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On SI2-convergence in T0-spaces

  • Yi Yang and Xiaoquan Xu EMAIL logo
Published/Copyright: June 5, 2025
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Open Mathematics
From the journal Open Mathematics Volume 23 Issue 1

Abstract

Recently, Shen et al. showed that the SI2-topology on a T 0 -space can be described completely in terms of SI2-convergence, and the SI2-convergence is topological whenever the given space is SI2-continuous. In this article, we give a characterization of T 0 -spaces for the SI2-convergence being topological by introducing the notion of strongly I2-continuous spaces, which are strictly weaker than SI2-continuous spaces but are more closely related to the SI2-convergence. Moreover, as a common generalization of the irr-convergence and the S -convergence, we introduce the concept of SI 2 * -convergence in T 0 -spaces and the related concept of SI 2 * -continuous spaces. It is proved that if a T 0 -space X is SI 2 * -continuous, then the SI 2 * -convergence in X is topological.

Keywords: irreducible set; SI2-convergence; SI2-continuous space; strongly I2-continuous space; SI2 *-convergence; SI2 *-continuous space
MSC 2010: 54A20; 06B35; 06F30

1 Introduction

Convergence and convergence class play an important role in both order theory and general topology [1,2]. For a topological space ( X , τ ) and a class ℒ consisting of pairs ( ( x i ) i ∈ I , x ) , where ( x i ) i ∈ I is a net in X and x a point of X , the topology τ can naturally induce a convergence class as follows:

C ( τ ) = { ( ( x i ) i ∈ I , x ) : ( x i ) i ∈ I is a net in X , x ∈ X and for any U ∈ τ , x ∈ U implies that ( x i ) i ∈ I is eventually in U } .

And we can define a topology on X associated with ℒ :

O ( ℒ ) = { U ⊆ X : ( ( x i ) i ∈ I , x ) ∈ ℒ and x ∈ U imply x i ∈ U eventually } .

It is easy to verify that τ = O ( C ( τ ) ) . However, if ℒ is not a convergence class in the sense of Kelley [2], then the convergence class C ( O ( ℒ ) ) ≠ ℒ , that is, the class ℒ is not topological.

Numerous researchers have studied various types of convergences [1,3–11]. With different convergence, they have not only proposed the corresponding continuity of posets (more generally, topological spaces) but also presented some links between order theory and topology. In [1], it was proved that the lim-inf convergence in a dcpo P is topological iff the poset P is a continuous domain. This result was generalized to partially ordered sets (posets) in [3]. In [4], using the cut operator instead of joins, Ruan and Xu introduced and discussed S -convergence and GS -convergence in posets. They proved that a poset P is s2-continuous (resp., s2-quasicontinuous) iff the S -convergence (resp., the GS -convergence) in P is topological.

In the invited talk at the Sixth International Symposium on Domain Theory in 2013, Jimmie Lawson emphasized the need to develop the core of domain theory directly in T 0 -spaces to instead posets. In this direction, by using irreducible sets instead of directed sets, Zhao and Ho [12] introduced the SI-topology on T 0 -spaces as a generalization of the Scott topology on posets. In [5], Andradi et al. defined SI-convergence in T 0 -spaces and proved that for any T 0 -space X having condition (I * ), X is an I-continuous space iff SI-convergence in X is topological. Later, Lu and Zhao [6] gave a characterization of T 0 -spaces for the SI-convergence being topological. In [7], Zhao et al. provided a different way to define irreducible convergence in T 0 -spaces, which can be seen as a topological counterpart of lim-inf convergence in posets, and presented a sufficient and necessary condition for irreducible convergence to be topological in T 0 -spaces. By using the cut operator of irreducible sets and the specialization order of a given T 0 -space, Shen et al. [8] defined the SI2-topology on T 0 -spaces and proved that SI2-convergence on a T 0 -space X is topological whenever the space X is SI2-continuous. This naturally raises a question whether there is a characterization of T 0 -spaces for the SI2-convergence to be topological.

In this article, we introduce a new way-below relation on T 0 -spaces, called the I 2 -way-below relation. By using the I 2 -way-below relation, we introduce the notions of I2-continuity and strongly I2-continuity of T 0 -spaces, both of them are strictly weaker than the SI2-continuity but are more closely related to the SI2-convergence. We prove that the SI2-convergence in a T 0 -space X is topological iff X is strongly I2-continuous, giving an positive answer to the aforementioned question. Moreover, we define and study the SI 2 * -convergence in T 0 -spaces, which can be seen as topological counterparts of the S -convergence and the irr-convergence in posets. The related concept of SI 2 * -continuous spaces is also introduced. It is proved that if a T 0 -space X is SI 2 * -continuous, then the SI 2 * -convergence in X is topological.

2 Preliminaries

In this section, we briefly recall some basic concepts and results about ordered structures and T 0 -spaces that will be used in the article. For further details, we refer the reader to [1–2,13–14].

For a poset P and A ⊆ P , define ↑ A = { x ∈ P : a ≤ x for some a ∈ A } and ↓ A = { x ∈ P : x ≤ a for some a ∈ A } . For x ∈ X , let ↑ x = ↑ { x } and ↓ x = ↓ { x } . A subset A is called a lower set (resp., an upper set) if A = ↓ A (resp., A = ↑ A ). Define A ↑ = { u ∈ P : A ⊆ ↓ u } (the sets of all upper bounds of A in P ) and A ↓ = { v ∈ P : A ⊆ ↑ v } (the sets of all lower bounds of A in P ). The set A δ = ( A ↑ ) ↓ is called the c u t of A in P . If the set of upper bounds of A has a unique smallest element (that is, the set of upper bounds contains exactly one of its lower bounds), we call this element the least upper bound and write it as ∨ A or sup A (for supremum). Similarly the greatest lower bound is written as ⋀ A or inf A (for infimum).

The set of all natural numbers is denoted by N . When N is regarded as a poset (in fact, a chain), the order on N is the usual order of natural numbers. A nonempty subset D of a poset P is called directed if every finite subset of D has an upper bound in D . The set of all directed sets of P is denoted by D ( P ) . The poset P is called a directed complete poset, or dcpo for short, if for any D ∈ D ( P ) , ∨ D exists in P .

Let P be a poset and a , b ∈ P . We say that a is way below b , in symbols a ≪ b , if for all D ∈ D ( P ) for which ∨ D exists in P , b ≤ ∨ D implies a ∈ ↓ D . The poset P is called a continuous poset if for any a ∈ P , the set ↡ a ≔ { b ∈ P : b ≪ a } is directed and a = ∨ ↡ a . A subset U of P is Scott open if (i) U = ↑ U , and (ii) for any directed subset D for which ∨ D exists, ∨ D ∈ U implies D ∩ U ≠ ∅ . The topology formed by all the Scott open sets of P is called the Scott topology, written as σ ( P ) . The upper sets of P form the (upper) Alexandroff topology α ( P ) . The topology generated by the collection of sets P \ ↑ x (as subbasic open subsets) is called the lower topology and denoted it by ω ( P ) ; dually, the upper topology on a poset P , generated by the complements of the principal ideals of P , is denoted by υ ( P ) .

