## Abstract

In this paper, the α waybelow relation, which is determined by *O*_{2}-convergence, is characterized by the order on a poset, and a sufficient and necessary condition for *O*_{2}-convergence to be topological is obtained.

Show Summary Details# A result for *O*_{2}-convergence to be topological in posets

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## Abstract

## 1. Introduction

## 2. Preliminaries

## 3. *O*_{2}-doubly continuous posets

## 4. A sufficient and necessary condition for *O*_{2}-convergence to be topological

## Acknowledgement

## References

## About the article

More options …# Open Mathematics

### formerly Central European Journal of Mathematics

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Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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IMPACT FACTOR 2016 (Open Mathematics): 0.682

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In this paper, the α waybelow relation, which is determined by *O*_{2}-convergence, is characterized by the order on a poset, and a sufficient and necessary condition for *O*_{2}-convergence to be topological is obtained.

Keywords: O2-convergence; O2-doubly continuous poset; Topological, α-doubly continuous poset; α-doubly continuous poset

The *O*_{2}-convergence on a poset *P*, which is a generalization of *O*-convergence, is defined as follows [1–4]: Let *P* be a poset, a net (*x _{i}*)

- 1.
sup

*M*= inf*N*=*x*; - 2.
For each

*a*∈*M*and*b*∈*N*, there exists*k*∈*I*such that*a*≤*x*≤_{i}*b*holds for all*i*≥*k*.

By saying the *O*_{2}-convergence on a poset *P* is topological, we mean that there exists a topology 𝒯 on *P* such that a net (*x _{i}*)

In [5], Zhao and Li proved that if the *O*_{2}-convergence on a poset *P* is topological, then *P* is an α-doubly continuous poset. Also they give a sufficient condition: If *P* is an *α**-doubly continuous poset, then *O*_{2}-convergence is topological. Moreover, if *P* is a poset satisfying condition (*), *O*_{2}-convergence is topological if and only if *P* is an *α*-doubly continuous poset.

But as pointed in this paper, there exists poset which is *α*-doubly continuous but not *α**-doubly continuous. Then for such a poset, is the *O*_{2}-convergence topological? In other words, can we find a sufficient and necessary condition for *O*_{2}-convergence to be topological for any poset?

In this paper, the *α* waybelow relation is characterized just by the order on a poset. The concept of *O*_{2}-doubly continuous poset is given, which is strictly stronger than *α*-doubly continuous and strictly weaker than *α**-doubly continuous. Most important, we prove that for a poset *P*, the *O*_{2}-convergence on it is topological if and only if *P* is an *O*_{2}-doubly continuous poset.

This section will briefly mention some of the important results that Zhang and Li presented in [5]. For terms which are not defined here, please refer to [9, 10] and related references.

For a poset *P*, *a, b* ∈ *P*, we denote [*a, b*] = {*x* ∈ *P* : *a* ≤ *x* ≤ *b*}. We use *M*_{0} ⊆_{fin}*M* to indicate *M*_{0} is a finite subset of *M*.

([5]).*Let P be a poset. For x, y, z* ∈ *P, define x* ≪_{α}*y if for every net* (*x _{i}*)

It follows from Definition 2.1 that *x* ≪* _{α} y* ⟹

([5]). *A poset P is called an α-doubly continuous poset if a* = sup {*x* ∈ *P* : *x* ≪* _{α} a*} = inf {

- (1)
*Every finite lattice is α-doubly continuous, every chain or antichain is α-doubly continuous*. - (2)
*If P is α-doubly continuous, then for every a*∈*P, the set*{*x*∈*P*:*x*≪}_{α}a*and*{*y*∈*P*:*z*⊳}_{α}a*are both nonempty*.

([5]). *Let P be a poset. For x, y, z* ∈ *P, define x* ≪_{α*} *y if for every net* (*x _{i}*)

([5]). *A poset P is called an α* ^{*}*-doubly continuous poset if a*= sup {*x* ∈ *P* : *x* ≪_{α*}*a*} = inf {*y* ∈ *P*: *z* ⊳_{α*}*a*} *for every a* ∈ *P*.

([5]).

- (1)
*Obviously, x*≪_{α*}*y*⟹*x*≪⊳_{α}y and z_{α*}*y*⟹*z*⊳._{α}y, then an α*-doubly continuous poset is α-doubly continuous - (2)
*If P is an α*-doubly continuous poset, then for each x*∈*P*,

*x* = sup{*a* ∈ *P* : ∃*a*′ ∈ *P*, *a* ≪_{α*} *a*′ ≪_{α*} *x*} = inf{*b* ∈ *P* : ∃*b′* ∈ *P*, *b* ⊳_{α*} *b′* ⊳_{α*} *x*}.

([5]). *For any poset P, if O*_{2}-*convergence is topological, then P is α-doubly continuous*.

([5]). *If P is an α***-doubly continuous poset, then O*_{2}-*convergence is topological*.

