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PAGE 1 S.T. 1967 T58 PAGE 2 A Oonatruotive roundation tor Topolog7 Bill Thur a ton June 14, 1967 New College Ben1or Thesis, submitted to Roger Renne /C I TS PAGE 4 Introdu.c tion In modern mathematicsabstract mathematics, that is, tor applied mathematics uses as its criterion workabilitycertain things are generally taken fat granted. Classical logic is taken as the standard for proots, in particular, the principle ot tertium non datur: either a proposition is true, or its denial is true; in another form, to prove a proposition it is sutticient to show that its denial would be absurd. A strong torm of set theory is taken as the underlying substance upon which a .mathematical structure is imposed. It is assumed that trom any set, a new set can be tormed which contains all the that set ; the axiom ot subsets. The axiom of choice is also assumed, which together with the axiom or subsets implies the wellordering principle: the elements in any set can be put in an order analogous to a counting order. That is, in any subset there is one element you come to first in the orderingelements are adjoined one at a time to the back end ot the order. Finally, a mathematical theory proceeds in its development abstractly and tormalistically. In the beginning there was chaos, and then certain axioms were created. The rest ot the history of a mathematical theorr is the exploration ot the con sequences or these original decrees. PAGE 5 2 ODe outstanding example of the achievements of this approach is modern general topology. In this paper I will develop an alternate foundation topology, based on a nonclassical approach to mathematics: roughly, intuitionism, but since the primary sources intuitionism are unavailable to me (Brouwer and Bey ting), the underlring philosophy will partlr be based on some of my own ideas. A development of a formal srstem for basic intuitionistio logic, along with classical logic, can be found in Kleene, Introduction to }\letamathematics, pp 69213 and a more philosophical discussion of mathematical reasoning pp 3668. That will be the basis of logical reasoning used here. It will not be developed for.mallJ, but occasional discussions will point out deviations from classical reasoning. The criteria for the reasoning can be taken as intuitive clearness and constructi bilitr: together, these might be summarized as imaginability. Intuitionists claim it to be impossible to lar out ahead of time all valid principals of reasoning; Godel's theorem on the incom pleteness of any formal system which includes number theorr has pr9vided strong backing for this claim. I will use four different mathematical concepts to replace classical set theorr. 1) Finite set theory. This is much like the finite ease or classical set theory. The principal ot tertium datur holds tor subsets of a finite set: given a finite set and a PAGE 6 3 subset or it, a complementary subset oan be tormed, so that any element in the set is either in the subset or in its com plementary subset (but not both). It is supposed that a list could be formed which would be known to contain exactly the members ot a finite set. Finite sets are to be distinguished from noninfinite denumerated sets, and even tram finitely bounded denumerated sets. An illustrative example will be given soon. 2) Denumerated sets. The basic kind ot denumerated set I call an array, to emphasize its dependence on its ordering (and thus it trom the classical denumerable set). The intuitionist image of a denumerated set is not the classical completed infinite, but instead a potential infinity: a denumer ated set is seen in an unterminating process or growing. One fairly liberal formal system tor arrays could be developed as follows. The basic pool of symbols is ( ) 0 1. The basic pool ot which mar belong to an array consists ot the finite sequences or these symbols. This pool is cut down by requiring that the subsequences (, ,) ,, do not occur, and that parentheses be closed in the proper manner. Then an array is defined as an algorithm which decides to accept or reject any sequence or symbols from the pool. In order for it to be possible to form a disjoint copy of any array, it is also required that there be a (known) tinite upper bound to the number ot compositions ot parentheses. An algorithm can be tound which determines the ordered pairs ot ot two given arrays: this is the PAGE 7 4 Cartesian product of the two given arrays. Functions can be defined either as in classical set theorya subarray ot the Cartesian product or two arrays fulfilling the obvious conditionor as a rule (algorithm) which assigns to any member of one array a member of a second array. One divergence from classical set theory at this point is that the image or an array under a function need not be a suparray of the range. Godel1s theorem provides an example of one such tunctioq, if we limit our reasoning to a given formal system: this function maps the natural numbers into the finite set consistiqg of 0 and 1, and it is impossible to prove within the formal system either that 0 is in the image ot the function or that it is not. Intuitionistic reasoning shows however that it is not. It we had not stumbled upon this reasoning, the function would be an example the assertion. There are examples, however, of functions whose images we know not to be arrays (see, for instance, Kleenes IntDo. to Metamathematics, section 61), and there are even Diophantine equations whose tmages are not arrays. There is no general method for finding an algorithm to determine the image or a function, even when image is bounded (as for the function in Godel's theorem); so intuitionistically, it is denied that tor any function, the image of the function can bedenumerated as an arrar. However, the image of a function is a denumerated set, tor it can be denumerated in the order of its domain. PAGE 8 5 The intractability of such a denumerated set is due to the basic divergence between the order of its denumeration and the order of the array which contains it. It is not in general possible, instance, to rearrange a denumerated subset or the natural numbers to coincide with the order of the natural numbers. By the very definition of array, given a subarray it is possible to find a complementary subarray. But it is not in general possible, given a denumerated subset of an array, to find a complement, even in the form of a denumerated subset. 