A net ( x i ) i ∈ I in a set X is a mapping from a directed set I to X . For each x ∈ X , one can define a constant net with the value x by x i = x for all i ∈ I . We denote this constant net by ( x ) i ∈ I . If Q ( x ) is a property of the elements x ∈ X , we say that Q ( x ) holds eventually in the net ( x i ) i ∈ I if there is a i 0 ∈ J such that Q ( x i ) is true whenever i 0 ≤ i .

Definition 2.1

[1] We say a net ( x i ) i ∈ I lim-inf converges to x in a poset P if there exists a directed subset D of P such that

  1. ∨ D exists and x ≤ ∨ D , and

  2. for every d ∈ D , d ≤ x i holds eventually, i.e., there exists i 0 ∈ I such that d ≤ x i for all i ≥ i 0 .

Definition 2.2

[4] Let P be a poset and ( x j ) j ∈ J a net in P .

  1. A point y ∈ P is called an eventual lower bound of a net ( x j ) j ∈ J in P , if there exists a k ∈ J such that y ≤ x j for all j ≥ k , i.e., ( x j ) j ∈ J is eventually in ↑ y .

  2. Let S ( P ) denote the class of those pairs ( ( x j ) j ∈ J , x ) such that x ∈ D δ for some directed set D of eventual lower bounds of the net ( x j ) j ∈ J . For each such pair, we again say that x is an S -limit of ( x j ) j ∈ J or ( x j ) j ∈ J S -converges to x , and write ( x j ) j ∈ J ⟶ S x .

As in [9], an upper subset U of a poset P is called s 2 -open if for any directed subset D of P , D δ ∩ U ≠ ∅ implies D ∩ U ≠ ∅ . The collection of all s 2 -open subsets of P forms a topology, called s 2 -topology, and is denoted by s 2 ( P ) . It is easy to see that s 2 ( P ) = O ( S ( P ) ) = { U ⊆ P : whenever x i ⟶ S x and x ∈ U , then eventually x i ∈ U } . The way-below relation ≪ 2 on P is defined by x ≪ 2 y iff for any directed subset D of P , y ∈ D δ implies x ∈ ↓ D .

Lemma 2.3

[5] Let P be a poset, D a nonempty subset of P, and ( x i ) i ∈ I a net in P. Then the following conditions are equivalent:

  1. D is a set of eventual lower bounds of ( x i ) i ∈ I .

  2. For every upper set U of P, D ∩ U ≠ ∅ implies x i ∈ U eventually.

For a T 0 -space X , let ≤ X denote the specialization order on X : x ≤ X y iff x ∈ { y } ¯ . In the following, when a T 0 -space X is considered as a poset, the order always refers to the specialization order if no other explanation. The pair ( X , ≤ X ) is denoted by Ω X or simply by X if no confusion arises, and sometimes we briefly write ≤ instead ≤ X . Let O ( X ) (resp., Γ ( X ) ) be the set of all open subsets (resp., closed subsets) of X . Clearly, each open set is an upper set and each closed set is a lower set with respect to the specialization order ≤ X . For a subset of X , denote the closure of A in X by cl X A or simply by cl A and the interior of A in X by int X A in X or simply by int A . We also simply use A ¯ to denote the the closure of A if no confusion arises.

A nonempty subset A of a T 0 -space X is called an irreducible set if for any F 1 , F 2 ∈ Γ ( X ) , A ⊆ F 1 ∪ F 2 implies A ⊆ F 1 or A ⊆ F 2 . We denote by Irr ( X ) (resp., Irr c ( X ) ) the set of all irreducible (resp., irreducible closed) subsets of X . Clearly, every subset of X that is directed under ≤ X is irreducible and the nonempty irreducible sets of a poset equipped with the Alexandroff topology are exactly the directed sets of P . And we said that X is irreducible complete space if every irreducible subset of X has a sup.

Lemma 2.4

[15] If f : X → Y is continuous and A ∈ Irr ( X ) , then f ( A ) ∈ Irr ( Y ) .

For a set X and a class ℒ consisting of pairs ( ( x i ) i ∈ I , x ) , where ( x i ) i ∈ I is a net in X and x is a point of X , the topology on X associated with ℒ is denoted by O ( ℒ ) , that is, O ( ℒ ) = { U ⊆ X : ( ( x i ) i ∈ I , x ) ∈ ℒ and x ∈ U imply x i ∈ U eventually } .

Definition 2.5

[5,7] Let X be a T 0 -space.

  1. A net ( x i ) i ∈ I of X is said to irreducibly converge to a point x of X , if there exists an irreducible set F of X with ∨ F existing such that x ≤ ∨ F , and for each e ∈ F , e ≤ x i holds eventually. In this case, we write ( x i ) i ∈ I ⟶ Irr x .

  2. A net ( x i ) i ∈ I of X is said to SI-converge to a point x of X , if there exists an irreducible set F of X with ∨ F existing such that x ≤ ∨ F , and for every U ∈ O ( X ) , F ∩ U ≠ ∅ implies x i ∈ U eventually. In this case, we write ( x i ) i ∈ I ⟶ SI x .

An open subset U of T 0 -space X is called SI-open if for any F ∈ Irr ( X ) , ∨ F ∈ U implies F ∩ U ≠ ∅ whenever ∨ F exists. The collection of all SI-open sets, denoted by O SI ( X ) , is a topology on X , called the irreducibly-derived topology (shortly SI-topology). The space ( X , O SI ( X ) ) will also be simply written as SI ( X ) . In [7], Zhao et al. denoted by τ Irr the topology induced by irr-convergence.

Proposition 2.6

[5] For any T 0 -space X, the SI-topology coincides with the topology induced by SI-convergence, namely, V ∈ O SI ( X ) iff for every net ( x i ) i ∈ I in X , ( x i ) i ∈ I ⟶ SI x and x ∈ V imply x i ∈ V eventually.

Definition 2.7

[5,7,12] Let X be a T 0 -space and x , y ∈ X . We say

  1. x is SI-way-below y , in symbols x ≪ SI y , if for any irreducible set F of X , y ≤ ∨ F implies x ∈ ↓ F whenever ∨ F exists.

  2. x is I-way-below y , in symbols x ≪ I y , if for every irreducible set F of X with ∨ F existing, y ≤ ∨ F implies x ∈ cl F .

  3. x is Irr-way-below y , in symbols x ≪ Irr y , if for every net ( x i ) i ∈ I in X irreducibly converging to y , x ≤ x i holds eventually.

Definition 2.8

[8] Let X be a T 0 -space. A subset U of X is called SI2-open if the following two conditions are satisfied:

  1. U is an open set in X , and

  2. for any F ∈ Irr ( X ) , F δ ∩ U ≠ ∅ implies F ∩ U ≠ ∅ .

The set of all SI2-open sets in X is denoted by O SI 2 ( X ) . It is straightforward to verify that O SI 2 ( X ) is a topology on X , called the SI2-topology. The space ( X , O SI 2 ( X ) ) will also be simply written as SI 2 ( X ) .

Definition 2.9

[8] Let X be a T 0 -space and x , y ∈ X .

  1. We say that x is SI2-way-below y , in symbols x ≪ SI 2 y , if for all irreducible set F of X , the relation y ∈ F δ always implies x ∈ ↓ F . We write ↡ SI 2 a = { x ∈ X : x ≪ SI 2 a } and ↟ SI 2 a = { x ∈ X : a ≪ SI 2 x } .

