Elsevier

Topology and its Applications

Volume 289, 15 February 2021, 107402
Topology and its Applications

Atoms in the lattice of covering operators in compact Hausdorff spaces

https://doi.org/10.1016/j.topol.2020.107402Get rights and content

Abstract

Let Comp be the category of compact Hausdorff spaces with continuous maps. A cover of a space in Comp is an irreducible preimage; equivalent covers are identified. A covering operator (co) is a function c assigning to each X in Comp a cover XcXcX which is minimum among covers Y of X with Y=cY. The family of all such c is denoted coComp. This is a complete lattice (albeit a proper class), with bottom the identity operator id and top the Gleason (extremally disconnected, projective) cover operator g. Here, we completely determine the atoms (minimal elements above id) in the lattice coComp, show that any cid in coComp is above an atom, and show that coComp is not atomic. At the end, we make some remarks about what the present paper does and does not tell us about several other categories related to Comp.

Section snippets

Introduction and preliminaries

A minimum proper cover (mpc) is a proper cover XY such that every proper cover of X is a cover of Y. The main theorems of this paper are: First (§2), the characterization of the mpc's as the covers XfY with Y infinite and extremally disconnected and f identifying only two different non-isolated points of Y. Second (§3), the atoms in the lattice coComp correspond one-to-one with the mpc's. Third (§3), every element of coComp different from the identity operator lies above an atom. Finally (§4

The mpc's

Before the theorem here (2.2), we review our situation.

Definitions, etc. 2.1

(a) Let L be a lattice (not necessarily a set) with a bottom 0. An atom in L is a minimal element of L{0}. A strong atom in L (we coin the term) is an atom in L which is minimum in L{0} (thus unique). Birkhoff [4, p. 194] defines an element y of a lattice L (not assuming L has a bottom) to be strictly meet irreducible just in case {xL:y<x} has a least element, say l(y). Thus, if L has 0, then a is the strong atom if and only if 0 is

mpc's and atoms in coComp

Recall (1.1(c)) that coComp is a complete lattice.

Theorem 3.1

  • (a)

    If E˙γγE is an mpc, then an atom a(γ) in coComp is defined for YComp by(a(γ)Y,a(γ)Y)={(E,γ)if Y=E˙γ(Y,idY)if YE˙γ. (Note: fix(a(γ))=Comp{E˙γ}.)

  • (b)

    If a is an atom in coComp, then there is an mpc γ with a=a(γ).

Proof

(a) We show first that a(γ)coComp via the equivalence given in 1.1(c). Let YComp. We show that a(γ)Yfix(a(γ)). If a(γ)YE, then YE˙γ by the definition of a(γ), so a(γ)Y=Y and thus a(γ)(a(γ)Y)=a(γ)Y. So suppose a(γ)Y=E. Since EE˙γ, one

coComp is not atomic

We shall establish another fact about the lattice coComp, made possible by the characterization in Theorem 3.1.

Theorem 4.1

The Gleason covering operator gcoComp is not the supremum of the atoms of coComp. The lattice coComp is not atomic.

We are denoting an mpc γ as E˙γ=EpqγE. Recall that for s,ccoComp, sc if and only if fix(s)fix(c).

Lemma 4.2

Suppose ccoComp. Lets(c)={a(γ):a(γ)c}, the supremum in the complete lattice coComp. Then s(c)<c if and only if there exists Yfix(s(c))fix(c). The latter condition

Final remarks

“Covering theory” as described in Comp here, can be described in general categories and, as such, has an opposite/dual theory, which is that of essential extensions; the dual of a covering operator is an essential extension operator, frequently called a hull operator (ho). See [2] and [1, §9] for discussions of this duality. It remains a question that, with appropriate assumptions about the category C, to what extent the dual of Theorem 3.1 carries over, giving the same correspondence between

Acknowledgements

We thank the referee for several helpful comments.

We are profoundly in the debt of Bernhard Banaschewski, who identified a gap in our original proof of 3.1(b), and was kind enough to discuss extensively the situation with us.

References (16)

  • J. Adámek et al.

    Abstract and Concrete Categories

    (2009)
  • B. Banaschewski et al.

    Epimorphisms and maximal covers in categories of compact spaces

    Appl. Gen. Topol.

    (2013)
  • R.N. Ball et al.

    The quasi-Fκ cover of a compact Hausdorff space and the κ-ideal completion of a Archimedean -group

  • G. Birkhoff

    Lattice Theory

    (1979)
  • R. Carrera et al.

    A classification of hull operators in Archimedean lattice-ordered groups with weak unit

    CGASA

    (2020)
  • F. Dashiel et al.

    Order-Cauchy completions of rings and vector lattices of continuous functions

    Can. J. Math.

    (1980)
  • R. Engelking

    General Topology

    (1989)
  • L. Gillman et al.

    Rings of Continuous Functions

    (1960)
There are more references available in the full text version of this article.
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