Oliver Kullmann Computer Science Department Swansea University. MRes Seminar Swansea, November 17, 2008

Transcription

1 Computer Science Department Swansea University MRes Seminar Swansea, November 17, 2008

2 Introduction In this lecture some fundamental aspects of set theory and related to ideals (and their existence) are discussed: First we reflect upon access to sets in sets theory: Normally framed in considerations around the axiom of choice, I regard the original Cantorian intuition as more adequate, and thus we ll discuss (rather briefly) well-orderings. A simple applications yields the existence of (many) maximal ideals in rings. Then we discuss fundamental notions of, focusing on topologies and. Of central importance is the notion of a subbasis. Finally, we discuss (quasi-)compactness, and prove Tychonoff s theorem.

3 Often no proofs are given, but emphasise is put on the definitions and properties, leaving proofs as relatively straightforward exercises. You must fill the gaps yourself!!

4 Overview

5 George Boole

6 Georg Cantor

7 Felix Hausdorff

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12 Wellordered sets Unter einer wohlgeordneten Menge ist jede wohldefinirte Menge zu verstehen, bei welcher die Elemente durch eine bestimmt vorgegebene Succession mit einander verbunden sind, welcher gemäss es ein erstes Element der Menge giebt und sowohl auf jedes einzelne Element (falls es nicht das letzte in der Succession ist) ein bestimmtes anderes folgt, wie auch zu jeder beliebigen endlichen oder unendlichen Menge von Elementen ein bestimmtes Element gehört, welches das ihnen allen nächst folgende Element in der Succession ist (es sei denn, dass es ein ihnen allen in der Succession folgendes überhaupt nicht giebt). (Georg Cantor, Über unendliche, lineare Punktmannichfaltigkeiten, Teil 5; Mathematische Annalen, Band 21, 1883)

13 Translation (using modern terminology): A wellordered set ist a set M together with a linear order, such that every subset, which has a strict upper bound, has a smallest strict upper bound. Thus there is a smallest element of M if M is not empty; every element, which is not the largest element (if there is one), has a successor. Equivalently: A wellordering of a set M is a linear order on M such that every non-empty subset has a smallest element.

14 Exhausting sets In the same article: Der Begriff der wohlgeordneten Menge weist sich als fundamental für die ganze Mannichfaltigkeitslehre aus. Dass es immer möglich ist, jede wohldefinirte Menge in die Form einer wohlgeordneten Menge zu bringen, auf dieses, wie mir schient, grundlegende und folgenreiche, durch seine Allgemeingültigkeit besonders merkwürdige Denkgesetz werde ich in einer späteren Abhandlung zurückkommen. Simplified translation: Every set can be wellordered, and this is a fundamental property of sets. (Where according to Cantor a mathematical object exists iff it is present in the omnipotent intellect of god.)

15 More on exhausting sets The principle behind well-ordering a set is that we grab elements out the bag, one after another, and if this doesn t exhaust the set, then at limit steps we make a jump, assume that the infinitely many choices have been made, and start with a new chosen element which is posited as successor of all the elements previously chosen.

16 Fundamental intuitions on the notion of set are: 1 A set presents an actual infinity, not a potential infinity. 2 Better said, the set is fixed ( frozen ), it is being, not (like proper classes(!)), becoming. 3 Thus the set is measurable, where its size is measured by a cardinal number. 4 The successor of this cardinal number provides a powerful enough exhaustion process (by all smaller ordinal numbers ). 5 Here the cardinality of sets just measures its size, while an ordinal number provides the details of how we (transfinitely) enumerated it. (So well-ordering of sets are possible iff measurement tools are powerful enough!)

17 Exhausting sets, more precisely With Hausdorff s Grundzüge der Mengenlehre (1914): Aus einer unendlichen Menge A greife man willkürlich ein Element heraus, daß man mit a 0 bezeichne, dann aus A {a 0 } ein Element a 1, aus A {a 0, a 1 } ein weiteres Element a 2 usf. Dies ist für jede endliche Zahl möglich. Wenn die Menge {a 0, a 1, a 2,...} noch nicht die ganze Menge A ist, so läßt sich aus A {a 0, a 1, a 2,...} ein weiteres Element a ω auswählen, wenn damit A noch nicht erschöpft ist, ein Element a ω+1 usw.

18 Continuation Dies Verfahren muß einmal ein Ende nehmen, denn über der Menge W der Ordnungszahlen, denen man Elemente von A zuordnen kann, gibt es größere Zahlen, und diesen kann man also keine Elemente von A mehr zuordnen. Man kann nun leicht zeigen (s.u.), daß dann auch alle Elemente von A verbraucht sind, also A mit W äquivalent ist. In short: If we take a sufficiently large ordinal number, then by choosing for each smaller element (by transfinite induction) some new element of A we can exhaust A, and the obtained initial segment of ordinal numbers yields a well-ordering of A. Using for the choices the Axiom of Choice, we see that the Axiom of Choice is equivalent to the Wellordering Axiom.

