Open quotients of trivial vector bundles
aa r X i v : . [ m a t h . F A ] A p r Open quotients of trivial vector bundles ∗ Pedro Resende and
Jo˜ao Paulo Santos
Abstract
Given an arbitrary topological complex vector space A , a quo-tient vector bundle for A is a quotient of a trivial vector bundle π : A × X → X by a fiberwise linear continuous open surjection.We show that this notion subsumes that of a Banach bundle over alocally compact Hausdorff space X . Hyperspaces consisting of lin-ear subspaces of A , topologized with natural topologies that includethe lower Vietoris topology and the Fell topology, provide classifyingspaces for various classes of quotient vector bundles, in a way thatgeneralizes the classification of locally trivial vector bundles by Grass-mannians. If A is normed, a finer hyperspace topology is introducedthat classifies bundles with continuous norm, including Banach bun-dles, and such that bundles of constant finite rank must be locallytrivial. Keywords:
Vector bundles, Banach bundles, Grassmannians, lowerVietoris topology, Fell topology.2010
Mathematics Subject Classification : 46A99, 46M20, 54B20, 55R65
Contents ∗ Work funded by FCT/Portugal through projects EXCL/MAT-GEO/0222/2012 andPEst-OE/EEI/LA0009/2013, and by COST (European Cooperation in Science and Tech-nology) through COST Action MP1405 QSPACE. Classifying spaces 176 Finite rank and local triviality 247 Grassmannians 32A Appendix 37
The reduced C*-algebra C ∗ red ( G ) of a locally compact Hausdorff ´etale groupoid G (see [10, 11]), whose elements can be regarded as complex valued functionson G , can be “twisted” by considering instead the reduced C*-algebra C ∗ red ( π )of a Fell line bundle π : E → G , where now the elements of the algebra areidentified with sections of the bundle; see [5, 12]. This provides one way ofgeneralizing to C*-algebras the notion of Cartan subalgebra of a Von Neu-mann algebra [1]: Cartan pairs ( A, B ), consisting of a C*-algebra A and asuitable abelian subalgebra B , correspond bijectively to a certain class of´etale groupoids with Fell line bundles on them [12].The motivation for the present paper stemmed from studying the con-struction of a Fell bundle from a Cartan pair, in particular in an attempt togeneralize the class of groupoids to which it applies by taking into accountthat both a C*-algebra A and an ´etale groupoid G have associated quantalesMax A [4, 7, 8] and Ω( G ) [13], respectively, the former consisting of all theclosed linear subspaces of A and the latter being the topology of G . In doingso it became evident that it is useful to study bundles whose construction isbased on a preexisting object of global sections, such as the C*-algebra A ofa Cartan pair, in a way that in fact is independent of the algebraic structureof groupoids, but which instead applies to Banach bundles to begin with, andindeed to more general bundles. So the original endeavour has naturally beensplit into several parts, of which the present paper is the first one, where nofurther mention of C*-algebras or groupoids will be made. Instead the bulkof this paper will deal with completely general topological vector spaces, andon occasion locally convex or normed spaces.If A is a topological vector space and X is any topological space, then π : A × X → X is a trivial vector bundle on X . The vector bundlesstudied in this paper, termed quotient vector bundles , consist of those bundles π : E → X that arise as quotients of trivial bundles by a fiberwise linear2ontinuous open surjection q : A × X q / / / / π ●●●●●●●●● E π ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ X We shall see that any Banach bundle π : E → X (in particular, any finiterank locally trivial vector bundle) on a locally compact Hausdorff space is ofthis kind, where A can be taken to be the space C ( π ) of continuous sectionsvanishing at infinity.Although now without any quantale structure (since A is not even analgebra), the set Max A of all the closed linear subspaces of A plays animportant role: equipped with suitable topologies it provides a notion ofclassifying space for quotient vector bundles. Concretely, these are obtainedby pullback along continuous maps κ : X → Max A (or, even more generally, maps into Sub A , the space of all the linear sub-spaces) of a universal bundle π A : E A → Max A : E A × Max A X π / / π (cid:15) (cid:15) E Aπ A (cid:15) (cid:15) X κ / / Max A In this paper we study three topologies on Max A . Perhaps surprisingly, twoof them are well known hyperspace topologies: • The (relative) lower Vietoris topology [9, 14] classifies all the quotientvector bundles with Hausdorff fibers (the restriction on the fibers dis-appears if we use Sub A instead of Max A ). • The topology of Fell [2] classifies the quotient vector bundles whose zerosection is closed — at least provided both A and X are first countable.If A is normed its quotient vector bundles π : E → X are naturally equippedwith an upper semicontinuous norm k k : E → R . In this case a thirdtopology on Max A , referred to as the closed balls topology , coarser than the(full) Vietoris topology but finer than the Fell topology, classifies the bundlesfor which the norm on E is continuous. In particular, if A is a Banach spaceit classifies Banach bundles over first countable Hausdorff spaces.3uotient vector bundles are not necessarily locally trivial, and no clas-sification of the locally trivial ones for arbitrary A is provided in general.But something can be said about quotient vector bundles whose fibers areall of the same finite dimension d . These are classified by the subspaceMax d A ⊂ Max A that contain the closed linear subspaces of codimension d in A . We show that such bundles are locally trivial if A is normed andMax d A is equipped with the closed balls topology. As a corollary, this yieldsa fact that is stated but not proved in [3, p. 129], namely that Banach bun-dles on locally compact Hausdorff spaces whose fibers are of constant finitedimension are locally trivial.The role played by our spaces Max A with respect to quotient vectorbundles is analogous to that of Grassmannians for locally trivial vector bun-dles, and at the end of the paper we provide a detailed comparison betweenthem, in particular showing explicitly that Gr ( d, C n ), the Grassmannian of d -dimensional subspaces of C n , is homeomorphic to Max n − d C n . We begin by fixing basic terminology and notation.
Topological vector spaces.
By a topological vector space will be meant acomplex topological vector space without assuming any topological separa-tion axioms except where specified otherwise. We recall that if the topologyof a topological vector space is T then it is Hausdorff (in fact completely reg-ular), and that a Hausdorff vector space is finite dimensional if and only if itis locally compact. Moreover, the topology of a finite dimensional Hausdorffvector space is necessarily the Euclidean topology. We define the followingnotation, for an arbitrary topological vector space A : • Sub A is the set of all the linear subspaces of A ; • Max A is the subset of Sub A consisting of all the topologically closedlinear subspaces.The notation Max A is borrowed from quantale theory, where Max A , equippedwith a natural quantale structure, plays the role of spectrum of a C*-algebra A . See [4, 6–8].For each subset S ⊂ A , we denote by h S i the linear span of S . If S is afinite subset { a , . . . , a n } we also write h a , . . . , a n i instead of h S i .4 xample 2.1 Sub A and Max A may coincide, for instance if A has thediscrete topology, or if A is a Hausdorff finite dimensional vector space as inthe following examples: • Sub C = Max C = {{ } , C } ; • Sub C = Max C = {{ } , C } ∪ C P ; • Sub C n = Max C n = {{ } , C n } ∪ ` n − r =1 Gr ( r, C n ). Bundles.
Let X be a topological space. By a bundle on X will always bemeant a topological space E equipped with a continuous surjection π : E → X , which is referred to as the projection of the bundle. For each element x ∈ X we refer to the subspace π − ( { x } ) ⊂ E as the fiber of π over x , and we usethe following notation for all x ∈ X and open sets U ⊂ X : E x = π − ( { x } ) ,E U = π − ( U ) ,π U = π | E U : E U → U . If ρ : F → X and π : E → X are bundles, by a map of bundles over X , h : ρ → π , will be meant a continuous map h : F → E such that π ◦ f = ρ . The categoryof bundles over X and their maps is denoted by Bun ( X ).If π : E → X is a bundle, Y is a topological space and g : Y → X isa continuous map, the pullback E × X Y together with its second projection π : E × X Y → Y defines a bundle g ∗ ( π ) on Y : E × X Y π / / g ∗ ( π )= π (cid:15) (cid:15) E π (cid:15) (cid:15) Y g / / X .
Continuous sections. A continuous section of a bundle π : E → X is acontinuous function s : X → E such that π ◦ s is the identity on X . The setof all the continuous sections of the bundle is denoted by C ( π ). We say thatthe bundle has enough sections if for all e ∈ E x there is a continuous section s such that s ( x ) = e . 5ny map h : ρ → π in Bun ( X ) induces a mapping h ∗ : C ( ρ ) → C ( π )given by postcomposition, F h / / EX h ∗ ( s )= h ◦ s > > ⑦⑦⑦⑦⑦⑦⑦⑦ s ` ` ❅❅❅❅❅❅❅❅ and thus we obtain a functor ( ) ∗ : Bun ( X ) → Set . In addition, given apullback E × X Y π / / π (cid:15) (cid:15) E π (cid:15) (cid:15) Y g / / X there is a mapping g ∗ : C ( π ) → C ( π ), defined for all s ∈ C ( π ) and y ∈ Y by g ∗ ( s )( y ) = (cid:0) s ( g ( y )) , y (cid:1) , so we obtain a contravariant functor ( ) ∗ : ( Top /X ) op → Set . Moreover, thefollowing diagram commutes: E × X Y π / / EY g / / g ∗ ( s ) O O X s O O We note that if π has enough sections then so does the pullback g ∗ ( π ). Bundles with linear structure.