  2. The space X is called SI2-continuous if for any x ∈ X , ↟ SI 2 x ∈ O ( X ) , ↡ SI 2 x ∈ Irr ( X ) and x = ∨ ↡ SI 2 x .

By Remark 5.1(1) and Proposition 5.6 of [8], we obtain the following result.

Proposition 2.10

For a T 0 -space X, the following conditions are equivalent:

  1. X is SI2-continuous.

  2. For all x ∈ X , ↟ SI 2 x ∈ O ( X ) , ↡ SI 2 x ∈ Irr ( X ) , and x = ( ↡ SI 2 x ) δ .

  3. For all x ∈ X , ↟ SI 2 x is SI2-open, ↡ SI 2 x ∈ Irr ( X ) and x = ∨ ↡ SI 2 x .

  4. For all x ∈ X , ↟ SI 2 x is SI2-open, ↡ SI 2 x ∈ Irr ( X ) and x = ( ↡ SI 2 x ) δ .

Throughout this article, when we say X is a space, it always means X is a T 0 -space. For x ∈ X and a net ( x i ) i ∈ I in X , we use the symbols ( x i ) i ∈ I ⟶ x to represent that the net ( x i ) i ∈ I converges to x in the space X .

3 I2-continuous spaces and strongly I2-continuous spaces

In this section, we introduce the notions of I2-continuous spaces and strongly I2-continuous spaces, and discuss some basic properties of these spaces. Especially, we prove that a T 0 -space X is strongly I2-continuous iff SI2-convergence on X is topological.

We first recall the definition of SI2-convergence and give some its properties.

Definition 3.1

[8] We say a net ( x i ) i ∈ I SI2-converges to a point x in a T 0 -space X if there exists an irreducible set F in X such that

  1. x ∈ F δ and

  2. for any U ∈ O ( X ) , F ∩ U ≠ ∅ implies x i ∈ U eventually.

And in this case, we write ( x i ) i ∈ I ⟶ SI 2 x . Let Sℐ 2 ( X ) = { ( ( x i ) i ∈ I , x ) : ( x i ) i ∈ I is a net in X , x ∈ X and ( x i ) i ∈ I ⟶ SI 2 x } .

Remark 3.2

For a T 0 -space X , we have the following statements:

  1. The constant net ( x ) i ∈ I in X with value x SI2-converges to x .

  2. If ( x i ) i ∈ I ⟶ SI 2 x in X , then ( x i ) i ∈ I ⟶ SI 2 y for any y ≤ x . Thus, the SI2-convergence points of a net are generally not unique.

  3. Let P be a poset. Then the SI2-convergence in ( P , α ( P ) ) coincides with the S -convergence in P .

  4. If X is irreducible complete, then for any net ( x i ) i ∈ I in X , ( x i ) i ∈ I SI-converges to x ∈ X iff ( x i ) i ∈ I SI2-converges to x .

Lemma 3.3

[8] For any T 0 -space X, the two topologies O ( Sℐ 2 ( X ) ) and O SI 2 ( X ) coincide, that is, O SI 2 ( X ) = { U ⊆ P : whenever ( x i ) i ∈ I ⟶ SI 2 x and x ∈ U , then eventually x i ∈ U } .

Recall that a net ( y j ) j ∈ J is a subnet of ( x i ) i ∈ I if (i) there exists a function g : J → I such that y j = x g ( j ) for all j ∈ J , and (ii) for each i ∈ I there exists j ′ ∈ J such that g ( j ) ≥ i whenever j ≥ j ′ .

Proposition 3.4

Let X be a T 0 -space and A ⊆ X . Then the following conditions are equivalent:

  1. A is an SI 2 -closed set.

  2. A is a closed subset of X, and for any irreducible set F in X, F ⊆ A implies F δ ⊆ A .

  3. For any net ( x i ) i ∈ I in A , if ( x i ) i ∈ I ⟶ SI 2 x , then x ∈ A .

Proof

(1) ⇔ (2): See [8, Proposition 3.6].

(1) ⇒ (3): Let ( x i ) i ∈ I be a net in A and ( x i ) i ∈ I ⟶ SI 2 x . If x ∉ A , then x ∈ X \ A . Since A is an SI2-closed set, X \ A is SI2-open, and hence, X \ A ∈ O SI 2 ( X ) by Lemma 3.3. Then the net ( x i ) i ∈ I must be eventually in X \ A , being a contradiction with the fact that ( x i ) i ∈ I is in A . Thus, x ∈ A .

(3) ⇒ (1): We show that X \ A is SI 2 -open. Let x ∈ X \ A and ( x i ) i ∈ I ⟶ SI 2 x . Then the net ( x i ) i ∈ I is eventually in X \ A . Otherwise, for each i ∈ I , there exists a φ ( i ) ∈ I with φ ( i ) ≥ i such that x φ ( i ) ∈ A . Let J be the subset of I consisting of all j ∈ I such that x j ∈ A . Then J is cofinal in I , and ( x j ) j ∈ J is a subnet of ( x i ) i ∈ I . As ( x i ) i ∈ I ⟶ SI 2 x , we have ( x j ) j ∈ J ⟶ SI 2 x , and hence, x ∈ A by (3), which contradicts x ∈ X \ A . Then we conclude that the net ( x i ) i ∈ I is eventually in X \ A . Hence, X \ A ∈ O ( Sℐ 2 ( X ) ) . By Lemma 3.3, A is SI2-closed.□

Lemma 3.5

Let X be a T 0 -space and F be an irreducible set of X with x ∈ F δ . Then there exists a net ( x i ) i ∈ I in X such that all of its terms are in F and ( x i ) i ∈ I SI 2 -converges to x.

Proof

Let I = { ( U , n , e ) ∈ O ( X ) × N × F : e ∈ U } and define an order on I by the lexicographic order on the first two coordinates, that is, ( U , m , a ) < ( V , n , b ) iff V is a proper subset of U or U = V and m < n . For any ( U 1 , n 1 , e 1 ) , ( U 2 , n 2 , e 2 ) ∈ I , we have e 1 ∈ F ∩ U 1 and e 2 ∈ F ∩ U 2 . By the irreducibility of F , we have F ∩ U 1 ∩ U 2 ≠ ∅ . Select e 3 ∈ F ∩ U 1 ∩ U 2 . Then ( U 1 , n 1 , e 1 ) , ( U 2 , n 2 , e 2 ) < ( U 1 ∩ U 2 , n 1 + n 2 + 1 , e 3 ) . Hence, I is a directed set. We let x ( U , n , e ) = e for any ( U , n , e ) ∈ I . Now we show that the net ( e ) ( U , n , e ) ∈ I SI 2 -converges to x . We firstly have that F ∈ Irr ( X ) and x ∈ F δ by the assumption. For any U ∈ O ( X ) with F ∩ U ≠ ∅ , select a d ∈ F ∩ U . Then ( U , 1 , d ) ∈ I and e ∈ U for all ( V , n , e ) ∈ I with ( V , n , e ) ≥ ( U , 1 , d ) , proving that ( e ) ( U , n , e ) ∈ I SI 2 -converges to x .□

Proposition 3.6

Let X , Y be T 0 -spaces and f be a continuous mapping from X to Y. Then the following two conditions are equivalent:

  1. f is a continuous mapping from SI 2 ( X ) to SI 2 ( Y ) .

