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. filters Bases and τ is called the topology of the space. subbases Topology: SAT in The elements of τ are called the open sets of the disguise topological space. Compactness The complements of the elements of τ are called the closed sets of the topological space. The closed sets form a dual topology, which is stable under finite unions and arbitrary intersections. Ideal objects in set Neighbourhood filters theory and topology
Oliver Kullmann
Given a topological space (X, τ), the neighbourhood Historical remarks filter N (x) of a point x is the filter coarsening the Enumerating sets Topologies and principal filter at x by considering all supersets of open filters
sets containing x. Bases and subbases For the finest topology τ = (X), the discrete P Topology: SAT in topology, the neighbourhood filter at every point x is disguise the principal filter at x. Compactness For the coarsest topology τ = {∅, X}, the indiscrete topology, the neighbourhood filter at every point x is just {X}. The system of neighbourhood filters of a topological space specifies how “close” we can come to each point. Ideal objects in set Neighbourhood axioms theory and topology
Oliver Kullmann
Historical remarks
Given a set X, a neighbourhood system N on X Enumerating sets
associates to every x ∈ X a filter N (x) coarsening the Topologies and principal filter at x such that: filters Bases and subbases For every x ∈ X and every U ∈ N (x) there exists W ⊆ U Topology: SAT in such that x ∈ W and ∀ y ∈ W : W ∈ N (y). disguise Compactness 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). Ideal objects in set theory and topology
Oliver Kullmann
Historical remarks
Enumerating sets Pairs (X, N ), where N is a neighbourhood system Topologies and on X, were the original “topological spaces”. filters Bases and Topological spaces via open sets (as we defined subbases them, and as it is common now) are cryptomorphic to Topology: SAT in disguise topological spaces via neighbourhood system Compactness (“encoding” and “decoding” are inverse operations, from both starting points). Ideal objects in set Reminder: Morphisms theory and topology
Oliver Kullmann
Historical remarks
Enumerating sets
Topologies and filters
Filtered sets and topological spaces are special Bases and cases of structures given by a set and a set system subbases Topology: SAT in (of subsets). disguise As usual in mathematics, the morphism are the Compactness “backward morphisms” (recall last lecture). Ideal objects in set Continuity theory and topology
Oliver Kullmann
A map f : X → Y is called continuous, if for every x ∈ X Historical remarks
Enumerating sets f (x) and f (x0) will be as “close” as we wish (guaranteed), 0 Topologies and if only we make x sufficiently “close” to x. filters Bases and For X, Y being topological spaces, closeness is subbases Topology: SAT in “measured” by neighbourhoods, and thus disguise
Compactness 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). Ideal objects in set Closures of set systems theory and topology
Oliver Kullmann 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 Historical remarks Enumerating sets system on M, i.e., S ⊆ (M). P Topologies and Consider the set S ⊆ P(P(M)) of all possible set filters Bases and systems S (as given by the species). subbases A very important property of (nearly indispensable) Topology: SAT in S disguise 0 ⊆ is to be a closure system, that is, for all S S we Compactness T 0 have M S ∈ S. Then every set system S0 ⊆ P(M) generates a structure hS0i as the smallest set system in S containing S0: \ hS0i = {S ∈ : S0 ⊆ S}. M S Ideal objects in set Filter subbases theory and topology
The system of all filters on a set M is a closure system. Oliver Kullmann Given F0 ⊆ P(M), the filter F generated by F0 is obtained by the following 2-step process: Historical remarks Enumerating sets 1 Let F0 be the closure of F0 under finite intersections. Topologies and 2 Then F is the closure of F0 under formation of filters super-sets. Bases and subbases
Given a filter F on M, a filtersubbasis for F is a set Topology: SAT in disguise F0 ⊆ F such that the filter generated by F0 is F. Compactness A filterbasis for F is a set F0 ⊆ F such that the closure of F0 under formation of super-sets yields F. So from a filtersubbasis first a filterbasis is created, and then the filter.
Every F0 is a filtersubbasis (of some filter), while F0 is a proper filtersubbasis (a filtersubbasis of some proper filter) iff no finite intersection with elements of F0 is empty. Ideal objects in set Ultrafilters theory and topology
An ultrafilter is a proper filter U on M such that no proper Oliver Kullmann filter F on M with U ⊂ F exists. Every trivial filter is an Historical remarks ultrafilter, and for finite sets the converse also holds. For a Enumerating sets filter F on M the following conditions are equivalent: Topologies and filters 1 F is an ultrafilter on M. Bases and 2 For every subset T ⊆ M either T ∈ F or M \ T ∈ F subbases Topology: SAT in holds (but not both). disguise 3 F is proper, and for all A, B ⊆ M with A ∪ B ∈ F we Compactness have A ∈ F ∨ B ∈ F. Since the system of proper filters 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 ultrafilters containing it.) Ideal objects in set Subbases for the open sets theory and topology
Oliver Kullmann
The system of all topologies on a set X is a closure Historical remarks system. Enumerating sets Topologies and Given T ⊆ P(X), the topology τ generated by T is filters obtained by the following 2-step process: Bases and subbases 1 Let τ0 be the closure of T under finite intersections. 2 Topology: SAT in Then τ is the closure of τ0 under arbitrary unions. disguise Given a topology τ on X, a subbasis of τ is a set Compactness T ⊆ τ such that the topology generated by T is τ.
