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Concepts of General Topology in Constructive Mathematics and in Sheaves, !1 Annals of Mathematical Logic 23 (1982) 55-98 55 North-Holland Publishing Company CONCEPTS OF GENERAL TOPOLOGY IN CONSTRUCTIVE MATHEMATICS AND IN SHEAVES, !1" R.J. GRAYSON lnstitut fi~r Mathematische Log'ik, Einsteinstrasse 64., 4400 Miinster, West Germany Received 5 May 1981 Topological properties of spaces of sections of bundles are investigated in detail, and connected with results in constructive topology, in particular for notions of compactness, connectedness and uniformity. 1. Preliminaries 1.1. Introduction and summary This paper is to some extent a sequel to Grayson [6], hereafter referred to as CGT I. One should be tamiliar with CGT I, or at least have access to it, since frequent reference is made to it. The emphasis here however is more on the (sheaf) models and less On the constructive theory of topology; another difference from CGT I is that the methods used here in reasoning about the models are often (necessarily) classical, whereas in CGT I we were concerned to reason about them constructively as far as possible. In the present section we cover some preliminary topological notions. In Section 2 a more general class of models is presented, the spaces of sections of bundles; in CGTI we only considered 'constant' examples of these. The Rep- resentation Theorem 2.1.1 of [5] shows that all internal topological spaces in 'topological models' (i.e., in topoi c~ sheaves over spaces) arise as subspaces of such spaces of sections; thus this clat,~ of models has an intrinsic interest, and the bulk of the paper is concerned with establishing connections between internal properties of these spaces and external properties of their fibres. Our main tool is the Spreading Lemma (Section 2.3), whose first applications include an internal characterisation of subsets of a space of sections which are 'representable' by external subsets of th~ bundle (Section 2.4). In Sections 3 and'4 we consider notions of compactness and connectedness, and interpret them in spaces of ~ections. This leads to two main results, on the validity for all spaces in all topological models, of two theorems on connected and chain-connected metric spaces. We acknowledge a gap here, in that we have not * Research completed with the support of the Alexander yon Humboldt-Stiftung. 0003-4843/82[0000-0000/$02.75 © 1982 North-Holland 56 R.J. Grayson yet found non-topological counter-examples to these results (in the style of [4] and CGTI, for properties of the reals, for example). Note also that we are concerned to deduce both internal properties from external ones, and vice versa; this is in contrast to CGT I, where we only spoke of properties being 'preserved' `from a space X to the space Xr. In Section 3.4 we give a brief account of compact-open topologies on function spaces. The final section contains an account of uniform spaces, where we show a connection with the 'fields of uniform spaces' of Dauns and Hofmann [3]. 1.2. Some topological notions We introduce here some of the topological notions needed later, relying on CGT I for the more basic notions. Definition 1.2.1. A topology on X i'~ a set O(X) of subsets of X, containing X and closed under finite intersections and arbitrary unions. Concepts such as: interior, neighbourhood (basis), closure, continuous function, product, are as usual; see CGT I for those of: separated, Hausdorff, regular. For a subset A of a space X (i) -hA ={x Ix # A}=Int(X\A); (ii) A cc B (A well-contained in B) iff X= (-hA Ulnt B); (iii) A is well-closed (or: w-closed) lit (A c_ U---~ Acc U); (iv) A is located iff V UVxeU[~ye(AOU)vx # A). Definition 1.2.2. For X a metric space we have the notion of filter-completeness, which, in the absence of the principle of countable choice (CAC), is stronger than that of sequential-completeness. See Definition 5.1.7 for the extension to uniform spaces. For a subset A of a metric space (X, p) (v) A is metrically located iff for every x eX the distance p(x,A)= inf{p(x, y) lyEA} exists, and, if p(x,A)<~, then 3 y~ A" p(x, y)<& (vi) A is totally bounded iff for each k there are xo, ..,x. eA such that A = Ui~,, S(xi; k-l). As is well known from the intuitionistic literature (e.g., [9]), a subset of I~" is totally bounded iff it