Let π : E → X be a bundle. By a linear structure on the bundle will be meant a structure of vector spaceon each fiber such that the operations of scalar multiplication and vectoraddition are globally continuous when regarded as maps C × E → E and E × X E → E , respectively, and such that the zero section of π , which sendseach x ∈ X to 0 x (the zero of E x ), is continuous. Hence, a bundle equippedwith a linear structure is a very loose form of vector bundle. We shall referto bundles with such linear structures as linear bundles . The set C ( π ) ofcontinuous sections of a linear bundle π is a vector space whose operationsare computed fiberwise. 6he category of linear bundles over X , LinBun ( X ), has the linear bundlesas objects and, given linear bundles π : E → X and ρ : F → X , a morphism h : ρ → π is a map h : ρ → π in Bun ( X ) which is fiberwise linear: for each x ∈ X , h restricts to a linear map F x → E x . Hence, the induced mapping on sections h ∗ : C ( ρ ) → C ( π ) is linear, so we obtain a functor ( ) ∗ : LinBun → Vect . Inaddition, if g : Y → X is a continuous map the pullback g ∗ ( π ) is a linearbundle, and we obtain a contravariant functor ( ) ∗ : ( Top /X ) op → Vect .A trivial vector bundle is a linear bundle of the form π : A × X → X forsome topological vector space A , with the obvious algebraic structure, anda locally trivial vector bundle is a linear bundle π : E → X for which each x ∈ X has an open neighborhood U such that the restricted bundle π U isisomorphic to the trivial vector bundle π : E x × U → U . The pullback g ∗ ( π )of a locally trivial vector bundle π : E → X along g : Y → X is itself locallytrivial.The following simple fact will be useful later on. Lemma 2.2
For any locally trivial vector bundle π : E → X with Hausdorfffibers, the image of the zero section is a closed set of E .Proof. Let v ∈ E , v = 0, and let U be a neighborhood of π ( v ) ∈ X such that E U is isomorphic to a trivial vector bundle. Let f : V × U → E U be an isomorphism in LinBun ( U ). Then V is a Hausdorff space and f (cid:0) ( V \ { } ) × U (cid:1) ⊂ E \ { } is an open set in E containing v . Now we introduce the central notion of vector bundle in this paper.
Basic definitions and properties.
The condition that a bundle π : E → X has enough sections is equivalent to the requirement that the evaluationmapping C ( π ) × X ( s,x ) s ( x )eval / / E be surjective. This fact suggests the following definition:7y a quotient bundle will be meant a triple ( π, A, q ) consisting of a bundle π : E → X , a topological space A and a map q of bundles over X , A × X q / / π ●●●●●●●●● E π ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ X referred to as the quotient map , which is both surjective and open. Thismakes E homeomorphic to a topological quotient of A × X , hence the ter-minology. For each x ∈ X we also write q x : A → E x for the map defined by q x ( a ) = q ( a, x ). Lemma 3.1
Let ( π : E → X, A, q ) be a quotient bundle. The followingconditions hold:1. π is an open map.2. The topology of each fiber E x as a subspace of E coincides with thetopology of E x regarded as a quotient of A .3. The maps q x are continuous and open surjections.Proof. If ( π, A, q ) is a quotient bundle then q is a morphism in Bun ( X )from π : A × X → X to π , and thus the openness of π is a consequence ofthe openness of π plus the continuity and surjectivity of q : if U ⊂ E is openthen π ( U ) = π ( q ( q − ( U ))) = π ( q − ( U )) . The second and third properties follow from a simple property of open maps:if f : Y → Z is an open map then for every subset S ⊂ Z the restriction f | f − ( S ) : f − ( S ) → S is itself an open map if S and f − ( S ) are equippedwith their respective subspace topologies, because for every open set U ⊂ Y we have f ( f − ( S ) ∩ U ) = S ∩ f ( U ). Pullbacks.
Quotient bundles are well behaved under pullbacks:
Lemma 3.2
The class of quotient bundles is closed under pullbacks alongcontinuous base maps. roof. Let ( π : E → X, A, q ) be a quotient bundle, and g : Y → X a continuous map. The universal property of the pullback g ∗ ( π ) ensuresthat there is a unique continuous map q ′ that makes the following diagramcommute: A × Y q ′ & & id × g / / π " " A × X q $ $ ❍❍❍❍❍❍❍❍❍❍ E × X Y g ∗ ( π ) (cid:15) (cid:15) π / / E π (cid:15) (cid:15) Y g / / X The map q ′ is defined by q ′ ( a, y ) = ( q ( a, g ( y )) , y ) for all y ∈ Y and a ∈ A ,and thus it is surjective. Moreover, the outer rectangle A × Y id × g / / π (cid:15) (cid:15) A × X π ◦ q (cid:15) (cid:15) Y g / / X is itself a pullback diagram, and thus so is the upper rectangle A × Y id × g / / q ′ (cid:15) (cid:15) A × X q (cid:15) (cid:15) E × X Y π / / E .
Hence, q ′ is open because it is the pullback of the open map q along thecontinuous map π , and therefore ( g ∗ ( π ) , A, q ′ ) is a quotient bundle. Continuous sections.
Let ( π : E → X, A, q ) be a quotient bundle. Foreach a ∈ A we define the continuous sectionˆ a : X → E by ˆ a ( x ) = q ( a, x ). We denote the set of such sections by ˆ A , and regard it asa topological space whose topology is the quotient topology obtained from A . Then, for each pair of open sets U ⊂ X and Γ ⊂ A we have the followingsubset of E : E Γ ,U = [ a ∈ Γ ˆ a ( U ) . Lemma 3.3
Let ( π : E → X, A, q ) be a quotient bundle. The followingconditions hold: . The family of subsets E Γ ,U as defined above is a basis for the topologyof E .2. ( π, ˆ A, eval) is a quotient bundle.Proof. The sets E Γ ,U form a basis for the topology of E because E Γ ,U = q (Γ × U ) and q is open. For the second property let χ : A → ˆ A be thequotient a ˆ a , and let U ⊂ ˆ A × X be open. Then, denoting by φ thesurjective map χ × id, we haveeval( U ) = q ( φ − ( U )) , and thus eval : ˆ A × X → E is open.If A = ˆ A and q = eval we say that the quotient bundle is sectional , andwe denote it simply as the pair ( π, A ). Quotient bundles with linear structure.