19 Zorn s Lemma A poset (M, ) is called inductive if every chain of M (a linearly ordered subset) has an upper bound. For every inductive poset (M, ) and every x M there exists a maximal element y M with x y. Proof idea: Just grab bigger elements (as long as they exist), and for the limit steps (where we jump ) use inductiveness this process must stop with a maximal element y above the start element x.

20 Maximal ideals in rings Consider a ring R and a proper ideal I of R (that is, I R). Then there exists a maximal proper ideal I of R with I I. Proof: The poset of proper ideals of R (ordered by subset-inclusion) is inductive.

21 Reminder: Filters A filter on a set M is a set system F P(M) stable under finite intersection and superset formation. F is called proper if F P(M), which is equivalent to / F. A principal filter (or trivial filter) consists exactly of the supersets of {x} for some x M. Since a filter does not care about big sets, every filter captures some ( robust ) notion of small sets. The notion of a filter is too general to allow some form of measurement, but to express that something holds for small sets, we say that it holds for all elements of the respective filter.

22 Reminder: Topologies Recall: A topological space is a pair (X, τ) where X is a set and τ is a set system on X stable under finite intersections and arbitrary unions. τ is called the of the space. The elements of τ are called the open sets of the topological space. The complements of the elements of τ are called the closed sets of the topological space. The closed sets form a dual, which is stable under finite unions and arbitrary intersections.

23 Neighbourhood Given a topological space (X, τ), the neighbourhood filter N (x) of a point x is the filter coarsening the principal filter at x by considering all supersets of open sets containing x. For the finest τ = P(X), the discrete, the neighbourhood filter at every point x is the principal filter at x. For the coarsest τ = {, X}, the indiscrete, the neighbourhood filter at every point x is just {X}. The system of neighbourhood of a topological space specifies how close we can come to each point.

24 Neighbourhood axioms Given a set X, a neighbourhood system N on X associates to every x X a filter N (x) coarsening the principal filter at x such that: For every x X and every U N (x) there exists W U such that x W and y W : W N (y). Given a neighbourhood system N on X, the open sets w.r.t. N are defined as the subsets O X such that O is a neighbourhood of every element of O (so the above condition just asks for an open neighbourhood W of x inside U).

25 Pairs (X, N ), where N is a neighbourhood system on X, were the original topological spaces. Topological spaces via open sets (as we defined them, and as it is common now) are cryptomorphic to topological spaces via neighbourhood system ( encoding and decoding are inverse operations, from both starting points).

26 Reminder: Morphisms Filtered sets and topological spaces are special cases of structures given by a set and a set system (of subsets). As usual in mathematics, the morphism are the backward morphisms (recall last lecture).

27 Continuity A map f : X Y is called continuous, if for every x X f (x) and f (x ) will be as close as we wish (guaranteed), if only we make x sufficiently close to x. For X, Y being topological spaces, closeness is measured by neighbourhoods, and thus f : X Y is continuous (by definition) if for all x X and every neighbourhood W of f (x) there exists a neighbourhood U of x with f (U) W. It is an important exercise to show that continuous maps between topological spaces are exactly the morphisms between topological spaces (as defined before).

28 Closures of set systems Recall: A special species of structures is given by a pair (M, S), where M is a set (of course), and S is a set system on M, i.e., S P(M). Consider the set S P(P(M)) of all possible set systems S (as given by the species). A very important property of S (nearly indispensable) is to be a closure system, that is, for all S S we have M S S. Then every set system S 0 P(M) generates a structure S 0 as the smallest set system in S containing S 0 : S 0 = M {S S : S 0 S}.

29 Filter The system of all on a set M is a closure system. Given F 0 P(M), the filter F generated by F 0 is obtained by the following 2-step process: 1 Let F 0 be the closure of F 0 under finite intersections. 2 Then F is the closure of F 0 under formation of super-sets. Given a filter F on M, a ubbasis for F is a set F 0 F such that the filter generated by F 0 is F. A filterbasis for F is a set F 0 F such that the closure of F 0 under formation of super-sets yields F. So from a ubbasis first a filterbasis is created, and then the filter. Every F 0 is a ubbasis (of some filter), while F 0 is a proper ubbasis (a ubbasis of some proper filter) iff no finite intersection with elements of F 0 is empty.