3) Spreads. Finite sets can be thought of as lists of objects, and denumerated sets are the generalizations of such lists obtained by allowing the lists to grow indefinitely (pos. sibly with repetitions): by adjunction of objects. Finite 3ets have other natural structures. For instance, the natural way to describe the (all too finite) set of books in the New College Library would be to make use of the cataloguing system. That is, first to divide the books into rough categories, then to divide each of these categories still further, and so on. The set is thus described by division, rather than by adjunction. It more books were to be added, they would go into existing sections, but these sections could be refined into further categories. The generalization ot this dividing process obtained by allowing the refinements or categories to continue indefinitely is a spread. This particular kind or spread, for which the branches at aDJ stage rorm a finite set, ia called a tan. In the general spread, PAGE 9 6 the branches at any stage or the cataloguing form an array, and branches may terminate after a finite number of stages. 4) Species. Sometimes it would be impractical either to compile a list or the members or a finite set, or to classify the m embers; but it is easy to one or the members, by certain characteristics. The generalization or this kind of set is the species. The intuitionistic notion ot a species is a rule for recognizing certain (preformed) objects. Thus, the algorithm ror an array would be a speciesa very special kind, because an array algorithm also recognizes when an object is not a member the array. Mathematical manipulation or a species is simply manipulation or the predicate (rule tor recognizing the members) which defines it. Notice that each or these generalizations or the finite set includes the previous generalizations, or includes mathematical systems which are equivalent to any example of one ot the previous generalizations. A mathematical theory has a historical development, and another, static, kind or development. The latter is the ordering ot a theory by the dependence of concepts within the theory. Neither kind or dev elopment should attempt to begin with full and then make specializations and applications, or even begin with special cases and yet proceed teleological, to the final theory. Such a method leads to stagnation. Instead, 1t should be recognized trom the beginning of a development that it will never be possible to formalize completely the ideas PAGE 10 7 involved in the theory, nor will it ever be possible to attain rull generality. Thus, eaoh step in the development or a theory must be wellmotivated and intuitively clear in its own right. The process is each developmental step is an I attempt to better deal with the environment or concrete mathematical ideas and images, and as the theory develops the environment meanwhile adapts to it. The terminology or classical general topology reflects the teleological approach. Thus, a topological space' is the most general, while ordinary cases ot topologies must be highly qualiried: 'first countable locally compact Hausdorfr topolo gical space'. Unnecessary provinciality in definitions should be avoided on the one hand, but too much generality on the other hand can cause glossing over or important local deviations. We have divided omnipresent set theory into tour part$, and b7 doing this we will tind it possible to avoid using the very glib axiom of choice. The same policynot always to try tor maxi mum generalitywill be carried out in the future. The defini tion of topology to be given has simple generalizations which, however, are hard to handle. Instead or applying qualif7ing adjectives to the practical forms or the definition, adjectives can be applied to such generalizations. PAGE 11 Topology: Basic Definitions The following logical notation will be used. The tour propositional connectives do not require explanation: Implies : ::::} And: & Or: V Not: , 8 Propositional sentence order is like English sentence order it "that" before a clause is replaced by parentheses around the clause, with the exception that not always precedes the clause it modifies. The two quantifiers and ]xXfS'are to be interpreted as tallows. S is supposed to be a species, and x is then a variable which ranges over the ot the species. Then \lxxt)(A(x)) means we have a method that will provide a demonstration, given an arbitrarJ X Of the species, of the predicate A(x). This does not coincide with the classical idea or the generality quantifier, which is interpreted more like, tor every object x in the species s, A(x) holds'. Intuitionistioally, it is not claimed that it is to examine every member ot a (potentially) infinite species, eo the analogue or the classical generality quantifier must make the stronger statement, that we can show that A(x} holds tor an object x using in our proof' only the assumption that x is in the species S: the defining predicate of the sfleoies s. 3xxE<;(A(x}) means we have in mind a specific object t such that A(t); or at least, we have a method ot tinding or constructing such an object in the species. Then VxxE;<;(3)(Xf:S(A(x,,.))) would mean that, given an arbitrary xE:.S, we can prove that we have a method tor finding a 1 such that A(x,7) PAGE 12 9 which means we have a rule which assigns to each x in the species a y in the such that A(x,y). This rule may be part ot the defining predicate of the species, tor instance. The formula would mean we cannot find a general method such that, given an arbitrarr x in s, we can demonstrate A(x); which is not aw much as to say that we can find a specific x such that not A(.x): 3xxfS{,A(.x)). However, the following provable tormulas are not hard to reason through intuitively: V x X'S . A(x) r 3 x )(t:S A Cx) 3xxe S .A (X)=} Vxxf: SA (X) Arrays trom now on will be under a tood as explained in the introduction. Notation tor arrays will be similar to notation tor classical set theory: an object is accepted if it is accepted by A or by B, rejected it rejected by A and by B. an object is accepted it it is accepted by A and by B, rejected it rejected by A or by a. A*B: ('A complement in B1 ) an object is accepted it accepted by B and rejected by A, rejected it accepted by A or rejected by B. x.E:B: x B: ACB: the object X is by a. the object x is rejected by B. x is rejected tor arbitrary x in the basic pool for arrays. The basic setlike structure upon which a topology is imposed is formed by division: a spread. The definition ot spread to be given tor the purposes or topology will be slightly degeneralized PAGE 13 10 by the requirement that a branch be nonterminating. This is to correlate with the image that a topology is indefinitely divisible. A line of descent in a spread could even so never split into multiple branches; but the onlr case when this turns out to be topologically nontrivial is when this line of descent is not topologically 11olose" to any ot.her line ot descent, that is, the line or descent represents an isolated potnt. Definition. A spread is a sequence or a.rra,a tA}, and n a rule S which assigns to each XEAn a nonempty rueray S(x)C'n.