  2. For any net ( x i ) i ∈ I and x ∈ X , ( x i ) i ∈ I ⟶ SI 2 x in X implies f ( x i ) i ∈ I ⟶ SI 2 f ( x ) in Y .

Proof

(1) ⇒ (2): First, f is order-preserving. In fact, if x ≤ X y , i.e., x ∈ cl { y } , then we have f ( x ) ∈ f ( { y } ¯ ) ⊆ f ( { y } ) ¯ by the continuity of f : X → Y , whence f ( x ) ≤ Y f ( y ) . Suppose that ( x i ) i ∈ I ⟶ SI 2 x in X . Now we show that f ( x i ) i ∈ I ⟶ SI 2 f ( x ) in Y . As ( x i ) i ∈ I ⟶ SI 2 x , there exists an irreducible set F in X such that conditions (i) and (ii) of Definition 3.1 are satisfied. Then f ( F ) ∈ Irr ( Y ) by Lemma 2.3. Since f is order-preserving, we obtain f ( x ) ∈ f ( F δ ) = f ( ( F ↑ ) ↓ ) ⊆ ( f ( F ↑ ) ) ↓ ⊆ ( f ( F ) ↑ ) ↓ = ( f ( F ) ) δ . For V ∈ O ( Y ) , if f ( F ) ∩ V ≠ ∅ , then F ∩ f − 1 ( V ) ≠ ∅ and f − 1 ( V ) ∈ O ( X ) by the continuity of f : X → Y , and consequently, x i ∈ f − 1 ( V ) eventually. Hence, f ( x i ) ∈ V eventually. Thus, f ( x i ) i ∈ I ⟶ SI 2 f ( x ) in Y .

(2) ⇒ (1): Let V ∈ O SI 2 ( Y ) . By the continuity of f : X → Y , we have f − 1 ( V ) ∈ O ( X ) . For any F ∈ Irr ( X ) , if F δ ∩ f − 1 ( V ) ≠ ∅ , then we can select a point a ∈ F δ ∩ f − 1 ( V ) . By Lemma 3.5, there exists a net ( a i ) i ∈ I F in F SI2-converging to a . By the assumption, the net ( f ( a i ) ) i ∈ I F SI2-converges to f ( a ) and f ( a ) ∈ V . Hence, by Lemma 3.3, f ( a i ) ∈ V eventually, or equivalently, a i ∈ f − 1 ( V ) eventually. It follows that F ∩ f − 1 ( V ) ≠ ∅ . We conclude that f − 1 ( V ) ∈ O SI 2 ( X ) , and therefore, (1) holds.□

In [8], Shen et al. proved that the SI2-convergence in a T 0 -space X is topological whenever the space X is SI2-continuous. This naturally raises a question whether there is a characterization of T 0 -spaces for the SI2-convergence to be topological. In the remainder of this section, we shall give such a characterisation.

First, we introduce a new notion of way-below relation.

Definition 3.7

Let X be a T 0 -space and x , y ∈ X . We say that x is I2-way-below y , in symbols x ≪ I 2 y , if for any irreducible set F in X , y ∈ F δ implies x ∈ cl F .

For a ∈ X , we write ↡ I 2 a = { x ∈ X : x ≪ I 2 a } and ↟ I 2 a = { x ∈ X : a ≪ I 2 x } .

Remark 3.8

For a T 0 -space X , the following statements hold for all u , x , y , z ∈ X :

  • (i) x ≪ I 2 y implies x ≤ y ;

  • (ii) u ≤ x ≪ I 2 y ≤ z implies u ≪ I 2 z ;

  • (iii) x ≪ I 2 y iff for every irreducible closed set F , y ∈ F δ implies x ∈ F ;

  • (iv) x ≪ SI 2 y implies x ≪ I 2 y . Hence, ↡ SI 2 x ⊆ ↡ I 2 x ⊆ ↓ x .

One can easily see that when X is a poset P endowed with the Alexandroff topology, the I2-way-below relation is exactly the way-below relation ≪ 2 (cf. [15, Fact 2.6]). When X is irreducible complete, we have x ≪ I 2 y iff x ≪ I y .

The following example shows that ≪ I 2 is different to ≪ SI and also different to ≪ I in general.

Example 3.9

Let Q = { a 1 , a 2 , … , a n , … } ∪ { b 1 , b 2 } ∪ { c } and define a partial order ≤ on Q as follows (see Figure 1):

  • (i) a 1 < a 2 < … < a n < a n + 1 < … ;

  • (ii) a n < b 1 , a n < b 2 for all n ∈ N ;

  • (iii) b 1 and b 2 are incomparable; and

  • (iv) c < b 1 and c < b 2 .

Consider the Alexandroff topology space ( Q , α ( Q ) ) . Then Irr ( ( Q , α ( Q ) ) ) = D ( Q ) (cf. [15, Fact 2.6]). It is easy to verify that for any D ∈ D ( Q ) , D has a largest element or D ⊆ { a n + 1 : n ∈ N } is countable infinite. Hence for any A ∈ Irr ( ( Q , α ( Q ) ) ) for which ∨ A exists, we have that c ≤ ∨ A implies c ∈ ↓ A . So c ≪ SI c and hence c ≪ I c . Let F = { a n + 1 : n ∈ N } . Then F ∈ Irr ( ( Q , α ( Q ) ) ) and c ∈ F δ = F ∪ { c } but c ∉ cl A = A . Thus, c ≪̸ I 2 c .

Figure 1 The poset Q Q in Example 3.9.
Figure 1

The poset Q in Example 3.9.

Example 3.15 shows that ≪ I 2 is different to ≪ SI 2 in general.

Proposition 3.10

Let X be a T 0 -space and x , y ∈ X . Then the following two conditions are equivalent:

  1. x ≪ I 2 y .

  2. For any net ( x i ) i ∈ I of X , ( x i ) i ∈ I ⟶ SI 2 y implies ( x i ) i ∈ I ⟶ x .

Proof

(1) ⇒ (2): Suppose that x ≪ I 2 y and ( x i ) i ∈ I is a net of X SI2-converging to y . We show that ( x i ) i ∈ I converges to x in the space X . As ( x i ) i ∈ I ⟶ SI 2 y , there exists an F ∈ Irr ( X ) such that y ∈ F δ , and F ∩ V ≠ ∅ implies x i ∈ V eventually for any V ∈ O ( X ) . Then we have x ∈ cl F by y ∈ F δ and x ≪ I 2 y . Hence, for any U ∈ O ( X ) with x ∈ U , it holds that F ∩ U ≠ ∅ , and then x i ∈ U eventually. Therefore, ( x i ) i ∈ I ⟶ x .

(2) ⇒ (1): Let F ∈ Irr ( X ) with y ∈ F δ . Then by Lemma 3.5, there exists a net ( x i ) i ∈ I F such that all of its terms are in F and it SI2-converges to y . So ( x i ) i ∈ I F ⟶ x . Then for any U ∈ O ( X ) with x ∈ U , we have x i ∈ U eventually. Since { x i : i ∈ I } ⊆ F , it holds that x i ∈ F ∩ U eventually, and hence, F ∩ U ≠ ∅ , proving that x ∈ cl F . Thus, x ≪ I 2 y .□

Definition 3.11

A T 0 -space X is called I2-continuous if for every x ∈ X , ↡ I 2 x ∈ Irr ( X ) and x ∈ ( ↡ I 2 x ) δ .