By a quotient vector bundle will be meant a quotient bundle ( π : E → X, A, q ) such that π is a linearbundle, A is a topological vector space, and the open surjection q : A × X → E is fiberwise linear. Example 3.4
As we shall see below, every Banach bundle on a locally com-pact Hausdorff space can be made a quotient vector bundle (cf. Theorem 4.2).Every quotient vector bundle ( π, A, q ) determines a kernel map κ : X → Sub A defined by κ ( x ) = q − (0 x ) = ker q x for each x ∈ X . We denote by B ( π, A, q )the intersection T x ∈ X κ ( x ), and call it the bundle radical of ( π, A, q ). Clearly,the quotient A → ˆ A defined by a ˆ a is linear, and its kernel is B ( π, A, q ).Hence, we have an isomorphism ˆ A ∼ = A/ B ( π, A, q ), and the bundle is iso-morphic to a sectional quotient vector bundle if and only if its bundle radicalis { } .The following simple property will be useful later: Lemma 3.5
Let ( π : E → X, A, q ) be a quotient vector bundle with kernelmap κ , let a , . . . , a n ∈ A be linearly independent, and let V ⊂ A be the linear subspace spanned by a , . . . , a n . If x ∈ X and κ ( x ) ∩ V = { } then ˆ a ( x ) , . . . , ˆ a n ( x ) are linearlyindependent vectors of E x .Proof. There is a linear isomorphism E x ∼ = A/κ ( x ). Hence, if P z i ˆ a i ( x ) = 0in A/κ ( x ) for some z , . . . , z n ∈ C , we have P z i a i ∈ V ∩ κ ( x ), and thus z = · · · = z n = 0. 10 onstruction of quotient vector bundles. Let A be a topological vectorspace, X a topological space, and κ : X → Sub A an arbitrary map. Define E to be the quotient space of A × X by theequivalence relation given, for all a, b ∈ A and x, y ∈ X , by( a, x ) ∼ ( b, y ) ⇐⇒ x = y and a − b ∈ κ ( x ) . Let us fix some notation and terminology.1. We shall write [ a, x ] for the equivalence class of ( a, x ) ∈ A × X .2. The quotient map induced by κ is the map q : A × X → E given by q ( a, x ) = [ a, x ] for each ( a, x ) ∈ A × X .3. The bundle induced by κ is the continuous surjection π : E → X givenby the universal property of q ; that is, for each ( a, x ) ∈ A × X we have π ([ a, x ]) = x . Theorem 3.6
Let A be a topological vector space, X a topological space, and κ : X → Sub A a map. Let also π : E → X and q : A × X → E be the bundleand quotient map induced by κ . The following conditions are equivalent:1. q is an open map;2. ( π : E → X, A, q ) is a quotient vector bundle with kernel map κ .Proof. The implication (2) ⇒ (1) is true by definition of quotient vectorbundle. For the implication (1) ⇒ (2) we begin by noticing that the conti-nuity of π is due to the universal property of q as a quotient map, and allthat we have left to do is prove that π is a linear bundle. The fact that q isopen also implies that each fiber E x is, as a subspace of E , homeomorphic tothe quotient A/κ ( x ) (cf. Lemma 3.1). Hence, the fibers of π are topologicalvector spaces, and we are left with proving that the linear operations areglobally continuous on E . For addition we begin by noting that q × q : A × X × A × X → E × E is an open map, and thus, since ( A × X ) × X ( A × X ) = ( q × q ) − ( E × X E ),so is the restriction e q = q × q | ( A × X ) × X ( A × X ) : ( A × X ) × X ( A × X ) → E × X E , E is defined fiberwise by[ a, x ] + [ b, x ] = [ a + b, x ] , and thus it is the continuous map obtained from the addition on A × X viathe universal property of e q as a quotient map:( A × X ) × X ( A × X ) ∼ = A × A × X + / / e q (cid:15) (cid:15) A × X q (cid:15) (cid:15) E × X E + / / E Analogously, id × q : C × A × X → C × E is an open map, and thus scalarmultiplication on E is well defined as a continuous map on C × E : C × A × X m × id / / id × q (cid:15) (cid:15) A × X q (cid:15) (cid:15) C × E / / E To conclude, the zero section is obviously continuous because for each x ∈ X the sets E Γ ,U , with Γ an open neighborhood of 0 in A , form a local basisaround 0 x , and the preimage of each such set is the open set U .In addition, we note the following: Corollary 3.7
If the equivalent conditions of Theorem 3.6 hold, the follow-ing assertions are equivalent:1. κ is valued in Max A ;2. The fibers E x are Hausdorff spaces.Proof. The map κ is valued in Max A if and only if the kernels ker q x areclosed, which is equivalent to the singletons { x } being closed in the fibers E x , and in turn is equivalent to all the singletons { e } ⊂ E x being closed.This means E x is a T topological vector space, hence Hausdorff. Hausdorff bundles.
Given a quotient vector bundle ( π : E → X, A, q ),a natural question regards the kind of topology which is carried by E . Inparticular, in general E should not be expected to be a Hausdorff space evenif A and X are, but the property of being Hausdorff is closely related to thezero section of the bundle: 12 heorem 3.8 Let ( π : E → X, A, q ) be a quotient vector bundle. If E isHausdorff the image of the zero section is a closed set in E . In addition, ifboth A and X are Hausdorff spaces then E is Hausdorff if and only if theimage of the zero section is closed. Before we begin the proof, notice that the image of the zero section of E is closed if and only if q − (0) is closed in A × X . Furthermore, ( a, x ) ∈ q − (0)is equivalent to a ∈ κ ( x ). Proof.
Assume first that E is Hausdorff. We will show that q − (0) is closed,which is equivalent to the image of the zero section being closed. Let ( a, x ) / ∈ q − (0). Then q ( a, x ) = q (0 , x ), whence there are disjoint open sets U a , U ⊂ E such that q ( a, x ) ∈ U a and q (0 , x ) ∈ U . It follows that there are open sets W , W a ⊂ A and U ⊂ X such that a ∈ W a , 0 ∈ W , x ∈ U , q ( W × U ) ⊂ U and q ( W a × U ) ⊂ U a . Therefore we have q (cid:0) ( W a × U ) ∩ q − (0) (cid:1) ⊂ q ( W a × U ) ∩ q ( W × U ) ⊂ U a ∩ U = ∅ . Hence, ( W a × U ) ∩ q − (0) = ∅ and thus q − (0) is closed.Now assume that q − (0) is closed and that both A and X are Hausdorff.We will show that E is Hausdorff. Let ( a, x ) , ( b, y ) ∈ A × X be such that q ( a, x ) = q ( b, y ). We may assume that b = 0 and, since X is Hausdorff,that x = y . Then ( a, x ) / ∈ q − (0) and, since q − (0) is closed, there is aneighborhood W of a and a neighborhood U of x such that ( W × U ) ∩ q − (0) = ∅ . Then, for all y ∈ U we have W ∩ κ ( y ) = ∅ . Let W ′ be a neighborhood ofthe origin such that a + W ′ − W ′ ⊂ W . For all b ∈ W ′ and c ∈ a + W ′ wehave c − b ∈ a + W ′ − W ′ ⊂ W , and thus c − b / ∈ κ ( y ), which means that q ( b, y ) = q ( c, y ). Hence, q ( W ′ × U ) and q (cid:0) ( a + W ′ ) × U (cid:1) are disjoint opensets, and it follows that E is Hausdorff. Corollary 3.9
Let ( π : E → X, A, q ) be a quotient vector bundle with Haus-dorff fibers such that π is locally trivial and both X and A are Hausdorffspaces. Then E is a Hausdorff space.Proof. Immediate consequence of Lemma 2.2 and Theorem 3.8.
Normed bundles.
We shall say that a quotient vector bundle( π : E → X, A, q )is normed if A is a normed linear space. Then we define a mapping k k : E → R , quotient norm on E , by, for each x ∈ X and a ∈ A , k q ( a, x ) k = d ( a, κ ( x )) = inf p ∈ κ ( x ) k a + p k , where κ is the kernel map of the bundle. Theorem 3.10
Let ( π : E → X, A, q ) be a normed quotient vector bundle.The quotient norm on E is upper semicontinuous.Proof. Let ε >
0, and let Γ = B ε (0) ⊂ A . The basic open set E Γ ,X consistsof all the elements q ( a, x ) with k a k < ε , so we have k q ( a, x ) k ≤ k a k < ε . Hence, E Γ ,X ⊂ { e ∈ E | k e k < ε } . Now let e = q ( a, x ) ∈ E be suchthat k e k < ε . The condition k d ( a, κ ( x )) k < ε implies that k a + p k < ε for some p ∈ κ ( x ). Then a + p ∈ Γ and q ( a + p, x ) = e , showing that E Γ ,X = { e ∈ E | k e k < ε } . Hence, { e ∈ E | k e k < ε } is open for arbitrary ε >
0, which means precisely that k k : E → R is upper-semicontinuous.We shall say that the normed quotient vector bundle ( π, A, q ) is contin-uous if moreover the norm of E is also lower-semicontinuous. In this section we shall see that classical Banach bundles can always beregarded as continuous quotient vector bundles, at least if the base space islocally compact Hausdorff.
Basic definitions and facts.
Let X be a Hausdorff space. By a Banachbundle over X [3, II.13.4] is meant a Hausdorff space E equipped with acontinuous open surjection π : E → X such that:1. for each x ∈ X the fiber E x has the structure of a Banach space;2. addition is continuous on E × X E to E ;3. for each λ ∈ C , scalar multiplication e λe is continuous on E to E ;4. e
7→ k e k is continuous on E to R ;14. for each x ∈ X and each open set V ⊂ E containing 0 x , there is ε > U ⊂ X containing x such that E U ∩ T ε ⊂ V , where T ε is the “tube” { e ∈ E | k e k < ε } .Condition (5) is equivalent to stating that for each x ∈ X the open“rectangles” E U ∩ T ε with x ∈ U form a local basis of 0 x . It is also equivalent to the statementthat for every net ( e α ) in E , if π ( e α ) → x and k e α k → e α → x (theaxiom of choice is needed for the converse implication).Hence, in particular, the zero section of a Banach bundle is continuous.It can also be shown that scalar multiplication as an operation C × E → E is continuous, and thus Banach bundles are linear bundles.We further recall that if X is locally compact then any Banach bundleover X has enough sections [3, Appendix C]. Banach bundles with enough sections.
Let π : E → X be a Banachbundle with enough sections. For each s ∈ C ( π ) and each ε > T ε ( s ) = { e ∈ E | k e − s ( π ( e )) k < ε } . Lemma 4.1
Let π : E → X be a Banach bundle with enough sections. Thecollection of all the sets of the form E U ∩ T ε ( s ) , where s ∈ C ( π ) and U ⊂ X is open, is a basis for the topology of E .Proof. T ε ( s ) is the image of T ε by the homeomorphism h s : E → E which is defined by h s ( e ) = e + s ( π ( e )) (whose inverse h − s is h − s ), and thusit is an open set of E . Then, for each e ∈ E and each continuous section s through e , the collection { E U ∩ T ε ( s ) | ε > , U ⊂ X is open, π ( e ) ∈ U } is a local basis at e . 15 anach bundles on locally compact spaces. Now we shall see thatBanach bundles on locally compact Hausdorff spaces yield quotient vectorbundles (cf. Example 3.4).