30 Ultra An ultrafilter is a proper filter U on M such that no proper filter F on M with U F exists. Every trivial filter is an ultrafilter, and for finite sets the converse also holds. For a filter F on M the following conditions are equivalent: 1 F is an ultrafilter on M. 2 For every subset T M either T F or M \ T F holds (but not both). 3 F is proper, and for all A, B M with A B F we have A F B F. Since the system of proper on M is inductive, by Zorn s Lemma we obtain: For every proper filter F on M there exists an ultrafilter U on M with F U. (Furthermore every proper filter is the intersection of all ultra containing it.)

31 Subbases for the open sets The system of all topologies on a set X is a closure system. Given T P(X), the τ generated by T is obtained by the following 2-step process: 1 Let τ 0 be the closure of T under finite intersections. 2 Then τ is the closure of τ 0 under arbitrary unions. Given a τ on X, a subbasis of τ is a set T τ such that the generated by T is τ. A basis for τ is a set τ 0 τ such that the closure of τ 0 under arbitrary unions yields τ. So from a subbasis first a basis is created, and then the.

32 Subbases for the closed sets The system of all dual topologies (given by the closed subsets ) on a set X is a closure system. Given T P(X), the dual τ generated by T is obtained by the following 2-step process: 1 Let τ 0 be the closure of T under finite unions. 2 Then τ is the closure of τ 0 under arbitrary intersections. Given a dual τ on X, a subbasis of τ is a set T τ such that the dual generated by T is τ. A basis for τ is a set τ 0 τ such that the closure of τ 0 under arbitrary intersections yields τ. So, again, from a subbasis first a basis is created, and then the dual.

33 Continuity and Given topological spaces X, Y, a map f : X Y and a subbasis S of Y, it is easy to see: f is continuous iff for all O S the set f 1 (O) is open in X. Stronger: f is a quotient map (recall last lecture) iff f is surjective and {f 1 (O) : O S} is a subbasis (of the open sets) of X. In other words, quotient maps are surjective maps such that the on the domain space is the coarsest to make the map continuous.

34 Products of topological spaces For a given family (X i ) i I of sets, the product is the set X i := {f : I X i i I : f (i) X i } i I i I together with the canonical projections pr j : i I X i X j. Now we assume that all X i are topological spaces. The product has as subbasis the set of all pr 1 i (O) for i I and O open in X i. If for each X i a subbasis S i is given, then a subbasis of the product is given by the set of all pr 1 i (O) for i I and O S i. Thus the product is the coarsest on the product set such that all projections are continuous.

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38 Quasicompact spaces A topological space (X, τ) is quasicompact if every open cover of X contains a finite subcover, i.e., for every O τ with O = X there exists a finite O O with O = X. (X, τ) is quasicompact iff every proper ubbasis F 0 consisting of (some) closed sets has nonempty intersection (i.e., F 0 ). A compact space is a quasicompact space which is also hausdorff (T 2 ), i.e., every two different points have disjoint neighbourhoods. The Hausdorff separation axiom is most important in the context of the notion of (quasi)compactness. Thus, following Bourbaki, it is integrated into the notion of compactness.

39 Basic properties of compactness A subset A of a topological space X is called (quasi)compact, if it is (quasi)compact as a topological subspace. A is quasicompact iff every open cover of A in X has a finite subcover. If X is (quasi)compact and A is closed, then also A is (quasi)compact. If X is hausdorff and A is compact, then A is closed. The image of a quasicompact set under a continuous map is quasicompact. Thus every continuous map from a quasicompact space to a Hausdorff space is closed (the image of a closed set is closed).

40 Alexander s Subbasis Theorem Consider a topological space (X, τ) and a subbasis S: X is quasicompact iff every open cover of X by elements of S has a finite subcover. Using the formulation of compactness by closed sets, we obtain the following equivalent statement: Consider a subbasis S of the closed subsets. Then X is quasicompact iff A for every proper ubbasis A S.

41 Proof Consider a subbasis S of the closed subsets. Consider a set A of closed sets of X, such that A is a proper ubbasis. Assume A =. So for every x X there exists A x A with x / A x. Since S is a subbasis, for each x X there is now a finite S x S with S x A x and x / S x. Let U be an ultrafilter with A U. So for every x X there exists S x S x with S x U. Now {S x : x X} would be a proper ubbasis with x X S x =.

42 Tychonoff s Theorem Given a family (X i ) i I of sets, and a family (A i ) i I of subsets A i X i, we have ( i I X i ) \ ( i I pr 1 i (A i )) = i I pr 1 i (X i \ A i ) = X i \ A i. i I Thus, an easy application of the subbasis theorem yields: The product of quasicompact topological spaces is again quasicompact. And since the product of T 2 -spaces is again T 2 : The product of compact spaces is compact. (Also the reverse directions hold, provided that all spaces are non-empty.)

43 (finally) End