Ideal objects in set theory and topology Oliver Kullmann Historical remarks Ideal objects in set theory and topology Enumerating sets Topologies and filters Bases and Oliver Kullmann subbases Computer Science Department Topology: SAT in disguise Swansea University Compactness MRes Seminar Swansea, November 17, 2008 Ideal objects in set Introduction theory and topology In this lecture some fundamental aspects of set theory Oliver Kullmann and topology related to “ideals” (and their existence) are Historical remarks discussed: Enumerating sets First we reflect upon “access” to sets in sets theory: Topologies and filters Normally framed in considerations around the “axiom Bases and of choice”, I regard the original Cantorian intuition as subbases Topology: SAT in more adequate, and thus we’ll discuss (rather briefly) disguise well-orderings. Compactness A simple applications yields the existence of (many) maximal ideals in rings. Then we discuss fundamental notions of topology, focusing on “topologies” and “filters”. Of central importance is the notion of a “subbasis”. Finally, we discuss (quasi-)compactness, and prove Tychonoff’s theorem. Ideal objects in set theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and Often no proofs are given, but emphasise is put on the filters definitions and properties, leaving proofs as relatively Bases and straightforward exercises. subbases Topology: SAT in disguise You must fill the gaps yourself!! Compactness Ideal objects in set Overview theory and topology Oliver Kullmann Historical remarks 1 Historical remarks Enumerating sets Topologies and 2 Enumerating sets filters Bases and subbases 3 Topologies and filters Topology: SAT in disguise Compactness 4 Bases and subbases 5 Topology: SAT in disguise 6 Compactness Ideal objects in set George Boole theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and filters Bases and subbases Topology: SAT in disguise Compactness Ideal objects in set Georg Cantor theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and filters Bases and subbases Topology: SAT in disguise Compactness Ideal objects in set Felix Hausdorff theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and filters Bases and subbases Topology: SAT in disguise Compactness Ideal objects in set theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and filters Bases and subbases Topology: SAT in disguise Compactness Ideal objects in set theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and filters Bases and subbases Topology: SAT in disguise Compactness Ideal objects in set theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and filters Bases and subbases Topology: SAT in disguise Compactness Ideal objects in set theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and filters Bases and subbases Topology: SAT in disguise Compactness Ideal objects in set Wellordered sets theory and topology Unter einer wohlgeordneten Menge ist jede Oliver Kullmann wohldefinirte Menge zu verstehen, bei welcher Historical remarks die Elemente durch eine bestimmt vorgegebene Enumerating sets Succession mit einander verbunden sind, Topologies and welcher gemäss es ein erstes Element der filters Bases and Menge giebt und sowohl auf jedes einzelne subbases Element (falls es nicht das letzte in der Topology: SAT in Succession ist) ein bestimmtes anderes folgt, disguise Compactness 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) Ideal objects in set theory and topology Translation (using modern terminology): Oliver Kullmann A “wellordered set” ist a set M together with a Historical remarks linear order ≤, such that every subset, which Enumerating sets has a strict upper bound, has a smallest strict Topologies and filters upper bound. Thus Bases and there is a smallest element of M if M is not subbases Topology: SAT in empty; disguise every element, which is not the largest Compactness 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. Ideal objects in set Exhausting sets theory and topology In the same article: Oliver Kullmann Der Begriff der wohlgeordneten Menge weist Historical remarks sich als fundamental für die ganze Enumerating sets Topologies and Mannichfaltigkeitslehre aus. Dass es immer filters möglich ist, jede wohldefinirte Menge in die Bases and Form einer wohlgeordneten Menge zu bringen, subbases Topology: SAT in auf dieses, wie mir schient, grundlegende und disguise folgenreiche, durch seine Allgemeingültigkeit Compactness 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.) Ideal objects in set More on exhausting sets theory and topology Oliver Kullmann Historical remarks Enumerating sets The principle behind well-ordering a set is Topologies and filters that we grab elements out the “bag”, one after Bases and another, subbases Topology: SAT in and if this doesn’t exhaust the set, then at “limit disguise steps” we make a jump, assume that the infinitely Compactness many choices have been made, and start with a new chosen element which is posited as “successor” of all the elements previously chosen. Ideal objects in set theory and Fundamental intuitions on the notion of “set” are: topology 1 A set presents an “actual infinity”, not a “potential Oliver Kullmann infinity”. Historical remarks 2 Better said, the set is “fixed” (“frozen”), it is “being”, Enumerating sets Topologies and not (like proper classes(!)), “becoming”. filters 3 Thus the set is “measurable”, where its “size” is Bases and subbases measured by a “cardinal number”. Topology: SAT in disguise 4 The successor of this cardinal number provides a Compactness 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!) Ideal objects in set Exhausting sets, more precisely theory and topology Oliver Kullmann Historical remarks With Hausdorff’s “Grundzüge der Mengenlehre” (1914): Enumerating sets Aus einer unendlichen Menge A greife man Topologies and filters willkürlich ein Element heraus, daß man mit a0 Bases and bezeichne, dann aus A − {a0} ein Element a1, subbases aus A − {a , a } ein weiteres Element a usf. Topology: SAT in 0 1 2 disguise Dies ist für jede endliche Zahl möglich. Wenn Compactness die Menge {a0, a1, a2,...} noch nicht die ganze Menge A ist, so läßt sich aus A − {a0, a1, a2,...} ein weiteres Element aω auswählen, wenn damit A noch nicht erschöpft ist, ein Element aω+1 usw. Ideal objects in set Continuation theory and topology Dies Verfahren muß einmal ein Ende nehmen, Oliver Kullmann denn über der Menge W der Ordnungszahlen, Historical remarks denen man Elemente von A zuordnen kann, gibt Enumerating sets es größere Zahlen, und diesen kann man also Topologies and keine Elemente von A mehr zuordnen. Man filters Bases and kann nun leicht zeigen (s.u.), daß dann auch alle subbases Elemente von A verbraucht sind, also A mit W Topology: SAT in äquivalent ist. disguise Compactness 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. Ideal objects in set Zorn’s Lemma theory and topology Oliver Kullmann Historical remarks Enumerating sets A poset (M, ≤) is called inductive if every chain of M (a Topologies and linearly ordered subset) has an upper bound. filters Bases and For every inductive poset (M, ≤) and every x ∈ M there subbases Topology: SAT in exists a maximal element y ∈ M with x ≤ y. disguise Compactness 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. Ideal objects in set Maximal ideals in rings theory and topology Oliver Kullmann Historical remarks Enumerating sets Topologies and filters Consider a ring R and a proper ideal I of R Bases and subbases (that is, I 6= R). Then there exists a maximal Topology: SAT in proper ideal I0 of R with I ⊆ I0. disguise Compactness Proof: The poset of proper ideals of R (ordered by subset-inclusion) is inductive. Ideal objects in set Reminder: Filters theory and topology Oliver Kullmann A filter on a set M is a set system F ⊆ P(M) stable Historical remarks under finite intersection and superset formation. Enumerating sets F is called proper if F 6= (M), which is equivalent Topologies and P filters to ∅ ∈/ F. Bases and A principal filter (or trivial filter) consists exactly of subbases Topology: SAT in the supersets of {x} for some x ∈ M. disguise Compactness 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. Ideal objects in set Reminder: Topologies theory and topology Oliver Kullmann Historical remarks Recall: A topological space is a pair (X, τ) where X is a Enumerating sets set and τ is a set system on X stable under finite Topologies and intersections and arbitrary unions.