Theorem 4.2
Every Banach bundle on a locally compact Hausdorff spacecan be made a continuous normed sectional quotient vector bundle whosequotient norm coincides with the Banach bundle norm.Proof.
Let π : E → X be a Banach bundle with X locally compact Haus-dorff, and denote by C ( π ) the Banach space of continuous sections of π which vanish at infinity, equipped with the topology of the supremum norm k k ∞ . The evaluation mapping q = eval : C ( π ) × X → E is surjective because there are enough sections due to the local compactnessof X , and thus there are enough sections in C ( π ) because there are enoughcompactly supported sections (from any continuous section s through e andany compactly supported continuous function f : X → C such that f ( π ( e )) =1 we obtain a compactly supported continuous section f s through e ). Theevaluation mapping is also continuous because the supremum norm topologycontains the compact-open topology. For each x ∈ X the mapping C ( π ) → E x s q ( s, x )is linear, and thus q is a map in LinBun ( X ). Now let us prove that q isan open map. A basis for the topology of C ( π ) × X consists of all thesets B ε ( s ) × U with s ∈ C ( π ), ε >
0, and U ⊂ X open. The image q ( B ε ( s ) × U ) is of course contained in T ε ( s ) ∩ E U . For the converse inclusionlet e ∈ T ε ( s ) ∩ E U , and let t ∈ C ( π ) be such that t ( π ( e )) = e . We have k t ( π ( e )) − s ( π ( e )) k < ε , and thus, by the upper semi-continuity of the normof E , for some open set V ⊂ U containing π ( e ) we have k t ( x ) − s ( x ) k < ε forall x ∈ V . Let f : X → [0 ,
1] be a continuous compactly supported functionsuch that both f ( π ( e )) = 1 and supp f ⊂ V , and let t ′ = s + f ( t − s ). Then t ′ ∈ C ( π ) and t ′ ( π ( e )) = e , and furthermore we obtain k t ′ − s k ∞ = k f ( t − s ) k ∞ ≤ max x ∈ supp f {k t ( x ) − s ( x ) k} < ε , so we conclude that e ∈ q ( B ε ( s ) × U ). Hence, q ( B ε ( s ) × U ) = T ε ( s ) ∩ E U ,and it follows that q is an open map. So we have a normed sectional quotient16ector bundle ( π, C ( π ) , eval). The quotient norm k k q on E is defined forall s ∈ C ( π ) and x ∈ X by (cf. definition preceding Theorem 3.10) k s ( x ) k q = inf (cid:8) k s + p k ∞ | p ∈ C ( π ) and p ( x ) = 0 x (cid:9) . Let us show that k k q coincides with the Banach bundle norm k k . First, forall x ∈ X and all s, p ∈ C ( π ) such that p ( x ) = 0 x we have k s ( x ) k = k s ( x ) + p ( x ) k ≤ k s + p k ∞ , and thus k k ≤ k k q . Now in order to prove that we have k k = k k q itsuffices to show that for all x ∈ X and s ∈ C ( π ) there is p ∈ C ( π ) suchthat p ( x ) = 0 x and k s + p k ∞ = k s ( x ) k . There are two cases: if s ( x ) = 0 x justlet p = − s ; otherwise let p = f s where f : X → ( − ,
0] is the continuousfunction defined by, for all y ∈ X , f ( y ) = ( k s ( x ) kk s ( y ) k − k s ( x ) k ≤ k s ( y ) k , k s ( x ) k ≥ k s ( y ) k . Finally, the definition of Banach bundle states that the norm on E is alsolower-semicontinuous, so it follows that the normed quotient vector bundle( π, C ( π ) , eval) is continuous. Now we shall look at conditions on a map κ : X → Sub A which ensure thatit is the kernel map of a quotient vector bundle. As we shall see, this is sowhen κ is continuous with respect to a suitable topology on Sub A . Lower Vietoris topology.
The Vietoris topology on the space of closedsubsets of a topological space [14] is often presented as the coarsest topologythat contains both the lower and the upper Vietoris topologies — see, e.g.,[9]. For the purposes of this section, given a topological vector space A weshall topologize Max A (and indeed also Sub A ) with the subspace topologywhich is obtained from the lower Vietoris topology.Let A be a topological vector space. For each open set U ⊂ A we shallwrite e U = { P ∈ Sub A | U ∩ P = ∅} . The collection of all the sets e U is a subbasis for a topology on Sub A , whichwe shall refer to as the lower Vietoris topology on Sub A .17e remark that Sub A with this topology is usually not a T space, sincefor all V ∈ Sub A the neighborhoods of a linear subspace V are the same asthe neighborhoods of its closure V . However, the subset Max A , equippedwith the subspace topology, is always a T space. Theorem 5.1
Let A be a topological vector space.1. Max A is a topological retract of Sub A .2. The map h i : A → Max A which to each a ∈ A assigns its span h a i = C a is continuous.Proof. The mapping V V from Sub A to Max A is continuous, so (1)holds. In order to prove (2), let a ∈ A and let W ⊂ Max A be an openneighborhood of h a i . It suffices to take W = e U ∩ Max A for some open set U ⊂ A . Then there is λ ∈ C such that λa ∈ U . Due to the continuity ofscalar multiplication in A there is V ⊂ A open such that a ∈ V and λc ∈ U for all c ∈ V . Hence, h c i ∈ W for all c ∈ V , showing that (2) holds. Classification of quotient vector bundles.
Let again A be a topologicalvector space, X a topological space, and κ : X → Sub A an arbitrary map. Let q : A × X → E be the quotient map induced by κ (cf. definition above Theorem 3.6), and, for each Y ⊂ A × X , write [ Y ] for q − (cid:0) q ( Y ) (cid:1) (the saturation of Y ). Lemma 5.2
The induced quotient map is open if and only if for every neigh-borhood U ⊂ A of the origin the set [ U × X ] = (cid:8) ( a, x ) ∈ A × X | a ∈ κ ( x )+ U (cid:9) is an open subset of A × X .Proof. The forward implication is trivial. In order to prove the reverseimplication let W ⊂ X and U ⊂ A be open sets. We want to show that[ U × W ] = q − (cid:0) q ( U × W ) (cid:1) is an open set. We easily see that[ U × W ] = [ x ∈ W (cid:0) κ ( x ) + U (cid:1) × { x } = (cid:8) ( a, x ) ∈ A × W | a ∈ κ ( x ) + U (cid:9) Given any v ∈ A the translation τ v : A × X → A × X given by τ v ( a, x ) =( a + v, x ) is a homeomorphism and τ v [ U × W ] = [( U + v ) × W ], so q ( U × W )is open if and only if q (cid:0) ( U + v ) × W (cid:1) is open, and thus we may assume that U is a neighborhood of the origin. Now let ( a, x ) ∈ [ U × W ] ⊂ [ U × X ]. Byhypothesis [ U × X ] is open, so there are open sets W ′ ⊂ X and U ′ ⊂ A suchthat ( a, x ) ∈ U ′ × W ′ ⊂ [ U × X ]. Hence, U ′ × ( W ′ ∩ W ) ⊂ [ U × W ].18 heorem 5.3 Let κ : X → Sub A be a map. The induced quotient map isopen if and only if κ is continuous with respect to the lower Vietoris topology.Proof. First we assume that κ is continuous. In order to prove that theinduced quotient map is open we shall use Lemma 5.2. Let U ⊂ A be aneighborhood of the origin and let ( a, x ) ∈ [ U × X ]. Then a ∈ κ ( x ) + U sowe can write a = a + a with a ∈ κ ( x ) and a ∈ U . Consider the map f : A × A → A given by f ( v , v ) = a + v − v and let U ′ ⊂ A be aneighborhood of the origin so that f ( U ′ × U ′ ) = a + U ′ − U ′ ⊂ U . Since κ is continuous and κ ( x ) ∈ ^ a + U ′ , there is a neighborhood W of x suchthat, for any y ∈ W , we have κ ( y ) ∩ ( a + U ′ ) = ∅ . Then ( a + U ′ ) × W is aneighborhood of ( a, x ), so we only have to show that ( a + U ′ ) × W ⊂ [ U × X ].Let ( a ′ , x ′ ) ∈ ( a + U ′ ) × W . We need to show that a ′ ∈ κ ( x ′ ) + U . Since x ′ ∈ W , we have κ ( x ′ ) ∩ ( a + U ′ ) = ∅ and hence a ∈ κ ( x ′ ) − U ′ . Since a ′ ∈ a + U ′ = a + a + U ′ , we have a ′ ∈ a + U ′ − U ′ + κ ( x ′ ) ⊂ U + κ ( x ′ ),which concludes the first half of the proof.Now assume that the induced quotient map is open. Let x ∈ X and let U ⊂ A be an open set such that κ ( x ) ∈ e U (equivalently: U ∩ κ ( x ) = ∅ ). Wewant to find a neighborhood W ⊂ X of x so that U ∩ κ ( y ) = ∅ for every y ∈ W . Let a ∈ U ∩ κ ( x ) and consider the neighborhood U ′ = a − U of the origin.Since a ∈ κ ( x ) and 0 ∈ U ′ , we have ( a, x ) ∈ [ U ′ × X ] which is, by hypothesis,open, so there is a neighborhood W ⊂ X of x such that { a } × W ⊂ [ U ′ × X ]and, for every y ∈ W , we obtain a ∈ κ ( y ) + U ′ = a + κ ( y ) − U . It followsthat 0 ∈ κ ( y ) − U , and therefore U ∩ κ ( y ) = ∅ . Universal bundles.