Remark 3.12

By Remark 3.8 (i), we can easily see that a T 0 -space X is I 2 -continuous iff for any x ∈ X , ↡ I 2 x ∈ Irr ( X ) and x = ∨ ↡ I 2 x .

Proposition 3.13

For a T 0 -space X, the following two conditions are equivalent:

  1. X is I 2 -continuous.

  2. For any x ∈ X , there exists F ∈ Irr ( X ) such that F ⊆ ↡ I 2 x and x = ∨ F .

Proof

(1) ⇒ (2): Let F = ↡ I 2 x . Then F ∈ Irr ( X ) and x = ∨ F .

(2) ⇒ (1): For x ∈ X , by the assumption, there exists an irreducible subset F ⊆ ↡ I 2 x such that x = ∨ F . Then F δ = ↓ ∨ F = ↓ x , and hence, x ∈ F δ . It follows that F ⊆ ↡ I 2 x ⊆ cl F . So cl ↡ I 2 x = cl F ∈ Irr c ( X ) . Then ↡ I 2 x ∈ Irr ( X ) and x = ∨ F = ∨ cl F = ∨ cl ↡ I 2 x = ∨ ↡ I 2 x . Thus, X is I 2 -continuous.□

By Remark 3.8(iv) and Proposition 3.13, we directly obtain the following corollary.

Corollary 3.14

Every SI 2 -continuous space is I 2 -continuous.

However, I2-continuous spaces are not SI2-continuous in general, as shown in the following example.

Example 3.15

Let X be a countable infinite set and X cof the space equipped with the co-finite topology (the empty set and the complements of finite subsets of X are open). Then

(a) X cof is a T 1 -space, and hence, its specialization order is the discrete order on X .

(b) Irr ( X cof ) = { { x } : x ∈ X } ∪ { A : A is a countable infinite set of X } and Irr c ( X cof ) = { { x } : x ∈ X } ∪ { X } .

(c) For any countable infinite set A of X , cl A = X .

(d) For x , y ∈ X , x ≪ I 2 y iff x = y by (b) and (c). So X cof is I2-continuous.

(e) For any x , y ∈ X , x ≪̸ SI 2 y .

In fact, if x ≪ SI 2 y , then as y ∈ { y } = { y } δ , we have x ∈ ↓ { y } = { y } and hence, x = y . By (b), X \ { x } ∈ Irr ( X cof ) and x ∈ X = ( X \ { x } ) δ , but x ∉ ↓ ( X \ { x } ) = X \ { x } , which is a contradiction with x ≪ SI 2 y . Thus, x ≪ SI 2 y for no x , y ∈ X .

(f) X cof is not SI2-continuous by (e).

Proposition 3.16

Let X be a T 0 -space, y ∈ X and ( x i ) i ∈ I be a net in X. Consider the following two conditions:

  1. ( x i ) i ∈ I ⟶ SI 2 y .

  2. For any x ≪ I 2 y , ( x i ) i ∈ I ⟶ x .

Then (1) ⇒ (2), and two conditions are equivalent if X is I 2 -continuous.

Proof

(1) ⇒ (2): By Proposition 3.10.

(2) ⇒ (1): Suppose that X is I 2 -continuous. Then ↡ I 2 y ∈ Irr ( X ) and y ∈ ( ↡ I 2 y ) δ . For any U ∈ O ( X ) , if ↡ I 2 y ∩ U ≠ ∅ , then there is x ∈ U such that x ≪ I 2 y , and hence, ( x i ) i ∈ I ⟶ x by (2). So x i ∈ U eventually. Thus, ( x i ) i ∈ I ⟶ SI 2 y .□

Proposition 3.17

Let X be a T 0 -space. If SI 2 -convergence in X is topological, then X is I 2 -continuous.

Proof

By Lemma 3.3, O ( Sℐ 2 ( X ) ) = O SI 2 ( X ) . Thus, if SI 2 -convergence in X is topological, we must have

( x i ) i ∈ I ⟶ SI 2 x iff ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) .

Let x ∈ X . Define

I = { ( U , n , a ) ∈ N SI 2 ( x ) × N × X : a ∈ U } ,

where N SI 2 ( x ) consists of all open sets containing x in the space ( X , O SI 2 ( X ) ) , and define an order on I by the lexicographic order on the first two coordinates, that is, ( U , m , a ) < ( V , n , b ) iff V is a proper subset of U or U = V and m < n . For any ( U 1 , n 1 , a 1 ) , ( U 2 , n 2 , a 2 ) ∈ I , we have U 1 ∩ U 2 ∈ N SI 2 ( x ) , and hence, ( U 1 ∩ U 2 , n 1 + n 2 + 1 , x ) ∈ I . Clearly, ( U 1 , n 1 , a 1 ) < ( U 1 ∩ U 2 , n 1 + n 2 + 1 , x ) and ( U 2 , n 2 , a 2 ) < ( U 1 ∩ U 2 , n 1 + n 2 + 1 , x ) . Thus, I is a directed set. Let x i = a for i = ( U , n , a ) ∈ I . It is easy to see that the net ( x i ) i ∈ I converges to x in ( X , O SI 2 ( X ) ) , and hence, ( x i ) i ∈ I ⟶ SI 2 x . So there exists an irreducible set F ∈ Irr ( X ) such that ( x i ) i ∈ I and F satisfy conditions (i) and (ii) of Definition 3.1. Now we show that F ⊆ ↡ I 2 x .

Suppose that s ∈ F . We verify that s ≪ I 2 x . Let E ∈ Irr ( X ) and x ∈ E δ . Then by Lemma 3.5, there exists a net ( e j ) j ∈ J such that all of its terms are in E and ( e j ) j ∈ J SI 2 -converges to x , and hence, it converges to x in the space ( X , O SI 2 ( X ) ) by Lemma 3.3.

For V ∈ O ( X ) with s ∈ V , we have s ∈ F ∩ V , and hence, F ∩ V ≠ ∅ . As ( x i ) i ∈ I and F satisfy conditions (i) and (ii) of Definition 3.1, there is i 0 = ( U 0 , m 0 , z ) ∈ I such that x i ∈ V for all i ≥ i 0 . For any t ∈ U 0 , ( U 0 , m 0 + 1 , t ) > ( U 0 , m 0 , z ) , whence t = x ( U 0 , m 0 + 1 , t ) ∈ V . So x ∈ U 0 ⊆ V . Since U 0 ∈ N SI 2 ( x ) and ( e j ) j ∈ J converges to x in ( X , O SI 2 ( X ) ) , e j ∈ U 0 eventually, and consequently, e j ∈ V eventually. Hence, E ∩ V ≠ ∅ (note that ( e j ) j ∈ J is a net in E ), proving that s ∈ cl X E .

