Perfectness of 2-star compactifications of frames

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Abstract

We examine N-star compactifications of non-compact regular continuous frames and provide conditions under which these kinds of compactifications are perfect. Some properties and characterisations of such compactifications are established for the cases where N{1,2}. Connectedness of the remainder of a non-compact regular continuous frame in these compactifications is also investigated.

Introduction

It is a classical fact that a topological space has a smallest compactification if and only if it is a locally compact Hausdorff space.

Banaschewski [5] showed that a frame has a least compactification if and only if it is regular continuous. Therefore, we think of regular continuous frames as the analogue of locally compact Hausdorff spaces. Naturally, we also consider the least compactification of a frame as the frame counterpart of the Alexandroff one-point compactification of a locally compact Hausdorff space.

Some conditions under which the least compactification of a non-compact regular continuous frame is perfect were given by Baboolal in [2]. Notably, one of the characterisations is that the least compactification of a non-compact regular continuous frame L is perfect if and only if the remainder of L in any of its compactification is compact and connected. This is the case for spaces as well and noteworthy as it is related to some aspect of the connectedness of remainders of regular continuous frames in some of their N-star compactifications.

Magill [7] introduced the notion of an N-point compactification for topological spaces as any compactification ζ(X) such that the annex ζ(X)X consists of exactly N points, where N is a positive integer.

Baboolal [3] introduced N-star compactifications for frames as the analog of N-point compactifications for topological spaces and showed that the least compactification of a non-compact regular continuous frame is the 1-star compactification.

The purpose of this paper is to characterise N-star compactifications of frames which are perfect, for N>1. This work is strongly motivated by [2] where some characterisations and properties of an N-star compactification with respect to perfectness were established for the case where N=1. Herein, we study the conditions under which an N-star compactifications of a regular continuous frame are perfect. These conditions are studied under the setting where N=2 and we conjecture that the results can be generalised to any N>1. Up to equivalence, we show that a 2-star compactification of a non-compact regular continuous frame L is perfect if and only if it is the only 2-star compactification of L. Moreover, if a 2-star compactification is perfect, we prove that there exists no other N-star compactification of L for any N>2. The latter and its converse are also shown to hold true for the least (1-star) compactification of a non-compact regular continuous frame.

Some results concerning connectedness are also found. That is, we exhibit the connectedness of the remainder of any non-compact regular continuous frame in its least compactification. Some special elements of the remainder of a 2-star compactification are shown to be connected.

Section snippets

Preliminaries

We provide definitions of concepts and some basic results that are going to be used in a sequel. For more on the theory of frames, we refer the reader to [4], [6] and [8].

A frame L is a complete lattice which satisfies the infinite distributive law:xS={xs:sS} for every xL and every SL. The top element and the bottom element of L will be denoted by e and 0, respectively. A frame homomorphism is a map h:LM, between the frames L and M, that preserve finite meets, including e, and arbitrary

Connectedness of the remainder of the least compactification

The following result by Baboolal [2] characterises non-compact regular continuous frames whose least compactification is perfect. The aim is to extend the scope of this theorem to the case where the compactification at hand is an N-star compactification, with N>1.

Theorem 3.1

[2, Theorem 4.2] The following conditions are equivalent for a non-compact regular continuous frame L.

  • (1)

    The least compactification of L is perfect.

  • (2)

    Whenever (uv) is compact where u,vL and uv=0, then either k(u)J=L or k(v)J=L.

  • (3)

N-star compactifications of frames and their perfectness

N-star compactifications of frames were introduced by Baboolal [3] as the analog of the concept of N-point compactifications of topological space due to Magill [7]. We recall the definition of an N-star below.

Definition 4.1

Let L be a frame. For any positive integer N, an N-star is a collection, α={u1,u2,,uN}, of N mutually disjoint elements of L with the property that: (u1u2uN) is compact and (u1u2ui1ui+1uN) is not compact, for each i. In the case where N=1, the latter is interpreted to mean

Acknowledgement

I would like to thank D. Baboolal and P. Pillay for their valuable input. Furthermore, I would like to thank the referee for useful suggestions.

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The support from the National Research Foundation (S.A) under Grant Number 129590 is acknowledged.

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