A basis for τ is a set τ0 ⊆ τ such that the closure of τ0 under arbitrary unions yields τ. So from a subbasis first a basis is created, and then the topology. Ideal objects in set Subbases for the closed sets theory and topology
Oliver Kullmann The system of all dual topologies (given by the “closed subsets”) on a set X is a closure system. Historical remarks Enumerating sets Given T ⊆ (X), the dual topology τ 0 generated by T P Topologies and is obtained by the following 2-step process: filters 1 Let τ 0 be the closure of T under finite unions. Bases and 0 subbases 2 0 0 Then τ is the closure of τ0 under arbitrary Topology: SAT in intersections. disguise Given a dual topology τ 0 on X, a subbasis of τ 0 is a Compactness set T ⊆ τ 0 such that the dual topology generated by T is τ. 0 0 0 A basis for τ is a set τ0 ⊆ τ such that the closure of 0 0 τ0 under arbitrary intersections yields τ . So, again, from a subbasis first a basis is created, and then the dual topology. Ideal objects in set Continuity and subbases theory and topology
Oliver Kullmann
Given topological spaces X, Y , a map f : X → Y and a Historical remarks
subbasis S of Y , it is easy to see: Enumerating sets
Topologies and f is continuous iff for all O ∈ S filters the set f −1(O) is open in X. Bases and subbases
Topology: SAT in Stronger: f is a quotient map (recall last lecture) iff f is disguise
surjective and Compactness
{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 topology on the domain space is the coarsest topology to make the map continuous. Ideal objects in set Products of topological spaces theory and topology
Oliver Kullmann For a given family (Xi )i∈I of sets, the product is the set Historical remarks Y [ Enumerating sets Xi := {f : I → Xi | ∀ i ∈ I : f (i) ∈ Xi } Topologies and i∈I i∈I filters Q Bases and together with the canonical projections prj : i∈I Xi → Xj . subbases Topology: SAT in Now we assume that all Xi are topological spaces. disguise The product topology has as subbasis the set of all Compactness −1 pri (O) for i ∈ I and O open in Xi . If for each Xi a subbasis Si is given, then a subbasis of the product topology is given by the set of all −1 pri (O) for i ∈ I and O ∈ Si . Thus the product topology is the coarsest topology on the product set such that all projections are continuous. 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 Quasicompact spaces theory and topology
Oliver Kullmann A topological space (X, τ) is quasicompact if every open cover of X contains a finite subcover, i.e., for Historical remarks S Enumerating sets every O ⊆ τ with O = X there exists a finite 0 S 0 Topologies and O ⊆ O with O = X. filters Bases and (X, τ) is quasicompact iff every proper filtersubbasis subbases
F0 consisting of (some) closed sets has nonempty Topology: SAT in T disguise intersection (i.e., F0 6= ∅). Compactness A compact space is a quasicompact space which is also hausdorff (T2), 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”. Ideal objects in set Basic properties of compactness theory and topology A subset A of a topological space X is called Oliver Kullmann (quasi)compact, if it is (quasi)compact as a topological Historical remarks subspace. Enumerating sets Topologies and A is quasicompact iff every open cover of A in X has filters a finite subcover. Bases and subbases
If X is (quasi)compact and A is closed, then also A is Topology: SAT in (quasi)compact. disguise Compactness 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). Ideal objects in set Alexander’s Subbasis Theorem theory and topology
Oliver Kullmann
Historical remarks Consider a topological space (X, τ) and a subbasis S: Enumerating sets Topologies and X is quasicompact iff every open cover of X filters Bases and by elements of S has a finite subcover. subbases Topology: SAT in Using the formulation of compactness by closed sets, we disguise obtain the following equivalent statement: Compactness
Consider a subbasis S of the closed subsets. T Then X is quasicompact iff A 6= ∅ for every proper filtersubbasis A ⊆ S. Ideal objects in set Proof theory and topology
Oliver Kullmann
Historical remarks Consider a subbasis of the closed subsets. S Enumerating sets Consider a set A of closed sets of X, such that A is a T Topologies and proper filtersubbasis. Assume A = ∅. filters Bases and So for every x ∈ X there exists Ax ∈ A with x ∈/ Ax . subbases Topology: SAT in Since S is a subbasis, for each x ∈ X there is now a finite disguise S S Compactness Sx ⊆ S with Sx ⊇ Ax and x ∈/ Sx . Let U be an ultrafilter with A ⊆ U. So for every x ∈ X there exists Sx ∈ Sx with Sx ∈ U.
Now {Sx : x ∈ X} would be a proper filtersubbasis with T x∈X Sx = ∅. Ideal objects in set Tychonoff’s Theorem theory and topology
Oliver Kullmann Given a family (Xi )i∈I of sets, and a family (Ai )i∈I of Historical remarks subsets Ai ⊆ Xi , we have Enumerating sets Y [ \ Y ( ) \ ( −1( )) = −1( \ ) = \ . Topologies and Xi pri Ai pri Xi Ai Xi Ai filters
i∈I i∈I i∈I i∈I Bases and subbases
Thus, an easy application of the subbasis theorem yields: Topology: SAT in disguise The product of quasicompact topological spaces Compactness is again quasicompact.
And since the product of T2-spaces is again T2:
The product of compact spaces is compact.
(Also the reverse directions hold, provided that all spaces are non-empty.) Ideal objects in set theory and topology
Oliver Kullmann
Historical remarks
Enumerating sets
Topologies and filters
Bases and subbases End Topology: SAT in disguise
(finally) Compactness