Apartness and Uniformity: A Constructive Development

213 Pages·2011·1.9 MB·English

by  Douglas S. Bridges

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Douglas S. Bridges • Luminiţa Simona Vîţă
EE
Apartness and Uniformity
E
A Constructive Development
V
Douglas S. Bridges Luminiţa Simona Vîţă
Department of Mathematics Department of Mathematics
and Statistics and Statistics
University of Canterbury University of Canterbury
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EPIGRAPH
I may prefer my ideals. The fact that they may not survive is not
a reason for not fighting for them. Schumpeter rightly said that
people who believe that ideals have to be absolute are idolatrous
barbarians. Civilization means that you must allow the possibility
of change without ceasing to be totally dedicated to—and ready
to die for—your ideals so long as you believe in them.
Isaiah Berlin1
1FromConversationswithIsaiahBerlin (2nded.),byRaminJahanbegloo,HalbanPub-
lishersLtd,London,2007.

PREFACE
The problem of finding a suitable constructive framework for general
topology is important and elusive.
Errett Bishop2
In the mid-1970s, Douglas Bridges came across the article [35], advocating
the use of nearness in the teaching of first courses in analysis. This led him to
consider a constructive theory based on proximity—or, rather, the construc-
tivelymoreappropriateoppositenotionofapartness—asapossiblealternative
approach to topology. Not, at that stage, being sufficiently mathematically
experienced to make significant progress with this idea, he put it on one side
until February 2000, when he and Lumini¸ta Vˆı¸t˘a began the project on apart-
ness spaces that is discussed in the present monograph. Clearly, two heads
were better than one, as we were able to initiate a project that, in the inter-
veningyears,has grownsubstantially,withcontributionsfrommathematicians
in several countries and continents.
What do we mean by constructive in this context, and why might a con-
structive approach to topology be interesting or significant? The answer to
the first question is simple: by constructive mathematics we mean, roughly,
mathematics with intuitionistic logic and a set- or type-theoretic foundation
thatprecludesthederivationofthelawofexcludedmiddle;inotherwords,we
mean mathematics carried out in the style of the late Errett Bishop [9]. Every
proof that is constructive in this sense embodies an algorithm that can be,
and in several cases has been, extracted and then implemented; further, the
original proof shows that the implementable algorithm (if correctly extracted)
2LettertoDouglasBridges,dated14April1975.
vii
viii Preface
meets its specifications. One other important feature of Bishop’s constructive
mathematics is that it is completely consistent with classical mathematics—
mathematics using classical logic.
In regard to the second question, we recall Bishop’s deflationary remark [9]
(page 63):
Verylittleisleftofgeneraltopologyafterthatvehicleofclassicalmathe-
maticshasbeentakenapartandreassembledconstructively.Withsome
regret, plus a large measure of relief, we see this flamboyant engine
collapse to constructive size.
At the time he wrote [9], Bishop firmly believed that computation in analy-
sis required uniform continuity—normally on compact sets—rather than the
weaker notion of pointwise (let alone sequential) continuity:
The concept of a pointwise continuous function is not relevant. A con-
tinuousfunction[ontherealline]isonethatisuniformlycontinuouson
compact intervals. ([9], pages ix–x)
Since the theorem asserting the uniform continuity of every pointwise con-
tinuous, real-valued mapping on the closed interval [0,1] cannot be proved-
within Bishop-style constructive mathematics,3 general topology, essentially
a pointwise matter, would therefore have little constructive relevance; what
was needed was a framework (such as metric-space theory) in which uniform
notions could be expressed.
Bishop’s remark, taken with his suggestion, in Appendix A to [9], that con-
structive theories like that of distributions might rely on ad hoc topological
notions, may have served to deflect attention away from the problem of find-
ing a suitable constructive framework for general topology. The advantage of
solving this problem would presumably be that we would have a general, and
hencealmostcertainlyclarifying,frameworkfortopologicalmatters.Moreover,
workofIshihara[55]andothers,20yearsafterthepublicationof[9],hasshown
that even sequential continuity has an important role in Bishop-style mathe-
matics;sothereisdefinitelyaplaceforaconstructivetheoryofabstractspatial
relationships that covers such highly non-uniform types of continuity.
One can certainly tackle topology by simply constructivising the classical
development of the subject from the usual three axioms about open sets.4
However, the theory of apartness that we present in this book does much
more, by encompassing both point-set topology and the theory of uniform
spaces. This, we believe, enables us to see those subjects in a clearer light.
Wenowoutlinethecontentsofthethreechaptersinourbook,whichbegins
with one that introduces informal constructive (intuitionistic) logic and set
3ThetheoremisprovableinonemodelofBishop-stylemathematics—namely,intuition-
isticmathematics—butisfalseinthemodelinwhicheverythingisinterpretedrecursively.
4ThishasbeendonebyGrayson[46,47],Troelstra[87],andWaaldijk[91],thelasttwo
workingwithinBrouwer’sfullintuitionisticframework.
Preface ix
theory, as well as—very briefly—the basic notions and notations for metric
and topological spaces.
In Chapter 2 we introduce axioms for a point-set apartness, a more com-
putationally informative notion than nearness.5 We then explore some of the
elementary consequences of our axioms, and examine the associated topology
and various types of continuity of mappings. One interesting feature of our
constructive approach is that intuitionistic logic reveals distinctions that are
invisible to the classical eye. For example, we present three natural notions of
continuity for maps between point-set apartness spaces, notions that coalesce
classically but are quite distinct constructively.
After dealing with continuity, the chapter continues with a discussion of
convergencefornetsinanapartnessspace,followedbyastudyoftheapartness
structure on the product of two apartness spaces. It ends with some remarks
about impredicativity and how that phenomenon might be avoided.
The theory really gets interesting—and a lot harder—when, in Chapter 3,
we consider apartness between subsets. The five axioms on which this part of
the theory is based enable us to bring most of the work on point-set apartness
acrosstotheset-setenvironment.Everyset-setapartnessgivesrisetoapoint-
setapartnessintheobviousway:apointxisapartfromasetS ifthesingleton
subset {x} is apart from S.
The canonical example of a set-set apartness arises from a quasi-uniform
structure (which, if a certain symmetry condition holds, becomes a uniform
structure).Havingintroducedbothapartnessandquasi-uniformspaces,weare
well placed to examine the connection between, on the one hand, the uniform
continuityofamappingf :X →Y betweenquasi-uniformspacesand,onthe
other, strong continuity, in which if the two subsets of Y are apart, then their
pre-images under f are apart in X. It is straightforward to show that uniform
continuity entails strong continuity; but the converse direction of entailment
is much harder to deal with. Similar difficulties are found in our discussion of
the relation between Cauchy nets in the usual uniform-space sense, and nets
that have the property of total Cauchyness relative to the apartness structure
associated with a given uniform structure. (The corresponding property of
total completeness of an apartness space X, whereby every totally Cauchy
sequence inX hasa limitinX,is classicallyequivalent tothecompactness of
the underlying apartness topology on X.)
The difficulties, alluded to above, in the proofs of certain results are unified
byourintroductionofageneralprooftechniquewhichhasseveralapplications
inthetheoryandwhichresembles—butisconsiderablymorecomplexthan—a
well-known technique introduced into constructive analysis by Ishihara [55].
Notonlydoesourtechniqueapplytothestudyofstronganduniformcontinu-
ity,butalsoitleadstoaneatproofthat,undercertainconditions,theuniform
5Actually, our fundamental notion is that of a pre-apartness. An apartness is a pre-
apartnesswithanextradisjunctionpropertythatenablesustosplitargumentsintocases.