In summary, we have proved that for any E ∈ Irr ( X ) with x ∈ E δ , s ∈ cl X E . Hence, s ≪ I 2 x . Thus, F ⊆ ↡ I 2 x . Therefore, F ∈ Irr ( F ) , F ⊆ ↡ I 2 x and x ∈ F δ . So by Proposition 3.13, X is I 2 -continuous.□

Proposition 3.18

Let X be a T 0 -space. If SI 2 -convergence in X is topological, then for any x , y ∈ X with x ≪ I 2 y and U ∈ O ( X ) with x ∈ U , there exists an SI 2 -open set W such that y ∈ W ⊆ U .

Proof

Suppose that x ≪ I 2 y , U ∈ O ( X ) and x ∈ U . Then y ∈ ↟ I 2 x ⊆ U . Consider the net ( y j ) j ∈ J similarly defined in the proof of Proposition 3.17, where J = { ( V , n , b ) ∈ N SI 2 ( y ) × N × X : b ∈ V } with the lexicographic order on the first two coordinates and y ( V , n , b ) = b for any ( V , n , b ) ∈ I . Then ( y j ) j ∈ J ⟶ SI 2 y (see the proof of Proposition 3.17). Hence, there exists an irreducible set M such that y ∈ M δ , and for any O ∈ O ( X ) , O ∩ M ≠ ∅ implies y j ∈ O eventually. By x ≪ I 2 y , we have x ∈ cl X M , and consequently, U ∩ M ≠ ∅ . So y j ∈ U eventually, more precisely, there is j 0 = ( W , l , c ) ∈ J such that y j ∈ U for all j ≥ j 0 . Then W is SI 2 -open. For any z ∈ W , we have ( W , l + 1 , z ) > ( W , l , c ) , whence z = y ( W , l + 1 , z ) ∈ U . So y ∈ W ⊆ U .□

Motivated by Propositions 3.17 and 3.18, we introduce the following concept.

Definition 3.19

A T 0 -space X is called strongly I2-continuous if the following two conditions hold:

  1. for any x ∈ X , ↡ I 2 x ∈ Irr ( X ) and x ∈ ( ↡ I 2 x ) δ (i.e., X is I2-continuous), and

  2. for any x , y ∈ X with x ≪ I 2 y and U ∈ O ( X ) with x ∈ U , there exists an SI2-open set W with y ∈ W ⊆ U .

Proposition 3.20

Let X be an I 2 -continuous space such that ↟ I 2 x is SI 2 -open for all x ∈ X . Then X is a strongly I 2 -continuous space.

Proof

We only need to verify condition (ii) of Definition 3.19. Let x , y ∈ X with x ≪ I 2 y and U ∈ N ( x ) . Then by the assumption ↟ I 2 x is SI 2 -open. By Remark 3.8(i), we obtain y ∈ ↟ I 2 x ⊆ U . Thus, X is strongly I 2 -continuous.□

Proposition 3.21

If X is an SI 2 -continuous space, then X is strongly I 2 -continuous.

Proof

By Corollary 3.14, it is sufficient to verify condition (ii) of Definition 3.19. Let x , y ∈ X with x ≪ I 2 y and U ∈ N ( x ) . Since X is SI 2 -continuous, ↡ SI 2 y ∈ Irr ( X ) and y ∈ ( ↡ SI 2 y ) δ (note that y = ∨ ↡ SI 2 y is equivalent to y ∈ ( ↡ SI 2 y ) δ ). As x ≪ I 2 y , we have x ∈ cl X ↡ SI 2 y , and hence, ↡ SI 2 y ∩ U ≠ ∅ by U ∈ N ( x ) . Select a point z ∈ ↡ SI 2 y ∩ U . Then ↟ SI 2 z ∈ O SI 2 ( X ) by Proposition 2.10 and y ∈ ↟ SI 2 z ⊆ U . So X is strongly I2-continuous.□

The converse of Proposition 3.21 may not be true, as shown in the following example.

Example 3.22

Let X cof be the space in Example 3.15. Then by Example 3.15, we have the following conclusions:

  1. X cof is an I2-continuous T 1 -space.

  2. X cof is not SI2-continuous.

  3. Irr ( X cof ) = { { x } : x ∈ X } ∪ { A : A is a countable infinite set of X } .

  4. For any s , t ∈ X , s ≪ I 2 t iff s = t .

Now we show that X cof is strongly I2-continuous. Suppose that x ≪ I 2 y and U ∈ N ( x ) . We first verify that U is SI2-open. For F ∈ Irr ( X cof ) with F δ ∩ U ≠ ∅ , by (c) F = { z } for some z ∈ X or F is a countable infinite set of X . Then F δ = { z } or F δ = X , and hence, z ∈ F ∩ U or F ∩ U ≠ ∅ by ∣ F ∣ = ω and U is an co-finite open set. So U is SI2-open, and by (d), we have y = x ∈ U ⊆ U . Thus, X is strongly I2-continuous.

Proposition 3.23

If X is a strongly I 2 -continuous space, then SI 2 -convergence in X is topological.

Proof

Let ( x i ) i ∈ I be a net in X and x ∈ X . Obviously, ( x i ) i ∈ I ⟶ SI 2 x implies that ( x i ) i ∈ I converges to x in ( X , O ( Sℐ 2 ( X ) ) ) . Conversely, suppose that ( x i ) i ∈ I converges to x in ( X , O ( Sℐ 2 ( X ) ) ) . Then by Lemma 3.3, ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) . We will show that ( x i ) i ∈ I ⟶ SI 2 x . Let F x = ↡ I 2 x . Then by the strong I2-continuity of X , we have that F x ∈ Irr ( X ) and x ∈ F x δ . For any U ∈ O ( X ) , if F x ∩ U ≠ ∅ , then we can select a u ∈ F x ∩ U . Hence, u ≪ I 2 x and U ∈ N ( u ) . By the strong I2-continuity of X again, there is an SI2-open set W such that x ∈ W ⊆ U . Since ( x i ) i ∈ I converges to x in ( X , O SI 2 ( X ) ) , there is i 0 ∈ I such that x i ∈ W ⊆ U for all i ≥ i 0 , proving that ( x i ) i ∈ I ⟶ SI 2 x . Thus, SI 2 -convergence is topological.□

By Lemma 3.3, Propositions 3.17, 3.18, and 3.23, we obtain the main result of this article.

Theorem 3.24

For a T 0 -space, the following conditions are equivalent:

  1. SI 2 -convergence X is topological.

  2. For any net ( x i ) i ∈ I in X and x ∈ X , ( x i ) i ∈ I ⟶ SI 2 x iff ( x i ) i ∈ I converges to x with respect to the SI2-topology O SI 2 ( X ) .

  3. X is strongly I 2 -continuous.

From Proposition 3.21 and Theorem 3.24 we directly deduce the following [8, Proposition 5.13].

Corollary 3.25

[8] If X is an SI 2 -continuous space, then the SI 2 -convergence in X is topological.

4 SI 2 * -continuous spaces

In this section, as a common generalization of the irr-convergence and the S -convergence, we introduce the concept of SI 2 * -convergence in T 0 -spaces and the related concept of SI 2 * -continuous spaces. Some basic properties of them are discussed. It is proved that if X is SI 2 * -continuous, then the SI 2 * -convergence in X is topological.

Definition 4.1

We say a net ( x i ) i ∈ I SI 2 * -converge to a point x in a T 0 -space X if there exists an irreducible set F in X such that

  1. x ∈ F δ , and

  2. for each e ∈ F , e ≤ x i holds eventually.