+1 This rule must tulfill the conditions !> Vx A" Vv 1/e.ArJx F 1=>s PAGE 14 11 even though such a definition would be and equivalent, the definition given is more easily modified for the important special case that S(x) is a finite set, or tor the introduction ot more general Se, where x1 x2 doea not necessarily mean that Se(x1 ) and Se(x2 ) are disjoint. The rule S is called the successor assignment, and elements of the array S(x) are called successors, or immediate descendants, of x. A spread tor which S assigns a finite set to every x is said to be locally compact. It An is a finite set tor ever.1 n, then the spread is said to be compact. It ia easy to see that any subarray or a finite set not only is finite, but also is itself a tinlte set, so a compact spread is necessarily Compact spreads are also called fans. When we t alk or an object x in some An or a spread, we mean to consider x in its role as a member or An and not as a member or some other Am It this were to be formalized, it probably simplest to label the objects of An' so that they would be in the form (n,x). If this were to be done, there would be an algorithm to decide when object x is in some A and when it is not, and the objects in the spread could this n be tormed into an arrar. When we use the formula x Z tor a spread z, we will mean that Z is to be taken in the interpretation as an arra,.. PAGE 15 12 The Cartesian product ot n arrays A 1 An is defined as the array which accepts exactl7 the ordered ntuples (x1 ,Xn) with xi accepted by A1 The Cartesian or n spreads z1 is defined as the spread consisting of the sequence A A 1 (m = 0, 1, )and the successor assignment X Si r.f/\ ;m r;.11 which assigns to each ntuple (x1 z1 the Cartesian ,11 product or the component successor arrays: Here, Ai;m is the nth array in the sequence z1 and s1 is the successor assignment or zi. It is readiry verified that this sequence ot arrays with this successor assignment is a spread. Note that the Cartesian product ot spreads is not the Cartesian product ot the arrays which correspond to them. A subspread z of a spread Z is a sequence consisting or a aubarray A{ or each A1 and a successor tunction s by s: s(x) = The subarrays must be chosen in such a wa,that eaob x E. A 1 n+1 is a successor ot some '1 in A 1 n' and so that r or ant x Z s )('I z' F Detiniti on. A topology is a spread Z, and a sequence consisting ot a em or itnt(i) tor each m, where Z(i) = Z, to tultil1 the tollowing conditions: (1) c1 = z (ev&r'J piece is close to itselt'. The objects ot a a topology Z are called pieces.) PAGE 16 13 (2) (x1 any permutation p or {1, m ] (It a list or pieces is close, any permutation or the order ot the list forma a list which also is close.) ( 3) (X1' 1 Xi 1 1 Xm.) e_ cf4 (x1 1 1 1 1 Xm ) E Om where a caret over an ob.ject means that object is to be omitted. ( 4) Xm) somewhere ( 'any sublist 6t a close list is close,) (x1 x1 X:m) e. (x1 xi, m+1 C where here an additional xi has been inserted in the list. ('repetitions don1t alter the oloseness ot a list). (5) Local finiteness. There ia a finite set C(x) associated with each x which includes exactlr the objects y such tbat (x, y) E: (6) Transitivity in the limit. a) It (x, y) 02, and (y, z)E c2 and it tor every n there are nth order descendants xn, ,n, zn ot x, y and z respectively, such that(xn, yn)E c2 and PAGE 17 A set tultilling the condition that every pair or its members is close, is called pairwiee close. Any set that is 14 close must also be pairwise close, as is easily shown by repeated applicat ions or 3). Since en be a spread, everr ntuple which is close must have arbitra*ily distant close descendants, therefore arbitrarily distant pairwise close descendants: this is the converse or 6b). e1assicall7, it would tollow that all the em with m>2 would be redundant and unnecessary. But intui tionistically, it we had not explicitly demanded that the em be given, we would not generally be able to assert or deny the hypotheses of 6b). A topological theory is in tact feasible without the assumption or the higher order em, without the assumption that branches do not terminate and do not mingle, without the or aymmetl"1 ( 2). ), without the assumption or local finiteness. But the present definition is and easier to with. Example: the unit interv al I. Let the base spread ot the topology be the sequence Z ot Urays A 1 as tallows.; A 0 = o A 1 = 0, 1 A 2 = 001 011 10, 11 A = the tinite set ot ndigit binary numbers. n Let the successor assignment S assign to each binary number b the set consisting ot b tollowed by 0 and b followed by 1. PAGE 18 cl is defined by (1): c 1 is Z. Let c2 contain all repeated pairs (b, b) and all pairs adjacent in counting: (b, b+l), and (b+l, b). (Provided of course that they have the same number of digits.) Let en be generated by repetitions of components of pairs in c 2 It is not hard to verify that 15 all the ci thus defined form spreads, and in fact. fans. All the conditions (1)(6) are simply checked. Here is a diagram of this topology: Ao: Al: A3: A4: Arrows denote the successor assignment S; horizontal lines denote close pairs. The geometric interpretation of the topology just defined: Let A 0 correspond to the set containing a line segment of length 1. Let 0 as a member of A 1 correspond to the right half of this line segment; 1 corresponds to the left half. Whenever a correspondence has already been defined between some binary number b in A and a line segment, let the binary number bO correspond to the n right half of the line segment for b and bl correspond to the left half. It is easy to see that the closeness relation between pairs of these intervals corresponds to pairs of intervals of the same length which touch on the ends. Example: the real line E. Simply place copies of I endtoend, in a sequence that grows on both ends. PAGE 19 16 Definition. The Cartesian product X Ti of m topologies T1, Tm m is the Cartesian product of the corresponding spreads z1 