Let A be a topological vector space. We shall referto the quotient vector bundle( π A : E A → Sub
A, A, q )whose kernel map is the identity map id : Sub A → Sub A as the universalquotient vector bundle for A .The following is an immediate corollary of the previous results: Theorem 5.4
Every quotient vector bundle ( π : E → X, A, q ) is isomorphicto the pullback of the universal bundle π A along the continuous map κ : X → Sub A which is defined by κ ( x ) = ker q x . Moreover, the fibers E x are Hausdorffspaces if and only if κ is valued in Max A . ell topology. Let A be a topological vector space. For each compact set K ⊂ A we define ˇ K = { P ∈ Max A | P ∩ K = ∅} , and the Fell topology on Max A is the coarsest topology that contains thelower Vietoris topology and all the sets ˇ K [2] (see also [9]). We shall useanalogous notation for finite dimensional linear subspaces V ⊂ A , as follows:ˇ V = (cid:8) P ∈ Max A | P ∩ V = { } (cid:9) . Note that if K , K ⊂ A are compact and K = K ∪ K then ˇ K ∩ ˇ K = ˇ K ,and therefore a basis for the Fell topology consists of all the sets of the form f U ∩ . . . ∩ f U m ∩ ˇ K , where the U i ’s are open sets of A and K ⊂ A is compact. Lemma 5.5
Let A be a Hausdorff vector space. For any finite dimensionalsubspace V ⊂ A the set ˇ V is open in the Fell topology.Proof. Let V ⊂ A be a finite dimensional vector space, hence with theEuclidean topology, and let K be the unit sphere in V in some norm. Then K is compact and ˇ V = ˇ K . Bundles classified by the Fell topology. If X is a topological space,we shall say that a map κ : X → Max A is Fell-continuous if it is continuouswith respect to the Fell topology of Max A . The Fell topology contains thelower Vietoris topology and therefore Fell-continuous maps κ : X → Max A determine quotient vector bundles ( π, A, q ). Lemma 5.6
Let ( π : E → X, A, q ) be a quotient vector bundle with Fell-continuous kernel map κ : X → Max A , for some Hausdorff vector space A .For any x ∈ X such that E x is finite dimensional there is a neighborhood U of x such that dim E y ≥ dim E x for any y ∈ U .Proof. Let x ∈ X and a , . . . , a n ∈ A be such that ˆ a ( x ) , . . . , ˆ a n ( x ) is abasis of E x . Let also V = h a , . . . , a n i and U = κ − ( ˇ V ). Then x ∈ U and,by Lemma 5.5, U is open. For any y ∈ U we have κ ( y ) ∩ V = { } and thus,by Lemma 3.5, ˆ a ( y ) , . . . , ˆ a n ( y ) are linearly independent in E y .The following result provides a characterization of what it means for abundle to have a Fell-continuous kernel map.20 heorem 5.7 Let ( π : E → X, A, q ) be a quotient vector bundle with kernelmap κ : X → Max A . If the image of the zero section of the bundle is closedthen κ is Fell-continuous. In addition, if both X and A are first countable,the image of the zero section is closed if and only if κ is Fell-continuous.Proof. Assume that the image of the zero section is closed. Let K ⊂ A be a compact set and let x ∈ κ − ( ˇ K ) — that is, κ ( x ) ∩ K = ∅ . Then, foreach a ∈ K we have ( a, x ) / ∈ q − (0). By hypothesis q − (0) is closed, andthus there are open sets U a ⊂ X and W a ⊂ A such that ( a, x ) ∈ W a × U a and ( W a × U a ) ∩ q − (0) = ∅ . The collection { W a } forms an open cover of K , whence there is a finite subcover W a , . . . , W a n . Let U = T i U a i . Then( K × U ) ∩ q − (0) = ∅ . In other words, for every y ∈ U we have κ ( y ) ∩ K = ∅ ,and thus x ∈ U ⊂ κ − ( ˇ K ). We showed that κ is Fell-continuous.Now assume that κ is Fell-continuous and that A and X are both firstcountable. Suppose q − (0) is not closed. Then there is a converging sequence( a n , x n ) in q − (0) with limit ( a, x ) / ∈ q − (0). Since a n → a and a / ∈ κ ( x ),and κ ( x ) is closed, there is a positive integer p such that a n / ∈ κ ( x ) for all n > p . Consider the compact set K = { a n | n > p } ∪ { a } ⊂ A . Then,by the continuity of κ , the set U = { y ∈ X | κ ( y ) ∩ K = ∅} is open in X and x ∈ U . But x n → x and a n ∈ κ ( x n ) for all n , and thus for some n wehave ( a n , x n ) / ∈ q − (0), which is a contradiction. Therefore q − (0) is closedin A × X .We have thus obtained a new necessary condition for local triviality, interms of Fell-continuity: Corollary 5.8
Let ( π : E → X, A, q ) be a quotient vector bundle with kernelmap κ : X → Max A , such that π is locally trivial. Then κ is Fell-continuous.Proof. This is an immediate consequence of Lemma 2.2, Corollary 3.7, andTheorem 5.7.We also record some simplified consequences of Theorem 5.7 when thespaces involved are all good enough:
Corollary 5.9
Let ( π : E → X, A, q ) be a quotient vector bundle with kernelmap κ : X → Max A , such that both X and A are first countable Hausdorffspaces. The following are equivalent:1. κ is Fell-continuous.2. The image of the zero section of the bundle is closed.3. E is a Hausdorff space.Proof. Immediate consequence of Theorem 3.8 and Theorem 5.7.21 he closed balls topology.
For a normed vector space A we shall considera refinement of the Fell topology on Max A such that instead of defining opensets ˇ K for compact K we shall consider instead a family of open sets indexedby A × R > as follows: for each a ∈ A and r > U r ( a ) = { P ∈ Max A | d ( a, P ) > r } . The coarsest topology that contains the lower Vietoris topology and the opensets U r ( a ) has a basis consisting of sets of the form f U ∩ . . . ∩ f U m ∩ U r ( a ) ∩ . . . ∩ U r k ( a k ) , where the U i ’s are open sets of A . We refer to this topology as the closed ballstopology on Max A because for all P ∈ U r ( a ) we have P ∩ B r ( a ) = ∅ , andthus the definition of U r ( a ) resembles that of ˇ K if we replace the compactset K by the closed ball B := B r ( a ). Indeed, if A is reflexive (for instance aHilbert space) we have U r ( a ) = ˇ B , although more generally, for any normedspace A , only the inclusion U r ( a ) ⊂ ˇ B holds. Lemma 5.10
Let A be a normed vector space. For every ε > and everycompact K ⊂ A , the set ˇ K ε = { P ∈ Max A | d (cid:0) P, K (cid:1) > ε (cid:9) is open in the closed balls topology of
Max A .Proof. Let ε >
0, and fix P ∈ ˇ K ε . Choose ε ′ and δ so that ε < ε ′ < d ( P, K )and δ < ε ′ − ε . Cover K by a finite number of balls B δ ( a i ) with a i ∈ K andlet U = T i U ε ′ ( a i ). Then U is open in the closed balls topology, and P ∈ U .We will show that U ⊂ ˇ K ε . Let Q ∈ U . Given u ∈ Q and v ∈ K , there is i such that k v − a i k < δ . Then ε ′ < k u − a i k ≤ k u − v k + k v − a i k < k u − v k + δ , so k u − v k > ε ′ − δ . Since u and v are arbitrary we get d ( Q, K ) ≥ ε ′ − δ > ε ,and thus Q ∈ ˇ K ε . Hence, ˇ K ε is open in the closed balls topology. Lemma 5.11
Let A be a normed vector space. The closed balls topology of Max A contains the Fell topology.Proof. Let K ⊂ A be compact, and recall the definition of ˇ K ε from Lemma 5.10.Clearly, [ ε> ˇ K ε ⊂ ˇ K .