In this case, we write ( x i ) i ∈ I ⟶ SI 2 * x . Let Sℐ 2 * ( X ) = { ( ( x i ) i ∈ I , x ) : ( x i ) i ∈ I is a net in X , x ∈ X and ( x i ) i ∈ I ⟶ SI 2 * x } .

Remark 4.2

For a T 0 -space X a net ( x i ) i ∈ I in X , we have the following statements:

  1. The constant net ( x ) j ∈ J in X with value x SI 2 * -converges to x .

  2. If ( x i ) i ∈ I ⟶ SI 2 * x in X , then ( x i ) i ∈ I ⟶ SI 2 * y for any y ≤ x . So the SI 2 * -convergence points of a net are generally not unique.

  3. ( x i ) i ∈ I ⟶ SI 2 * x implies ( x i ) i ∈ I ⟶ SI 2 x . In fact, if ( x i ) i ∈ I ⟶ SI 2 * x , then there exists an irreducible set F of eventual lower bounds of ( x i ) i ∈ I such that x ∈ F δ . For any U ∈ O ( X ) , if F ∩ U ≠ ∅ , then we can select an e ∈ F ∩ U . Hence, e ≤ x i holds eventually, and consequently, x i ∈ ↑ U = U eventually. Thus, ( x i ) i ∈ I ⟶ SI 2 x .

  4. Let P be a poset and ( s j ) j ∈ J be a net in P . Then ( s j ) j ∈ J SI 2 * -converges to s in ( P , α ( P ) ) iff ( s j ) j ∈ J S -converges to s iff ( s j ) j ∈ J SI2-converges to s by Lemma 2.3.

Definition 4.3

Let X be T 0 -space. Then

O ( Sℐ 2 * ( X ) ) = { U ⊆ X : whenever ( x i ) i ∈ I ⟶ SI 2 * x and x ∈ U , then eventually x i ∈ U }

is a topology, called the SI 2 * -topology on X . A subset U of X is said to be SI 2 * -open if U ∈ O ( Sℐ 2 * ) . Complements of SI 2 * -open sets are called SI 2 * -closed sets.

Lemma 4.4

Let X be T 0 -space and A ⊆ X . Then the following two conditions are equivalent:

  1. A is SI 2 * -closed.

  2. For any net ( x i ) i ∈ I in A , ( x i ) i ∈ I ⟶ SI 2 * x implies x ∈ A .

Proof

(1) ⇒ (2): Let ( x i ) i ∈ I be a net in A and ( x i ) i ∈ I ⟶ SI 2 * x . If x ∉ A , then x ∈ X \ A ∈ O ( Sℐ 2 * ( X ) ) . Hence, the net ( x i ) i ∈ I must be eventually in X \ A , being a contradiction with the fact that ( x i ) i ∈ I is in A . Thus, x ∈ A .

(2) ⇒ (1): We show that X \ A is SI 2 * -open. Let ( x i ) i ∈ I ⟶ SI 2 * x and x ∈ X \ A . Then x i ∈ X \ A eventually. Otherwise, for each i ∈ I , there exists a φ ( i ) ∈ I with φ ( i ) ≥ i such that x φ ( i ) ∈ A . Let J be the subset of I consisting of all j ∈ I such that x j ∈ A . Then J is cofinal in I and ( x j ) j ∈ J is a subnet of ( x i ) i ∈ I . As ( x i ) i ∈ I ⟶ SI 2 * x , we have ( x j ) j ∈ J ⟶ SI 2 * x , and hence, x ∈ A by the assumption, which contradicts x ∈ X \ A . So x i ∈ X \ A eventually. Hence, X \ A ∈ O ( Sℐ 2 * ( X ) ) , that is, A is SI 2 * -closed.□

Remark 4.5

For a T 0 -space X , we have the following statements:

  1. If U ⊆ X is an SI2-open set, then U is SI 2 * -open, that is, O SI 2 ( X ) ⊆ O ( Sℐ 2 * ( X ) ) .

  2. If ( x i ) i ∈ I ⟶ SI 2 * x , then ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) .

Proof

(1) Let U ∈ O SI 2 ( X ) . Then U ∈ O ( Sℐ 2 ( X ) ) by Lemma 3.3. It follows from Remark 4.2(3) that U ∈ O ( Sℐ 2 * ( X ) ) .

(2) Suppose ( x i ) i ∈ I ⟶ SI 2 * x . Then ( x i ) i ∈ I converges to x in ( X , O ( Sℐ 2 * ( X ) ) ) . By (1), we have that ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) .□

The following example shows that for a T 0 -space X , O SI 2 ( X ) generally does not agree with O ( Sℐ 2 * ( X ) ) .

Example 4.6

Let X cof be the space in Example 3.15. Then we have the following conclusions:

  1. X cof is a T 1 -space and hence the specialization order of X c o c is the discrete order.

  2. Irr ( X cof ) = { { x } : x ∈ X } ∪ { A : A is a countable infinite set of X } .

  3. For any x ∈ X , { x } is not open in X cof , and hence, { x } ∉ O SI 2 ( X ) .

  4. For any x ∈ X , { x } ∈ O ( Sℐ 2 * ( X ) ) .

Suppose ( x i ) i ∈ I ⟶ SI 2 * x . Then there exist an F ∈ Irr ( X cof ) such that conditions (i) and (ii) of Definition 4.1 hold. For any two points e 1 , e 2 ∈ F , since F satisfies condition (ii) of Definition 4.1, there is ( i 1 , i 2 ) ∈ I × I such that e 1 ≤ x i and e 2 ≤ x j for any i ≥ i 1 and j ≥ i 2 . As I is directed, there is i 3 ∈ I such that i 3 ∈ ↑ i 1 ∩ ↑ i 2 . Then for any i ≥ i 3 , e 1 = x i = e 2 (note that the specialization order of X c o c is the discrete order). Hence, F is a single point set. So x ∈ F δ = F and x i ∈ { x } eventually. Thus, { x } ∈ O ( Sℐ 2 * ( X ) ) .

Now we give an example to show that for a T 0 -space X , O ( Sℐ 2 * ( X ) ) generally does not agree with O ( X ) .

Example 4.7

Let L = N ∪ { ⊤ } , where N is the set of all natural numbers N = { 1 , 2 , 3 , … , n , … } , as a poset with the partial order defined by for any n ∈ N , n < n + 1 and n < ⊤ . We consider the Alexandroff topological spaces ( L , α ( L ) ) . Obviously, { ⊤ } = ↑ ⊤ ∈ α ( L ) . For any n ∈ N , let x n = n . Set F = { n : n ∈ N } . Then F ∈ Irr ( L , α ( L ) ) , ⊤ ∈ L = F δ , and for each n ∈ F , n ≤ x m holds eventually. Thus, ( x n ) n ∈ N ⟶ SI 2 * ⊤ . But x n ∉ { ⊤ } for any n ∈ N . So { ⊤ } ∉ O ( Sℐ 2 * ( X ) ) .

Lemma 4.8

Let X be an SI 2 -continuous space, x ∈ X and ( x i ) i ∈ I be a net in X. Then the following three condition are equivalent:

  1. ( x i ) i ∈ I ⟶ SI 2 * x .