Zm, and for each n the subspread Cn of X (X Ti) determined as follows. Form the Cartesian j (j) product of the c corresponding to the Ti; elements of this spread are then mtuples whose components are ntuples. Each ntuple component can be labelled where lSiSm, Ai; j If the indics are interchanged: xi;j is relabelled xj;i' this yields an ntuple of mtuples. Let en consist exactly of the ntuples of mtuples obtained in this way. In other words, an ntuple is close exactly when the ntuple obtained from the first components of its components is close, the ntuple obtained from the second components of its components is close, and so on. Example. The unit ncube In is X I(i)' where each I(i) I. In particular, the geometric interpretation of I2 can be diagrammed as follows: (0,0) (0,1) r (OO,! (00 00, (00, OO)' 01) 101 11) (01, (01' (01, (01, i (1,0) (1,1) 00) 01) 10) 11) I (10, (10, (10 (10, OOl Oll 11) (0, 0) (11' (11, (11 (11, 00) 01) 10' 11) It is elementary to verify that the Cartesian product of topologies is a topology. PAGE 20 17 The Comparison of Topologies An interpretation will be presented as motivation for some further developments. Spreads were presented in the introduction as generalizations of finite cataloguing systems. We interpret a spread, then, as an infinite (growing at the branchinp; tips) cataloguing or describing system. The classical structure which ordinarily takes the place of a given spread is the set of all infinite branches,of the spread. Rut instead of reifying the ultimate descriptions of our system as objects, and then identifying a certain description or category with the set of objects which fir the description, we think of the descriptions and the structure of .. .. the descriptive system as the basic information which is given to us. More concretely, we imagine varidus observers, each able to make observations and give us reports of these observations. We also imagine objects which we may present to the observers to be observed, hut these are "objects" in a different sense than above: they function as standards of consistency for the observationeach observation refers to some object, and (we think of the observers as being perfectly reliable) descriptions which may be applied to the same object are consistent with each other. An observer may end up describing two distinct objects in exactly the same way; if we had thought of objects as the ultimate descriptions of a descriptive system, these objects would be the same, hut we suppose that we have ways of keeping track of the identity of particular objects without relyinp; on the descriptions of some observer (for instance, through spatiotemporal continuity). On the other hand, we presume our PAGE 21 18 selves incapable of observing the objects ourselves, apart from keeping track of their identity, and so we get all our information about them secondhand through the various observers. Each observer has a definite repertoire of reports, which consists of finite sequences from a finite pool of symbols. (This means he can communicate to us a finite amount of information, in the form of distinct, nonmistakable language.) At when presented with an object he will give a report of the object, and a definite pool of reports is allotted to such first impressions. Then he can he requested to elaborate, any number of times; for each degree of elaboration there is a definite pool of allotted reports. If his cataloguing system is sharp, he must respond with the same nth elaboration each time he is presented with an object (the categories are mutually exclusive), and thus is a certain definite pool of reports which can be of any given report. (This pool is considered known to us and to the observer, because it is part of his conceptual system). The 8tages of elaboration are considered strict elaborationsif not, we could make them so by replacing a report by the sequence of reports the observer gives up till that oneand so the reports are permitted to follow distinct reports must be distinct. The standards of consistency compare various observers, as well as the various degrees of elaboration of the reports of one observer. We considered ourselves to know the conceptual systems of individual observers, and we also consider ourselves to know the relations between conceptual systems of different observers (exampleeyes, ears and nose). In fact, the objects are completely unnecessary to the theory of the conceptual systems, for we can obviously not get reports on enough objects to complete PAGE 22 19 ly define the structure and relationships of the conceptual systems (alias cataloguing systems, descriptive systems); the objects are mentioned only because the cataloguing systems serve as vehicles for handling the objects, and their structure is motivated by this. The comparison of two observers is called a correlation, and is put in the following form. To each category of 9ne stage of elaboration (say the nth) of one system is assigned an array of categories in the same stage of elaboration (the nth) of the other; and vice versa. As usual, these arrays must given by some rule (since an infinite number of assignments must be made). Then, if on a given level (stage of elaboration) we have a rule which assigns to each category an array of categories of another categorizing (cataloguing, describing, conceptual, spread) system, the rule can be simply deriveda is in array iff b is in array. A correlation is interpreted as telling us, when observer A gives one report on a given level, exactly which reports it is possible that observer B may give. It may occur that there is only one such possible correlated report, and in that case it is obviously necessary whenever A gives the report that B give the correlated report. From our ideas of consistency, we see that if a report by A is correlated by B, then there must be elaborations of these two reports which are correlated (consistent) with each other; and also, the immediate ancestors of consistent reports must be consistent. Each stage of elaboration can be thought of as a descriptive crosssection of the universe, and a correlation really acts between such crosssections; so we can think of the successor assignment as a correlation, with the PAGE 23 20 special property that the correlate of any category in the later stage of elaboration is necessary. With this, the last condition is a special case of the condition that necessary correlates of consistent reports be consistent. For the present, we regard descriptive systems all as referring to the