22n order to see that also the converse inclusion holds let P ∈ ˇ K . The con-dition P ∩ K = ∅ implies d ( P, k ) > k ∈ K because P is closed,and this further implies d ( P, K ) > d ( P, − ) : K → R is continuousand K is compact. Hence, choosing ε such that d ( P, K ) > ε > P ∈ ˇ K ε , and thus [ ε> ˇ K ε = ˇ K .
Normed bundles and Banach bundles.
The closed balls topology clas-sifies continuous normed bundles, as we now show.
Theorem 5.12
Let ( π : E → X, A, q ) be a normed quotient vector bundlewith kernel map κ : X → Max A . The following are equivalent:1. The bundle is continuous;2. κ is continuous with respect to the closed balls topology.Proof. Let us prove (1) ⇒ (2). Let a ∈ A and ε >
0. The distance d (cid:0) a, κ ( x ) (cid:1) equals k q ( a, x ) k , so we have κ − (cid:0) U ε ( a ) (cid:1) = (cid:8) x ∈ X | d ( a, κ ( x )) > ε (cid:9) = (cid:8) x ∈ X | k ˆ a ( x ) k > ε (cid:9) = ˆ a − (cid:0) { e ∈ E | k e k > ε } (cid:1) , and thus κ − (cid:0) U ε ( a ) (cid:1) is open due to the continuity of the norm on E andthe continuity of ˆ a , and we conclude that κ is continuous with respect to theclosed balls topology.Now let us prove (2) ⇒ (1). We only need to check that the norm on E is lower semicontinuous; that is, given a ∈ A , x ∈ X , and r > k q ( a, x ) k > r , we need to prove that for some neighborhood W of q ( a, x ) wehave k e k > r for all e ∈ W , where W can be taken to be q ( B ε ( a ) × U ) forsome ε > U of x . Let r = k q ( a, x ) k andlet m = ( r + r ) /
2. Note that for all y ∈ X the condition k q ( a, y ) k > m isequivalent to the statement that y ∈ U for the open set defined by U = κ − ( U m ( a )) . In particular, x ∈ U . Let ε = ( r − r ) /
2. Then for all b ∈ B ε ( a ) and all y ∈ U we have k q ( b, y ) k = d ( b, κ ( y )) ≥ d ( a, κ ( y )) − d ( a, b )= k q ( a, y ) k − k a − b k > m − ε = r . orollary 5.13 Let A be a Banach space, X a first countable Hausdorffspace, and ( π : E → X, A, q ) a quotient vector bundle with kernel map κ : X → Max A . The following are equivalent:1. The quotient norm makes π : E → X is a Banach bundle;2. κ is continuous with respect to the closed balls topology.Proof. (1) ⇒ (2) is an immediate consequence of Theorem 5.12 becauseBanach bundles have continuous norm. Conversely, let us prove (2) ⇒ (1).If (2) holds then the quotient norm is continuous, again by Theorem 5.12. Inaddition, κ is Fell-continuous by Lemma 5.11, so by Corollary 5.9 the space E is Hausdorff. To conclude, the fibers E x are Banach spaces because theyare isomorphic to quotients A/κ ( x ) with κ ( x ) ∈ Max A and, by Lemma 3.3,the topology of E around the image of the zero section is the required one(cf. axiom 5 in the definition of section 4). Let ( π : E → X, A, q ) be a quotient vector bundle and κ : X → Sub A its kernel map. We say the bundle has rank n if all of its fibers E x havedimension n or, equivalently, if all the subspaces κ ( x ) have codimension n in A . In this section we study such bundles. We shall only be interested inbundles whose fibers have the Euclidean topology, so we shall take κ to bevalued in Max A , in addition assuming that κ is Fell-continuous because oneof our aims is to study locally trivial bundles (cf. Corollary 5.8). From hereon we shall denote by Max n A the topological space, equipped with the relative Fell topology, whose pointsare the closed linear subspaces P ∈ Max A such that dim( A/P ) = n . Fiber structures.
Let A be a Hausdorff vector space and let( π : E → X, A, q )be a rank- n quotient vector bundle with Fell-continuous kernel map κ : X → Max n A .
Given any n -dimensional subspace V ⊂ A and any x ∈ X , both V and E x have the Euclidean topology, so there is a linear homeomorphism E x ∼ = V .24his suggests that, in a suitable sense, V can be regarded as being “the fiber”of the bundle.In order to pursue this idea, first note that for each P ∈ ˇ V ∩ Max n A wehave V ∼ = A/P and P ∩ V = { } , and therefore A = V ⊕ P , where for each a ∈ A the component of a in V is the vector v such that V ∩ ( a + P ) = { v } (cf. Lemma A.1). This leads to the following definitions:1. The fiber domain of V (with respect to κ ) is the open set D V ⊂ X defined by D V = κ − (cid:0) ˇ V (cid:1) .
2. The projection family of V is the mapping p V : A × D V → V defined by the condition V ∩ ( a + κ ( x )) = { p V ( a, x ) } . (The map p V defines a family of projections a p V ( a, x ) of A onto V indexed by x ∈ D V , hence the terminology — cf. Lemma A.1.)3. The fiber family of V is the mapping f V : E D V → V defined by the condition f V ◦ q = p V . (This is well defined because p V ( a, x ) = p V ( b, x ) if a − b ∈ κ ( x ), and itdefines a family of isomorphisms E x ∼ = V indexed by x ∈ D V .) Lemma 6.1
Let A be a Hausdorff vector space, let ( π : E → X, A, q ) be arank- n quotient vector bundle whose kernel map κ : X → Max n A is Fell-continuous, and let V ⊂ A be an n -dimensional linear subspace. For allsubsets W ⊂ V , all a ∈ A , and all x ∈ D V , the following conditions areequivalent:1. f V ( q ( a, x )) ∈ W (equiv., p V ( a, x ) ∈ W );2. (cid:0) a + κ ( x ) (cid:1) ∩ ( V \ W ) = ∅ .Proof. Immediate consequence of the definitions of f V and p V .25 emma 6.2 Let A be a Hausdorff vector space, and let ( π : E → Max n A, A, q ) be the rank- n quotient vector bundle whose kernel map is the identity on Max n A . Let also V ⊂ A be an n -dimensional linear subspace, let W ⊂ V be an open subset (in the subspace topology of V ), and let a / ∈ V . Fix somenorm on V ⊕ h a i , let r V ⊕h a i : (cid:0) V ⊕ h a i (cid:1) \ { } → S V ⊕h a i be the retraction r V ⊕h a i ( v ) = v/ k v k onto the unit sphere S V ⊕h a i and let K W = r V ⊕h a i (cid:0) ( V \ W ) − a (cid:1) . Then K W is compact. Moreover, for all P ∈ ˇ V we have (6.2.1) p V ( a, P ) ∈ W ⇐⇒ P ∈ ˇ K W . Remark.
Since S V ⊕h a i is closed in A , the closure of r V ⊕h a i (cid:0) ( V \ W ) − a (cid:1) isthe same in A or in S V ⊕h a i . Proof.
In order to show that K W is compact notice that S V ⊕h a i is compactbecause V ⊕ h a i is finite dimensional. Since K W ⊂ S V ⊕h a i is closed, K W iscompact. Now let P ∈ ˇ V . The condition P ∈ ˇ K W is, by definition, equivalentto(6.2.2) P ∩ K W = ∅ , which implies(6.2.3) P ∩ r V ⊕h a i (cid:0) ( V \ W ) − a (cid:1) = ∅ , and, equivalently,(6.2.4) P ∩ (cid:0) ( V \ W ) − a (cid:1) = ∅ , which in turn is equivalent, by Lemma 6.1, to(6.2.5) p V ( a, P ) ∈ W .
So we have proved the ⇐ implication of (6.2.1), and also concluded thatthe conditions (6.2.3)–(6.2.5) are all equivalent. Hence, in order to prove the ⇒ implication of (6.2.1) we only need to show that (6.2.3) implies (6.2.2).Suppose the former holds, and let w ∈ K W . Then, since V ⊕ h a i is normed,there is a sequence ( v k ) in ( V \ W ) − a such that lim v k / k v k k = w . If ( v k )is not bounded we also have lim( v k + a ) / k v k + a k = w . In this case, since26he sequence ( v k + a ) has values in V , we obtain w ∈ V , which impliesthat w / ∈ P because P ∈ ˇ V , so (6.2.2) holds. Now assume that ( v k ) isbounded. Then, since V is finite dimensional and V \ W − a is closed, thereis a convergent subsequence ( w k ) such that lim w k ∈ V \ W − a , and thus w = r V ⊕h a i (lim w k ) ∈ r V ⊕h a i (cid:0) ( V \ W ) − a (cid:1) . Therefore w / ∈ P and again weconclude that (6.2.2) holds. Theorem 6.3 If A is a Hausdorff vector space then Max n A , with the Felltopology, is a Hausdorff space.Proof. Let
P, Q ∈ Max n A be such that P = Q . By Lemma A.2 there isan n -dimensional subspace V ⊂ A such that V ∩ P = V ∩ Q = { } . Let( π : E → Max n A, A, q ) be the rank- n quotient vector bundle whose kernelmap is the identity on Max n A . Let a ∈ P \ Q . Then ˆ a ( P ) = 0 and ˆ a ( Q ) = 0.Let s : D V → V be defined by s = f V ◦ ˆ a . Then s ( P ) = 0 and s ( Q ) = 0, and,since V is Hausdorff, we only need to show that s is continuous. But thisfollows from Lemma 6.2 because for all R ∈ D V we have s ( R ) = p V (cid:0) a, R (cid:1) ,and thus for each open set W of V there is a compact set K W ⊂ A such thatthe preimage s − ( W ) equals ˇ K W . Local triviality.
