: 2019-10-28

Published Online: 2019-12-31

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2019-0123

Schlagwörter für diesen Artikel

s2-continuous poset; weakly irreducible topology; SI2-continuous space; 𝓓-convergence

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BY 4.0

On a new convergence in topological spaces

Artikel

Abstract

1 Introduction

2 Preliminaries

Remark 2.1

Definition 2.1

Definition 2.2

Definition 2.3

Definition 2.4

3 SI2-continuous spaces

Definition 3.1

Remark 3.1

Definition 3.2

Lemma 3.1

Proof

Remark 3.2

Example 3.1

Definition 3.3

Lemma 3.2

Remark 3.3

Example 3.2

Example 3.3

Theorem 3.1

Proof

Remark 3.4

Lemma 3.3

Proof

Lemma 3.4

Proof

Theorem 3.2

Proof

Corollary 3.1

Corollary 3.2

4 𝓓-convergence in SI2-continuous spaces

Definition 4.1

Example 4.1

Proposition 4.1

Proof

Proposition 4.2

Proof

Theorem 4.1

Theorem 4.2

Corollary 4.1

Corollary 4.2

5 Conclusion

Acknowledgement

References

Artikel in diesem Heft

Artikel in diesem Heft

Artikel in diesem Heft

3 SI₂-continuous spaces

4 𝓓-convergence in SI₂-continuous spaces