same universe. Then, it must hold that the correlates in one crosssection of a category in another crosssection form a nonempty array, and an even stronger conditio must hold: if we say that one sequence of reports is correlated to another the representatives on each level of elaboration are correlated, then any sequence of reports by oneohserver must be correlated to at least one sequence of reports by a second observer. One way to compare vatious observers is to compare their power of distinguishing between objects. If observer A can distinguish between all pairs of objects which observer B can distinguish, then A is said to be more acute than B; if A and B are each more acute than the other, then they are said to have the same resolving power. Rephrasing in terms of conceptual systems and correlations, A is more acute than B whenever any pair of sequences of reports by A correlated to distinct sequences \ of reports by B are distinct. Description can be resolved into two processes: distinguishing and comparing. The categories of a descriptive system are not simply arbitrary groupings, or collections, of distinguished objectsthey are also comparisons, and the fact that the categories are formed in a certain way is an assertion in effect that everything within a single category is somehow similar. Observers can be compared in the way they categorize, as well as the way PAGE 24 21 they distinguish. Obviously if observer A classes an object in a certain category, observer R must place the object in a category of which Ks category is a possible correlate. But this does not mean that As category is ever a necessary correlate of the sequence of Ws reports, even when A and B have the same resolving power. For instance, let the objects to be observed be the natural numbers, and let Ns first report specify the natural number, while all elaborations are identical with his first report; and first report specify either 1 or 0,2,3, ; Bs second report specify either 1 or 2 or 0, 3, 4, ; and so on. Band A both distinguish between each pair of natural numbers, but when presented with the object 0, Ns first report 0 is never a necessary correlate of any of Rs reports. R always hedges about calling a 0 aO, while A comes right out and says it. This situation means that B has no way to translate As category 0, while A can translate every category of B. (Arrays must here be allowed as translations; since we think of arrays as being described by a rule, we can think of this rule as the translation, or at least an explica tion in terms comprehensible to A of Hs meaning). If every by B is a necessary correlate of classifications by A, then A is said to be more flexible than B; if A and B are each more flexible than the other, then they have the same conceptual power. It is not hard to see that being more flexible implies beins more acute. In many practical cases, a descriptive system is not sharp. One way to ohtain blurry descriptive systems is to allow overlapping categories on each level of elaboration. In this type of syetem, an observer always PAGE 25 22 chooses a definite category at each level of elaboration, but the category is not always uniquely determined by the object, and different reports at one level may have identical elaborations at another: we have a spread with mingling descendants. The description of such a cataloguing system would involve saying which categories at each level are consistent with one another, and for each initial sequence of reports, what following elaborations are consistent. We choose instead to obtain blurry spreads in a different way: by permitting an ohRerver to report indecision between several categories, but requiring that the formal system of permissible elaborations be sharpwith mutually exclusive permissible elaborations. An observer's report need not be uniquely determined by the object itself, so that when an observer makes a report, he is not saying that it is the only possible report (the necessary report), but only that he will be able to continue making elaborations of it indefinitely (like the Devil's Advocate): it is a plausible report. Part of the conceptual system must now involve stating which sets of alternatives are permissible as a report. We call such sets of elementary categories confusible sets, or in topology, close sets. We require that if a set of elementary eategories is confusible, then there must be at any stage of elaboration a confusible set of categories including at least one representative of the descendants of each of the original eategorieseonfusibility cannot be resolved. But confusion in the reports themselves can be resolved: an observer may report wavering between several eategories at one stage of elaboration, and yet at another stage make a report of which the categories are not all possible correlates. PAGE 26 23 This must be allowed, because we regard the process of observation, just like the process of reporting, as never completed. When two categories are not confusible, this means that after a finite amount of observation the observer can definitely place an object in the one or the other if it is in one of the two; hut when two categories are confusible, the sequence of reports consisting of an indecision between descendants of the two categories is permissible, and this sequence is never resolved into one or the other of the categories; so when an observer has an indecision between the two categories, he cannot tell whether he will be ahle to resolve this indecision or not, by a finite amount of observationso it is pointless for him to try. If he were ahle to tell exactly when he could resolve indecision between several categories, he might as well use the common borders of groups of categories as categories for then he would obtain a sharp system. The same discussion applies to larger sets of categories. From any category is derived a set of possible correlates of it on the same level: those categories confusible with it. When there are no other categories confusible with it, it is a necessary correlate of itself. Now we look at the first elaboration of this level. The possible correlates of the given category on this lower level form a smaller portion of the universe, in the sense that the only possible correlates of them on the former level I were the former possible correlates (from general properties of confusibility, or closeness). And so we can continue: the possible correlates of a given category on all levels together form a subspread of the original spread, and