is bounded and metrically located. As to the relationship between the two kinds of locatedness, it is easy to see that metrically located implies located; it is also not hard to construct a Brouwer-type counter-example to the converse implication among the subsets of Baire space. We consider the situatiot~ in more detail in Proposition 3.1.5. Remark 1.2.3. The main point of the notion of w-closed set is the characterisa- tion of compact subspaces in Section 3.1. For the moment we note the following Concepts o[ genert l topology, II 57 facts: w-closure and ordinary closure are incomparable: (a) In ~, ([0, 1] O [1, 2]) is w-closed bul not closed constructively (its closure is [0, 2]). (b) In the model R~ (see CGT I, Section 2, and Example 2.1.6 below), let ~ be the generic real (At • t) and a the partial element (At e (0, ~o) • 0). Internally the set {a} is closed, but not w-closed, as we now see: we have {a}~ ((0, ~o)×~) internally (see Section 2.1 for the definitions), but 0~[~ # {a}v~e((0,~)×R)]l. Note also that: (c) In this example, there is no least w-closed set containing {a}: let b be (At. 0) and c be (At-min{t, 0}), so that {b},{c} are w-closed internally, but {a} =({b}f~{c}) is not w-closed. Thus w-closure does not arise from a closure operator on P(X). Finally (d) For A a subset of a separated space X, if A is w-closed, then A is located: If A is w-closed and x e U, the separatedness of X gives Vy~A[y~Uvx# y] so that A_({z~XI3yeAtTU}f3~{x}). Then, as A is w-closed, either x ({z e X 13 y e A f3 U} t_J -7{x}), whence 3 y ~ A n U, or x # A. 1.3. Soberness and relative soberness The following constructive notion of a sober space was introduced in [5], to which we refer for more details. It is equivalenL classically, to every irreducible closed set being the closure of some point, which follows cktssicaily from Haus- dorffness. Fourman and Scott [5, 8.14] show that e,ven the rationals need not be sober constructively; constructively, in fact, soberness is a kind of 'completeness' property, as is borne out by Proposition 3.2.2. Definition 1.3.1. A super]ilter on a space X is a filter ~ on (Y(X) such that v @z~_~(x)[U 0u~,~ ~3 u~(~u n~)]. ~; converges to a point x iff VU[x ~ U~-~ U~]. X is sober iff every superfilter converges. We generally assume X to be To so that every superfilter converges to at most one point. As examples of sober spaces we have all filter-complete metric (or uniform, see Section 5) spaces. As a general method of forming sober spaces we have the following. Definition 1.3.2. The soberiJication Soo(X) of a space X consists of all superfilters on X, with the topology with opens {St I U~:}, for U~(X). X is embedded in Sob(N) by x ,-~ ~x ={UI x e U}. 58 R.I. Grayson Remark 1.3.3. The key facts about Sob(X) that we will need later are (i) Sob(X) is sober; (ii) ~7(X) and ~7(Sob(X)) are isomorphic; (iii) X is dense in Sob(X) (under the embedding). Fact (i) will allow us to apply Fourman's Representation Theorem 2.1.1 for sober spaces in sheaves over a space; facts (ii) and (iii) will ensure that X and Sob(X) share all the relevant topological properties (see Lemma 4.3.1). A generalisation of the above concepts will serve in Section 2./4 to characterise 'representable' subsets of spaces of sections. Definition 1.3,4. For A c_ X and ~: a superfilter on if(A) and x ~ X we say that converges to x iff VU[x~ U,,.~ (UN A)~]. Then A is sober relative to X iff, whenever a superfilter ~ in e?(A) converges to a point x ~ X, we have x ~ A. Correspondingly, we define the soberification of A relative to X, Sob(A, X), to consist of those points of X converged to by some superfilter on ~(A). One can think of relative soberness as saying that A contains all the points of .~( that it 'should do'. We give some examples below, after we have simplified the definitions in the following Lemma L3.5. Sob(A, X) consists of those x ~ X such that VU, V[xE UA(UNA)~_ V---~ xe V]. (*) Thus A is relatively sober iff, whenever (*) holds for x ~ X, then x ~ A. ProoL Firstly, if ~ is a superfilter on ~Y(A) converging to x, and (x~ U^ (UAA)c_ V), then (UN A)~: and (UO A)c_(VNA), so that (VA A)~5~, too. But then x~ V, so that (*) holds. Conve~:sely, if (*) holds for x, set ~;={(UAA)Ix~ U} and check that this is a superfilter on ~Y(A) converging to x.
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