That is to say, *α**-doubly continuous ⟹ *O*_{2}-convergence is topological⟹ *α*-doubly continuous.

**Condition** (*). *Let P be a poset and x, y, z* ∈ *P with x* ≪* _{α} y* ≤

([5]). *For any poset P which satisfies condition (*), O*_{2}-*convergence is topological if and only if P is α-doubly continuous*.

In this section, we give a characterization of the relations ≪_{α} and ⊳_{α}, then introduce the concept of *O*_{2}-doubly continuous poset which is strictly stronger than *α*-doubly continuous and strictly weaker than *α**-doubly continuous.

*Let P be a poset. y* ≪* _{α} x if and only if for every subset M and N such that* sup

*Dually, z* ⊳* _{α} x if and only if for every subset M and N such that* sup

Sufficiency, if there exists a net (*x _{i}*)

Necessary, suppose *y* ≪* _{α} x* and sup

*A poset P is called an O*_{2}-*doubly continuous poset if it satisfies the following condition*:

- (1)
*P is α-doubly continuous, i.e. x*= sup {*y*∈*P*:*y*≪_{α*}*x*} = inf {*z*∈*P*:*z*⊳_{α*}*x*}*for each x*∈*P*. - (2)
*If y*≪⊳_{α}x and z⊆_{α}x, then there exist finite A_{fin}{*a*∈*P*:*a*≪}_{α}x*and B*⊆_{fin}{*b*∈*P*:*b*⊳}_{α}x*such that y*≪⊳_{α}c and z._{α}$c\in \underset{m\in A}{\cap}\underset{m\in B}{\cap}[m,n]$

*Every finite lattice is O _{2}-doubly continuous, every chain or antichain is O_{2}-doubly continuous*.

*If P is an α**-*doubly continuous poset, then P is an O*_{2}-*doubly continuous poset*.

By Remark 2.6 we know an *α**-doubly continuous poset is *α*-doubly continuous. In an *α**-doubly continuous poset *P*, we denote *M _{x}* = {

If there exists a net (*x _{i}*)

An *α**-doubly continuous poset implies it is *O*_{2}-doubly continuous. An *O*_{2}-doubly continuous poset implies it is *α*-doubly continuous. But conversely both are untrue – see the following examples:

Let *P*_{1} = {1} ⋃ {*a _{j}*

- (1)
*a*≤_{i}*a*_{i+1};*b*≤_{i}*b*_{i+1};*c*≤_{i}*c*_{i+1};*for all i*∈ ℕ. - (2)
*a*≤_{i}*b*;_{i}*a*≤_{i}*c*∈ ℕ_{i}for all i - (3)
*b*≤_{ij}*b*_{i,j+1}≤*b*≤_{i}and c_{ij}*c*_{i,j+1}, ≤*c*∈ ℕ._{i}for all i, j - (4)
1

*is the greatest element in P*.

*It is straightforward to verify that P*_{1} *is α-doubly continuous*, {*x* ∈ *P* : *x* ≪_{α} 1 = {*a*_{i} : *i* ∈ ℕ} *and* {*x* ∈ *P* : *x* ⊳_{α} 1} = {1}. *Notice a*_{1} ≪_{α} 1, *for any finite A* ⊆_{fin} {*x* ∈ *P* : *x* ≪_{α} 1} = {*a*_{i} : *i* ∈ ℕ}, *there exists i*_{0} ∈ ℕ *such that* ${a}_{i0}\in \underset{{a}_{i}\in A}{\cap}[{a}_{i},1]$, *then* ${b}_{i0}\in \underset{{a}_{i}\in A}{\cap}[{a}_{i},1]$, *but a*_{1} ≪_{α} *b*_{i0} *does not hold, so P*_{1} *is not an O*_{2}-*doubly continuous poset*.

*Moreover, a*_{1} ≪_{α} *a*_{2} ≤ *b*_{2} *but a*_{1} ≪_{α} *b*_{2} *does not hold, that means P*_{1} *does not satisfy the condition*(*).

*Let P*_{2} = {1}⋃{*a*}⋃{*b*_{1}, *b*_{2}, *b*_{3}, ...}. *The order* “≤” *on P*_{2} *is defined as follows: (see Fig. 1 for the Hasse diagram): a* ≤ 1; *b*_{i} ≤ 1 *for all i* ∈ ℕ; *b*_{i} ≤ *b*_{j} , *whenever i* ≤ *j, for all i, j* ∈ ℕ.

*It is obvious that P is an α-doubly continuous poset and satisfies the condition (2) in Definition 3.2, then P is an O _{2}-doubly continuous poset. But it is not α*

In this section, we obtain the main result that *O*_{2}-convergence on a poset *P* is topological if and only if *P* is an *O*_{2}-doubly continuous poset.

We recall that saying the *O*_{2}-convergence on a poset *P* is topological means that there exists a topology 𝒯 on *P* such that a net (*x _{i}*)

*For any poset P, if the O*_{2}-*convergence is topological, then P is an O*_{2}-*doubly continuous poset*.