PAGE 27 24 this spread shrinks with respect to the original spread, in the sense that not all possible descendants of members of the spread are generally in the spread. Similarly, on each level there is an array consisting of the categories necessarily in a given category, i.e., the given category is a necessary correlate (on its level) of each member of the array. These do not form a subspread, because a category necessarily in the given category need not have an ancestor necessarily in the given category. However, by rearranging them so that each category which is the first in its line of descent to have the given category as a necessary correlate is placed on the first stage of elaboration, these categories necessarily in the given category do form a spread. And finally, the categories which are permissible or plausible of a given category (i.e., they are in a permissible line of descent with it,) together form a subspread. These three spreads are called the possibility, the necessity, and the plausibility, of the given category. The concept of a necessary correlate in the case of blurry spreads should be cleared up a little, before we continue. Suppose we two descriptive crosssections of the universe; this would be permissible for use in blurry classifying systems. But in such a case, any category wouln havP at leaAt two nossible correlates in this derived crosssection. The necessity ofi,any simple category would then be empty; for this and other cases, it is also to talk of an hdecision between several categories as a necessary correlate of another category whenever there are no other possible of this category within the crosssection of the indecision. In the present necessary correlates depend on PAGE 28 25 the particular crosssection in which a category or indecision between categories happens to be embedded, while possible correlates do not. The categories in the possibility of a given category but not in its necessity are those are correlated both to the category and to other categories in the crosssection of the given category. These also must form a subspread, evidently. This subspread is freedom of the given category. Freedom is locked between possibility and necessity; and as time progresses, necessity possibility shrinks, and freedomis squashed between them. More conventionally, these last three spreads could be called the interior subspread, the closure subspread, and the subspread of a piece. In the blurry cataloguing system, different do not in general distinguish between objects; objects, or rather, two sequences of permissible elaborations, are now whenever they are eventually not confusible with each otherwhen they are eventually in nonconfusible categories. The criterion for the acuity of observers now over from the sharp case. Observer A is more acute than observer B whenever any pair of sequences of reports by A correlated to distinguished sequences of reports by B are distinguished. The comparison of conceptual power also carries over without much change. Previously, every classification by an observer was a necessary classification, but we now need to explicitly require necessary classifications by B in: if every necessary classification by B is a necessary PAGE 29 correlate of classifications by A, then A is more flexible than B; it A and B are each flexible than the other, then A and B have the same conceptual power, or are topolo gically equivalent. (Under the given correlation) 26 The identity correlation ot a cataloguing system associates all pairs ot contusible categories (within constant levels). Under this correlation, a system should be more acute than itself and more flexible than itself. The acuteness condition means in particular that if arbitrary stages ot elaboration ot a pair ot categories confusible with a common category, then this pair of categories must be contusible (6a). This implies the condition, that it stages of elaboration ot a pair of categories have categories contusible with contus ible categories, then this pair of categories must be confueible under the assumption of local finiteness (S), tor then there must be two particular categories confusible reRpectively with the two given categories and contusible with each other, w1 th arbitrary pairs of descendants oontusible respectively w1thdeeoendants ot the two given categories and contuabile with each otheror more simply, there is a Qhain ot length tour having adjacent links contusible, and with arbitrary descendant chains of length tour having adjacent links con tusible. Then (6a) applied to the first tbree links makes PAGE 30 27 tho first and third links eonfuaible, and another application on the first, third and fourth links makes tho tirat and fourth oonfuaible. Brouwer1a tan theorem (see Kleeno and Vesley) states that if every growing branch of a tan (a spread which is finite on each level) is to eventually have some property, then there must be some finite n such that from level n on, every branch has tho property. With tho assumption of looal finiteness (S), the soquonoe of all categories con fusible with a given sequence of reports forma a tan with terminating branches. For any pair of sequences of elaborations which is eventually distinguished, any pair of branches troa tho oorreaponding tans (1. e., any pair of branches always cont"usible respectively with tho two given sequonees of reports) IIU&t eventually be distinguished as a consequence of (6a). It is easily seen, using a simple argument with the ran theorem, that there must be some finite n after which every pair of branches of the two epreade must be distinguished. Thus, (6a) and (S) imply that sequences of waverings oontusible respectively with two distinguished sequences of waveringa must eventually be distinguished. (6b) is derived readily from the assumption that tho identity correlation preserves acuteness, and w1 th an argument analogous to the above conditions (6a), (6b) .are equivalent to this assumption under the eondition (S). Soaewhat eurpr1zingly, the fan theorem and loeal finiteness (5) easily yield that (6a) PAGE 31 implies the condition that a categorizing system be more flexible than itself under the identity correlation. This oondition is the condition ot regularity, in topological terminologr, while (6a) is Hausdorttity and (5) is the con structive form of More general correlations can relate observers who do 28 not necessarily have the same scope of competence: observers may be permitted, when oontronted with certain obJects, to say, this is beyond