  2. ( x i ) i ∈ I ⟶ SI 2 x .

  3. ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) .

Proof

(1) ⇒ (2): By Remark 4.5.

(2) ⇔ (3): By Proposition 3.21 and Theorem 3.24.

(3) ⇒ (1): Let F = ↡ SI 2 x . Then F ∈ Irr ( X ) and x ∈ F δ by the SI 2 -continuity of X . For any e ∈ F , by Proposition 2.10, we have x ∈ ↟ SI 2 e ∈ O SI 2 ( X ) . As ( x i ) i ∈ I converges to x in ( X , O SI 2 ( X ) ) , x i ∈ ↟ SI 2 e eventually. Since ↟ SI 2 e ⊆ ↑ e , we obtain that x i ≥ e eventually. Thus, ( x i ) i ∈ I ⟶ SI 2 * x .□

By Lemmas 3.3 and 4.8, we obtain the following.

Corollary 4.9

For any SI 2 -continuous space X , O ( Sℐ 2 * ( X ) ) = O ( Sℐ 2 ( X ) ) = O SI 2 ( X ) .

Lemma 4.10

Let X be a T 0 -space and x , y ∈ X . If x ≪ SI 2 y , then for any net ( x i ) i ∈ I in X , ( x i ) i ∈ I ⟶ SI 2 * y implies x i ≥ x eventually.

Proof

Suppose that x ≪ SI 2 y and ( x i ) i ∈ I is a net of X SI 2 * -converging to y . We show that x ≤ x i holds eventually. Since ( x i ) i ∈ I ⟶ SI 2 * y , there is an F ∈ Irr ( X ) such that y ∈ F δ , and for each e ∈ F , e ≤ x i eventually. By x ≪ SI 2 y and y ∈ F δ , we have x ∈ ↓ F , and hence, there is e x ∈ F such that x ≤ e x . Consequently, x ≤ x i eventually.□

Proposition 4.11

Let X be a T 0 -space, y ∈ X and ( x i ) i ∈ I be a net in X. Consider the following two conditions:

  1. ( x i ) i ∈ I ⟶ SI 2 * y .

  2. For any x ≪ SI 2 y , x i ≥ x eventually.

Then (1) ⇒ (2), and two conditions are equivalent if X is SI 2 -continuous.

Proof

(1) ⇒ (2): By Lemma 4.10.

(2) ⇒ (1): Suppose that X is SI 2 -continuous. Let F = ↡ SI 2 y . Then by the SI 2 -continuity of X , F ∈ Irr ( X ) and y ∈ F δ . For any e ∈ F , x i ≥ e eventually by the assumption. Thus, ( x i ) i ∈ I ⟶ SI 2 * y .□

Definition 4.12

A T 0 -space X is called SI 2 * -continuous if for every x ∈ X , the following two conditions hold:

  1. ↟ SI 2 x is an SI 2 * -open set in X .

  2. ↡ SI 2 x is irreducible and x = ∨ ↡ SI 2 x (equivalently, x ∈ ( ↡ SI 2 x ) δ ).

Theorem 4.13

If X is an SI 2 * -continuous space, then SI 2 * -convergence in X is topological.

Proof

Suppose that ( x i ) i ∈ I converging to x in ( X , O ( Sℐ 2 * ( X ) ) ) . We need to show ( x i ) i ∈ I ⟶ SI 2 * x . Let F = ↡ SI 2 x . Then F ∈ Irr ( X ) and x ∈ F δ by the SI 2 * -continuity of X . For any e ∈ F , by the SI 2 * -continuity of X again, we have x ∈ ↟ SI 2 e ∈ O ( Sℐ 2 * ( X ) ) . As ( x i ) i ∈ I converges to x in ( X , O ( Sℐ 2 * ( X ) ) ) , x i ∈ ↟ SI 2 e eventually. Since ↟ SI 2 e ⊆ ↑ e , we obtain that x i ≥ e eventually. Thus, ( x i ) i ∈ I ⟶ SI 2 * x .□

But we do not know whether the converse of Theorem 4.13 is true. So naturally we asks the following question.

Question 4.14

Characterize those T 0 -spaces X for which the SI 2 * -convergence in X is topological.

Theorem 4.15

For a T 0 -space X, the following conditions are equivalent:

  1. X is SI2-continuous.

  2. X is SI 2 * -continuous and O ( Sℐ 2 * ( X ) ) = O SI 2 ( X ) .

  3. X is SI 2 * -continuous, and for any net ( x i ) i ∈ I in X and x ∈ X , ( x i ) i ∈ I ⟶ SI 2 * x iff ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) .

  4. X is SI 2 * -continuous, and for any net ( x i ) i ∈ I in X and x ∈ X , ( x i ) i ∈ I ⟶ SI 2 * x iff ( x i ) i ∈ I ⟶ SI 2 x .

Proof

(1) ⇒ (2): Suppose that X is SI2-continuous. Then by Proposition 2.10 and Remark 4.5 (1), X is SI 2 * -continuous. By Corollary 4.9, O ( Sℐ 2 * ( X ) ) = O ( Sℐ 2 ( X ) ) = O SI 2 ( X ) .

(2) ⇒ (1): Trivial.

(2) ⇒ (3): If ( x i ) i ∈ I ⟶ SI 2 * x , then ( x i ) i ∈ I ⟶ SI 2 x by Remark 4.5(2), and hence, ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) by Lemma 3.3. Conversely, if ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) , then it converges to x with respect to the topology O SI 2 * ( X ) by O ( Sℐ 2 * ( X ) ) = O SI 2 ( X ) . Then by Theorem 4.13, we obtain ( x i ) i ∈ I ⟶ SI 2 * x .

(3) ⇒ (4): Suppose ( x i ) i ∈ I ⟶ SI 2 * x . Then ( x i ) i ∈ I converges to x with respect to the topology O ( Sℐ 2 * ( X ) ) . It follows from Remark 4.5(1) that ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) . Conversely, if ( x i ) i ∈ I ⟶ SI 2 x , then ( x i ) i ∈ I converges to x with respect to the topology O SI 2 ( X ) by Lemma 3.3. By (3), we obtain ( x i ) i ∈ I ⟶ SI 2 * x .

(4) ⇒ (2): By (4) and Lemma 3.3, we have that O ( Sℐ 2 * ( X ) ) = O ( Sℐ 2 ( X ) ) = O SI 2 ( X ) .□

By Theorems 4.13 and 4.15, we obtain the following corollary.

Corollary 4.16

If X is an SI 2 -continuous space, then SI 2 * -convergence in X is topological.

Acknowledgments

The authors sincerely thank the anonymous reviewers for their valuable comments and suggestions that have improved the manuscript substantially.

  1. Funding information: This work was supported by the National Natural Foundation of China (Nos. 12471070 and 12071199).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. All authors contributed equally in this work.

  3. Conflict of interest: The authors state no conflicts of interest.

  4. Data availability statement: No datasets were generated or analyzed during the current study.

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Received: 2024-11-07
Accepted: 2025-04-23
Published Online: 2025-06-05

© 2025 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/math-2025-0154
Keywords for this article
irreducible set; SI2-convergence; SI2-continuous space; strongly I2-continuous space; SI2 *-convergence; SI2 *-continuous space
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