For a poset *P*, if the *O*_{2}-convergence is topological, then there exists a topology 𝒯 on *P* such that a net (*x*_{i})_{i∈I} *O*_{2}-converges to *x* ∈ *P* if and only if it converges to *x* with respect to the topology 𝒯. For every *x* ∈ *P*, let *I _{x}* = {(

For arbitrary *m* ∈ *M*_{0} and *n* ∈ *N*_{0}, if there exists a net (*x _{j}*)

Suppose *y* ≪_{α} *x* and *z* ⊳_{α} *x*, by Proposition 3.1 we know there exist *M*_{1} ⊆_{fin} *M*_{0} and *N*_{1} ⊆_{fin} *N*_{0} such that $\underset{m\in {M}_{1}}{\cap}\underset{n\in {N}_{1}}{\cap}[m,n]\subseteq [y,z]$. For arbitrary *m* ∈ *M*_{1} and *n* ∈ *N*_{1}, there exists *W _{mn}* ∈ 𝒩(

Since *P* is an *α*-doubly continuous poset, we know sup *A _{x}* = inf

For poset *P*, the *O*_{2}-convergence is topological, then (*x*_{(r,U)})_{(r,U)}∈𝒟 converges to *x* with respect to the topology 𝒯, hence *x*_{(r,U)} ∈ *W*_{1} holds eventually. There exist finite subsets *A*_{1} ⊆_{fin} *A _{x}* and

*The relations among* ${W}_{1},\underset{m\in {M}_{1}}{\cap}\underset{n\in {N}_{1}}{\cap}[m,n]$ *and* $\underset{a\in {A}_{1}}{\cap}\underset{b\in {B}_{1}}{\cap}[a,b]$ *are revealed in Fig 2*.

We review the general relation between convergence and topology. If on any set *P* a class 𝓛 of pairs ((*x _{i}*)

([11]). *Given a poset P, the class 𝓛 is topological if and only if the following axioms are satisfied*.

(Constant) If (*x _{i}*)

(Subnets) If ((*x _{i}*)

(Divergence) If ((*x _{i}*)

(Iterated Limits) If ((*x*_{i})_{i∈I}, *x*), ∈ 𝓛, and if ((*x*_{i, j})_{j∈J(i)}, *x*_{i}) ∈ 𝓛 for all *i* ∈ *I*, then ((*x*_{i, f(i)})_{i, f∈I×M},*x*) ∈ 𝓛, where $M=\prod _{{}_{i\in I}}J(i)$ is a product of directed sets.

*For an α-doubly continuous poset P, a net* (*x _{i}*)

Necessary, if net (*x _{i}*)

*For an O_{2}-doubly continuous poset P, a net* (

Necessary, if net (*x _{i}*)

*For any poset P, the class 𝓛 satisfies axiom (Constant)*.

Suppose *x _{i}* =

*For any poset P, the class 𝓛 satisfies axiom (Subnets)*.

Let ((*x _{i}*)

*For an α-doubly continuous poset P, the class 𝓛 satisfies axiom (Divergence)*.

Assume that ((*x _{i}*)

*For an O*_{2}-*doubly continuous poset P, the class 𝓛 satisfies axiom (Iterated Limits)*.

Let ((*x*_{i})_{i∈I}, *x*) ∈ 𝓛 and ((*x*_{i, j})_{j∈Ji}, ∈ 𝓛 for every *i* ∈ *I*. For any *y, z* ∈ *P* with *y* ≪_{α} *x* and *z* ⊳_{α} *x*, by Lemma 4.5 we know *y* ≪* _{α} x_{i}* and

From Propositions 4.6* –4.9* and Fact * 4.* 3, we have

*For any poset P, if P is O*_{2}-*doubly continuous, then O*_{2}-*convergence is topological*.

Combining Theorem 4.1 with Theorem 4.1 0, we obtain the following theorem.

*Let P be a poset. Then the O*_{2}-*convergence is topological if and only if P is O*_{2}-*doubly continuous*.

By Proposition 3.4 and Theorem 4.1 0, we have the following corollary.

([5]). *If P is an α*-doubly continuous poset, then O*_{2}-*convergence is topological*.

- (1)
*For the poset P*_{1}*showed in Example 3.5, the O*_{2}-*convergence is not topological*. - (2)
*In Example 3.6, the O*_{2}-*convergence on the poset P*_{2}*is topological*.

This work is supported by the Natural Science Foundation of China (Grant No: 11371130) and (Grant No: 2014GXNSFBA118015).

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**Received**: 2016-01-06

**Accepted**: 2016-03-09

**Published Online**: 2016-04-23

**Published in Print**: 2016-01-01

**Citation Information: **Open Mathematics, Volume 14, Issue 1, Pages 237–246, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0018.

© 2016 Li and Zou, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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