me, I can1t make it out. Whenever there is at least one sequence of reports by A correlated to every sequence of reports by B, A has a broader scope of competence than B. (Under the give. n correlation.) This might more precisely be oalled a broader scope up to the resolution of B, for there might be a pair of objects which B does'not distinguish but one ot which A classifies and the other of which is out of his soope of competence; such a situation does not count against A in the comparison ot scope w1 th B1a scope. From this exa.11ple 1 t is easy to see that having a broader scope is not generally transitive, except when the observers concerned have the same resolving power. Classical continuous tunotiona correspond to the oaae When the seoond observer is less flexible but has a broader soope than the first observer. PAGE 32 2.9 Roots and Sand (or, a comparison or the apread and the point set as foundations tor topology) A direct analogue or classical topology, using the point set as a foundation, canno t be defined w1 thin intui tionistic mathematics, because the machinery available is not strong enough. But blurry eystema can be defined and developed within claesioal mathematics, because classical aaohinery is usually at least as strong as intuitionistio. To Justiry our definitions or topology, then, we must make the analogous definitions within classical mathematics and show their equivalence to the ordinary classical definitions; unless, that is, we wished to make a direct metamathematical comparison or the theorems provable our topological system (using intuitionistic reasoning) with the theorems provable in topology, showing in what oases analogous theorems are provable. The latter course is beyond the scope ot this paper, since it involves too much metamathematics. Consider any classical system which fulfills the conditions tor a spreadtopology. Form the set or all infinite branches or the spread. (This can be done, because this is a subset I ot the Cartesian product or the arrays in the spread; this set is nonempty in the case or a nonempty spread bf the axiom of choice.) Certain pairs or these branches remain always close. This relation between branches 1s transitive, PAGE 33 30 by ( 6a}, symmetric by (2) and reflexive by (l) and (4 ). Hence it is an equivalence relation, so we can identify equivalence classes and call them points. We define piece" in a new sense to mean the set of points with at least one branch in the corresponding 1piece" in the original sense. (Or, in the terminology of the interpretation of the preceding section, we are defining category in a new sense). We impose a this set of points, by defining basic neighbourhoods: the union of the pieces representing all the branches of a point p at level n is a basic neighbourhood of p. These neighbourhoods for a fixed point p form a (settheoretically) decreasing sequence, so that tinite of basic neighbourhoods of a point contain a basic neighbourhood or that point. From a correlation under which A is more acute than B, and under whiCh B has a broader scope than A, a function can be derived: take any point of A to the intersection of the unions of the correlates of the ot any branch of the point. There is exactly one point in this intersection, because A is more acute than B. Even without the two hypotheses on the correlation, a correspondence between the pointsets of A and B is derived from a correlation, by letting p correspond to q when q is always in some correlate of some branch or p. PAGE 34 On the other hand, given any function taking the point set ot A to the point set ot B, a correlation is derived: correlate a category ot A with every category on the same level of B intersects the closare ot its image. It is easily verified that the derived correspondence between categories is a correlation, except for the condition that a category correspond to a an array, and that this corresondence be defined by a rule. However, this condition is unnecessary within classical mathematics, tor correspondences can be welldefined without being given constructively; so we drop that condition. How does the correspondence derived from this correlation compare with the original function? Form the Cartesian product A X B. The original function was a subseD ot the pointset Cartesian product. The derivedderived correspondence is the closure ot this subset: any point which has a neighbourhood which does not intersect the original subset is obviously not in the derivedderived correspondence, while any point tor which every neighuour hood intersects this original subset must always have representatives ot branchespairs ot contain points in the graph ot the function, and thus are correlated. Then, bf definition, this point corresponds to a pair ot points in the derived correspondence. Similarly the derivedderived correlation trom a correlation is a kind ot closure ot the PAGE 35 ' original correlationa pair of level categories are in the derivedderived correlation w.henever it always has des/ cendant pairs close to pairs of pieces in the correlation. JZ In summary, these derivations form a Galois connection (see Garrett Birkhorf, Lattice Theor1, p. 56) between the lattice of subsets or the Cartesian product or the pointsets and the lattice of subspreads or the Cartesian product or the spreads. In particular, the derivedderived correspondence isthe original correspondence exactly when the original correspondence had a closed graph; so from any continuous tunction is derivedderived the original function. Since being more flexible implies being more acute, whenever A is more flexible and narrower in scope than B under a given correlation, a function is derived. Such a function must be continuousan immediate corollary ot the definitions. The converse is also immediate. Taking into account other obvious connections between spreads and pointsets, such as the connection between the two kinds of Cartesian we have: The classical theory or topology on spreads is equivalent to the port,ion or the classical theory ot topology which involves structures and operations definable by spreadsin particular, continuous tunctions and homeomorphisms (topo logical equivalences) between topologies definable trom spreads, t1n1te (and indeed, denumerable) Cartesian products, and .. PAGE 36 JJ quite a large portion ot the other desirable things in point set topology. Now we investigate the question ot how large a class ot pointsettopologies are definable from spreads. First we get some definitions out ot the way: An essential cover ot a space is a family ot. sets such that the interiors ot finite unions ot closures ot sets in the family form a cover. Obviously, an open cover is an essential cover. family ot sets is close, it the intersection ot the closures ot its members is nonvoid. From any essential cover an open cover can be derived, by forming the family ot interiors ot unions ot closures ot members ot finite close familiesin other words, by limiting the tinite unions in the definition to the case ot close families. This torms a cover, tor given a point in the interior ot some finite union or closures ot members or the essential cover, the point is still in the interior or the union when 'those sets whose closures are disJoint trom the point are eliminated; and the remaining sets torm a close family, tor their closures intersect at least at the point. A olose refinement ot an \ essential cover is an essential cover, such that its derived open cover refines the derived open cover of the original essential cover. Obviously, an open refinement or an open cover is also a close refinement. A olose: simple refinement ot a cover is an essential oover whose derived open cover refines the original cover; etc. Close: simple refinements PAGE 37 )4 are similar to starrefinements (a star is the union of a close family, n?t necessarily finite. A cover V is a star refinement of U it the family or stars derived trom V refines U. See J. L. Kelley, General Topology, p. 170). A clean cover is an essential cover, such that the intersections of interiors or closures or pairs of ita members are void. A olean refinement o f a olean cover is a close refinement such that each set or the refinement is contained in the closure of one of the original seta. A family of seta is locally finite, if every point in the apace has a neighbournood w.hioh intersects only a finite number of seta in the family. A space is paracompact if every open cover has a locally finite refinement. Lemma l: Each paraoompact and Hausdorff space is regular, and each paracompaot and regular space is normal. This is a standard theorem, Whose proof is analogous to that of the corresponding theorem for compact spaces. See e.g. Hu, Elements of General Topology, propositions 2.19 and 2.20 (p. 68) or Kelley, PP 156, 159. Lemma 2: Each regular 2nd countable space is paraoompaot. See Kelley, p. 156 Theorem 28. Lemma 3: Each countable locally finite open cover or a normal space has an open olose:simple locally finite refinement. PAGE 38 For a sketch ot the proof, see my notes. to Lecture 2, uA Constructive of Topology,u, presented on May )0, 1967; p. 21. .35 Lemma 4. Every countable open essential cover has a close refinement by a olean cover. H Proo:t : From a cover define Bi = Au,( l) Ai ) "' Y\ f'=1 Then (l) Bif= PAGE 39 36 The sequence ot covers derived this way must be a locally close:simple refining sequence of essential covers; for, given a point artd a neighbourhood of the point, there must be some An which is contained in this neighbourhood, and by regularity. there must ee some Am, a neighbourhood of the point whose closure is contained in An; as soon as q exceeds An and Am the qth cover will have the property that every close containing the point must be contained in the neighbourhood. Call this sequence s1 b) Cons'truct, by Lemma 2, a locally finite simple: close refining sequence 82 d BJ_. Such a sequence must also be locally refining. c) Find, by Lemma), a locally finite close:simple refining sequence 83 of s2 d) Construct a locally finite closerefining sequence 84 of 83 consisting d clean covers, by Lemma 4. e) If 84 consists of clean covers Un1 define S5.to be the sequence Vn' where Vn consists of intersections of one set from each of U1, Un 85 is now a locally finite olean sequence of clean covers, of which the derived sequence of open covers is locally refining. Thus we can construct a spread from any topology. However, the spread we construct in this way is always compact, so the derived classical topology is compact. If we begin with a d d d rived topolog7 is the original, compact topology, the e PAGE 40 )7 It we have a locally compact topology, we can find a spread for it using this construction on each set ot some decomposition or the apace into compact .subspaces (possible any 2nd countable space. It is not possible to represent every paracompact 2nd countable regular space as a spread: a very simple example is the set of rational numbers with the topology as a subspace of the reals. If we attempt to represent them as a spread with this topology, we end up completing them, to form the reals. An example of a paracompact but not locally compact s.pace which it is possible to represent as a g,pread is the countable Cartesian product of copies of the real line. That is as tar as I have investigated the '81 tuation; t.o me, the condition or being representable as a spread seems a very natural way to designate the class ot topologies which are represent able as spreads. As a somewhat surprising development, it might be noted that 1t the local finiteness condition were dropped from the definition of a topology, it appears that classical topologies could be obtained which would not have countable bases, or even countable bases tor the neighbourhood systems or individual points. PAGE 41 B1 bl1ography Garrett Birkhoff, Lattice Theory, American Math. Soc. Colloquium Publications v. XXV, Providence: 1948 Samuel Eil.enberg and Nor.nan Steenrod, Foundations of Al0ebraic Topology; Princeton: 1952 s. T. Hu, Elements of General Topology, HoldenDay, San Francisco: 1964 John L. Kelley, General D. van Nostrand, Princeton: 1955 Stephen C. Kleene, Introduction to D. van Nostrand, Princeton: 1952 Stephen c. Kleene and R. s. Vesley, The Foundations of Intuitionistic Mathematics, NorthHolland Publishing Co., Amster 1965 s. Korner, The Philoso2h! of Mathematics, Hutchinson, London: 1960 CasimirKuratowsk1, Topolosie, v.1 and v.2 (Quatrieme Edition), Panstwowe Wydawnictwo NaUkOwe, Warszawa (Poland); 1958 E. J. Le:nmon, "New Foundation A for LewiR 1lodal Sysrbems", i. Symbolic Logic, 22 (1957) pp. 176186 J. c. c. McKinsey and A. Tarski, "Some Theorems about the Sentent,ial Calculi or Lewis and Heytine/', J. Syinbihlic Los1c 13 (1948) pp. 113 A. Mostowski, applications de la Topologie) la Logique Mathematique', appendice VIII a Topologie, v.l, C. Kuratowski. Bill Thurston, "Lecture 1: Two Standards for Mathematics" and "Lecture 2: A Constructive Theory for Topology" (mimeo graphed notes, New College, May 1967). 