Let A be a Hausdorff vector space, and let( π : E → X, A, q )be a rank- n quotient vector bundle with Fell-continuous kernel map κ : X → Max n A .
Let also V ⊂ A be an n -dimensional linear subspace. The restriction of q to A × D V is an open map because the fiber domain D V is open, andtherefore the projection family p V is continuous if and only if the fiber family f V is. The continuity of these maps is closely related to the existence of localtrivializations, as we now see: Lemma 6.4
With the bundle ( π, A, q ) and V as just defined, let q ′ : V × D V → E D V be the restriction of q to V × D V . Then q ′ is continuous and bijective, andthe following conditions are equivalent:1. q ′ is an isomorphism of bundles in LinBun ( X ) ; . f V is continuous (equiv., p V is continuous).Proof. The function q ′ is clearly continuous and it is bijective by Lemma 3.5.Its inverse is the pairing h f V , π D V i , and thus q ′ is a homeomorphism if andonly if f V is continuous. Theorem 6.5
Let A be a Hausdorff vector space and let ( π : E → X, A, q ) be a rank- n quotient vector bundle with Fell-continuous kernel map κ : X → Max n A .
The following conditions are equivalent:1. π is locally trivial;2. For all n -dimensional subspaces V ⊂ A the fiber family f V (equiv., theprojection family p V ) is continuous.Proof. The implication (2) ⇒ (1) is a consequence of Lemma 6.4: for each x ∈ X choose an n -dimensional subspace V ⊂ A such that V ∩ κ ( x ) = { } ;the fiber domain D V is an open neighborhood of x and the continuity of f V implies that there is a local trivialization ( q ′ ) − : E D V → V × D V , where q ′ is the restriction of q to V × D V .Now let us prove the implication (1) ⇒ (2). Suppose π is locally trivialand let V ⊂ A be an n -dimensional subspace. Let x ∈ D V , let q ′ : V × D V → E D V be the restriction of q to V × D V , and let φ : E U → V × U be atrivialization on some open neighborhood U of x . We may assume that U ⊂ D V , and thus obtain a continuous and bijective function φ ◦ q ′′ : V × U → V × U , which is a morphism in LinBun ( U ), where q ′′ is the restriction of q ′ to V × U . The function φ ◦ q ′′ is necessarily a bundle isomorphism becauseit restricts fiberwise to continuous linear isomorphisms φ x : V × { x } → V ×{ x } , which are necessarily homeomorphisms because V has the Euclideantopology, and thus q ′′ is a homeomorphism. Since q ′′ is a generic restrictionof q ′ and the domains of all such maps q ′′ cover the domain of q ′ , we concludethat q ′ is itself a homeomorphism. So, by Lemma 6.4, f V is continuous. Two technical lemmas.
In the following two lemmas A is an arbitraryHausdorff vector space, ( π : E → X, A, q ) is a rank- n quotient vector bundlewith Fell-continuous kernel map κ : X → Max n A , and V ⊂ A is an n -dimensional subspace. 28 emma 6.6 Let B be a basis of neighborhoods of the origin in the topologyof V . Then the map f V : E D V → V is continuous if and only if for any W ∈ B and for any x ∈ D V there is a neighborhood U ⊂ A of zero in thetopology of A such that x is in the open set N W,U = int (cid:8) y ∈ D V | κ ( y ) ∩ (cid:0) ( V \ W ) + U (cid:1) = ∅ (cid:9) . Proof.
First, assuming the hypothesis about the sets N W,U , we show that f V is continuous. Let [ a, x ] = q ( a, x ) ∈ E D V and let v = f V ([ a, x ]). Thecollection { v + W | W ∈ B} is a basis of neighborhoods of v . In order toprove continuity of f V we shall, given an arbitrary but fixed W ∈ B , find anopen set U ⊂ E D V containing [ a, x ] such that(6.6.1) f V ( U ) ⊂ v + W .
By hypothesis there is a neighborhood of zero U ⊂ A such that x ∈ N W,U ,and we define U = q (( v − U ) × N W,U ). So let us prove (6.6.1). Let y ∈ N W,U and b ∈ v − U . Then (cid:0) κ ( y ) − U (cid:1) ∩ ( V \ W ) = ∅ and, since b − v ∈ − U , wefind that (cid:0) κ ( y ) + b − v (cid:1) ∩ ( V \ W ) = ∅ , so (cid:0) κ ( y ) + b − v (cid:1) ∩ V ⊂ W , andthus (cid:0) κ ( y ) + b (cid:1) ∩ V ⊂ v + W . We have shown that f V ([ y, b ]) ∈ v + W for all[ y, b ] ∈ U , thus proving (6.6.1).Reciprocally, assume f V is continuous. Then, by the continuity of p V = q ◦ f V , for any W ∈ B and for any x ∈ D V there is an open neighborhood U ofzero in A and an open neighborhood N of x such that f V ( q (( − U ) × N )) ⊂ W .Hence, for any y ∈ N and b ∈ U we have (cid:0) κ ( y ) − b (cid:1) ∩ ( V \ W ) = ∅ . But thisimplies that (cid:0) κ ( y ) − U (cid:1) ∩ ( V \ W ) = ∅ for any y ∈ N , so we conclude that N ⊂ N W,U . Lemma 6.7
The following conditions are equivalent:1. For any b ∈ A and any x ∈ X such that ˆ b ( x ) = 0 x there is a neighbor-hood U of zero in A such that x ∈ int (cid:8) y ∈ X | κ ( y ) ∩ ( b + U ) = ∅ (cid:9) ;
2. For any compact set K ⊂ A and any x ∈ κ − ( ˇ K ) there is a neighbor-hood U of zero in A such that x ∈ int (cid:8) y ∈ X | κ ( y ) ∩ ( K + U ) = ∅ (cid:9) . Proof.
Trivially (2) ⇒ (1), so assume (1) holds. Then for any b ∈ K thereis a neighborhood U b ⊂ A of zero such that x ∈ int (cid:8) y ∈ X | κ ( y ) ∩ ( b + U b ) = ∅ (cid:9) . W b of zero in A such that W b + W b ⊂ U b . Then K ⊂ [ b ( b + W b ) , so we can pick a finite covering K ⊂ n [ i =1 ( b i + W b i ) . Let U = T i W b i . Then K + U ⊂ S i ( b i + W b i + U ) ⊂ S i ( b i + U b i ), andtherefore x ∈ \ i int (cid:8) y ∈ X | κ ( y ) ∩ ( b i + U b i ) = ∅ (cid:9) = int \ i (cid:8) y ∈ X | κ ( y ) ∩ ( b i + U b i ) = ∅ (cid:9) = int ( y ∈ X | κ ( y ) ∩ (cid:16)[ i ( b i + U b i ) (cid:17) = ∅ ) ⊂ int (cid:8) y ∈ X | κ ( y ) ∩ ( K + U ) = ∅ (cid:9) . Locally convex spaces.
Using the previous results we can provide anotherequivalent condition to local triviality, in the case where the Hausdorff vectorspace A is locally convex: Theorem 6.8
Let A be a Hausdorff locally convex space, and let ( π : E → X, A, q ) be a rank- n quotient vector bundle with Fell-continuous kernel map κ : X → Max n A . Then π is locally trivial if and only if for any ( b, x ) ∈ A × X such that ˆ b ( x ) = 0 x there is a neighborhood U of zero in A such that x ∈ int (cid:8) y ∈ X | κ ( y ) ∩ ( b + U ) = ∅ (cid:9) . [We remark that only the reverse implication uses local convexity.] Proof.
Assume first that π is locally trivial. Then, by Theorem 6.5, f V iscontinuous for any linear subspace V ⊂ A of dimension n . Let ( b, x ) ∈ A × X be such that ˆ b ( x ) = 0 x ; that is, b / ∈ κ ( x ). Then we can choose an n -dimensional vector subspace V ⊂ A such that b ∈ V and V ∩ κ ( x ) = { } (let V be spanned by b, a , . . . , a n − such that b + κ ( x ) , a + κ ( x ) , . . . , a n − + κ ( x )is a basis of A/κ ( x )). Let W ⊂ V be a neighborhood of zero in V with b / ∈ W . By Lemma 6.6, there is a neighborhood U of zero in A such that x ∈ int (cid:8) y ∈ D V | κ ( y ) ∩ (cid:0) ( V \ W ) + U (cid:1) = ∅ (cid:9) . b ∈ V \ W , we obtain x ∈ int (cid:8) y ∈ X | κ ( y ) ∩ (cid:0) ( V \ W ) + U (cid:1) = ∅ (cid:9) ⊂ int (cid:8) y ∈ X | κ ( y ) ∩ (cid:0) b + U (cid:1) = ∅ (cid:9) . In order to prove the converse let us assume that for any ( b, x ) ∈ A × X such that ˆ b ( x ) = 0 x there is a neighborhood U of zero in A satisfying x ∈ int (cid:8) y ∈ X | κ ( y ) ∩ ( b + U ) = ∅ (cid:9) . Equivalently, by Lemma 6.7, we assume that for any compact set K ⊂ A and any x ∈ κ − ( ˇ K ) there is a neighborhood U ⊂ A of zero such that(6.8.1) x ∈ int (cid:8) y ∈ X | κ ( y ) ∩ ( K + U ) = ∅ (cid:9) . Since A is locally convex we shall assume that U is convex. Let V ⊂ A be anarbitrary n -dimensional linear subspace, and choose some norm on V . Letalso B be the collection of balls around the origin of V in that norm, let W ∈ B , and let x ∈ D V . Now let K be the (compact) boundary of W in V .Then x ∈ κ − ( ˇ K ), and (6.8.1) holds. We claim that for any linear subspace P ⊂ A we have P ∩ (cid:0) ( V \ W ) + U (cid:1) = ∅ if and only if P ∩ ( K + U ) = ∅ . Oneimplication is clear because K ⊂ V \ W , so let b ∈ P ∩ (cid:0) ( V \ W ) + U (cid:1) . Then b = v + u with v ∈ V \ W and u ∈ U , so, since W is a ball, there is c ∈ (0 , cv ∈ K . Then cu ∈ U because U is convex and 0 ∈ U , and thus cb ∈ P ∩ ( K + U ), which proves the claim. Hence, (6.8.1) is equivalent to x ∈ int (cid:8) y ∈ X | κ ( y ) ∩ (cid:0) ( V \ W ) + U (cid:1) = ∅ (cid:9) , which in turn is equivalent to the condition x ∈ N W,U of Lemma 6.6 becausewe are assuming that x belongs to the open set D V . Hence, by Lemma 6.6,we have proved that f V is continuous for all n -dimensional subspaces V ⊂ A ,and thus, by Theorem 6.5, π is locally trivial. Normed bundles and Banach bundles.
Let us conclude our study oflocal triviality by looking at continuous normed bundles and Banach bundles.
Theorem 6.9
Let ( π : E → X, A, q ) be a normed rank- n quotient vectorbundle with kernel map κ : X → Max n A . If κ is continuous with respect tothe closed balls topology then:1. For every n -dimensional subspace V ⊂ A , the bundle E induced by κ is trivial on the open set D V = κ − (cid:0) ˇ V (cid:1) ; . π is locally trivial.Proof. The space A is Hausdorff and, by Lemma 5.11, the map κ is Fell-continuous. Hence, by Lemma 6.4 and Theorem 6.5, the two conditions (1)and (2) are equivalent. So let us apply Theorem 6.8 (since A is also locallyconvex). Given ( b, x ) ∈ A × X with ˆ b ( x ) = 0 x (equivalently, b / ∈ κ ( x )) let usdefine ε = d (cid:0) b, κ ( x ) (cid:1) / ,U = B ε (0) . Then x ∈ κ − ( U ε ( b )) ⊂ (cid:8) y ∈ X | κ ( y ) ∩ ( b + U ) = ∅ (cid:9) , and it follows from Theorem 6.8 that π is locally trivial.This implies the following result, which is mentioned but not proved in[3, p. 129]: Corollary 6.10
Every Banach bundle of constant finite rank on a locallycompact Hausdorff space is locally trivial.Proof.
Recall from Theorem 4.2 that every Banach bundle π : E → X with X locally compact Hausdorff can be made a quotient vector bundle( π, A, q ) by taking A = C ( π ) with the supremum norm and q = eval. ByCorollary 5.13 this bundle is classified by a kernel map κ which is continu-ous with respect to the closed balls topology. The conclusion follows fromTheorem 6.9. The role played by the spaces Sub A in the classification of quotient vectorbundles ( π, A, q ) is analogous to that of Grassmannians in the classificationof locally trivial finite rank vector bundles. However, the relations betweenSub A and Grassmannians in the preceding sections are obscured by thefact that A has in general been infinite dimensional, so our purpose in thepresent section is to investigate the extent to which the topologies of Sub A and Grassmannians coincide. This said, this section is largely independentfrom the preceding ones. We shall only need to consider the lower Vietoristopology on Sub A , and the main results of this section will require A to beHausdorff. 32 inearly independent open sets. Let A be a topological vector space.We say that a finite collection of open sets U , . . . , U k ⊂ A is linearly inde-pendent if any k vectors v , . . . , v k with v i ∈ U i are linearly independent. Lemma 7.1
Let A be a Hausdorff vector space, let S ⊂ A be a finite set,and for each v ∈ S let U v be a neighborhood of v . Let e , . . . , e m be a basisof the linear span h S i . Then there are linearly independent neighborhoods U ′ , . . . , U ′ m of e , . . . , e m such that, for any v ∈ S , if we write v = P j a j,v e j (with a j,v ∈ C ) then P j a j,v U ′ j ⊂ U v .Proof. We divide the proof into three steps:1. We claim that there are linearly independent neighborhoods U l.i. j of e j (with j = 1 , . . . , m ). Consider the unit sphere S m − ⊂ C m and let f : S m − × A m → A be the function f ( z , . . . , z m , u , . . . , u m ) = P j z j u j .Then S m − × { ( e , . . . , e m ) } ⊂ f − ( A \
0) so, since S m − is compact,there are (by the tube lemma) open neighborhoods U l.i. j of e j such that S m − × Q U l.i. j ⊂ f − ( A \ v = P a j,v e j ∈ S . We claim that there are neighborhoods U j,v of e j such that P j a j,v U j,v ⊂ U v . Consider the function f v : A m → A given by f v ( u , . . . , u m ) = P j a j,v u j . Since f v is continuous, there areneighborhoods U j,v of e j such that f v ( U ,v × · · · × U m,v ) ⊂ U v , whichproves the claim.3. Now, for each j = 1 , . . . , m , let U ′ j = U l.i. j ∩ \ v ∈ S U j,v ! . By (1), the sets U ′ j (which are open because S is finite) are linearlyindependent. And, by (2), we have P j a j,v U ′ j ⊂ U v , which concludesthe proof. Grassmannians as subspaces.
For any Hausdorff vector space A (of anydimension) and any integer k >
0, we shall write V ( k, A ) for the set ofinjective linear maps from C k to A with the product topology, and we shallwrite S ( v, U ), where v ∈ C k and U ⊂ A is open, for the sub basic open setconsisting of those φ ∈ V ( k, A ) such that φ ( v ) ∈ U . Denoting by Gr ( k, A )the set of all the k -dimensional subspaces of A , we have a surjective map p k : V ( k, A ) → Gr ( k, A ) , p k ( φ ) = Im φ . We shall refer to Gr ( k, A ), equipped with thequotient topology, as a Grassmannian of A (so we have a homeomorphism Gr ( k, A ) ∼ = V ( k, A ) / GL ( k, C )). This, of course, is a generalization of theusual definition of Grassmannian of a finite dimensional vector space. Theorem 7.2
Let A be a Hausdorff vector space. For any integer k > ,the Grassmannian Gr ( k, A ) is a topological subspace of Sub A .Proof. Let us first prove that, given an open set U ⊂ A , the set e U ∩ Gr ( k, A )is open in the quotient topology. We only need to show that the set p − k (cid:0) e U ∩ Gr ( k, A ) (cid:1) = (cid:8) φ ∈ V ( k, A ) | Im φ ∩ U = ∅} , is open in V ( k, A ). Given φ ∈ p − k (cid:0) e U ∩ Gr ( k, A ) (cid:1) , there is v ∈ C k such that φ ( v ) ∈ U . Then clearly φ ∈ S ( v, U ) ⊂ p − k (cid:0) e U ∩ Gr ( k, A ) (cid:1) , which completesthe proof.Now we prove that any set W ⊂ Gr ( k, A ) which is open in the quotienttopology is also open in the topology induced by Sub A . Let P be an arbitraryelement of W , with W open in the quotient topology. In order to show that W is also open in the subspace topology we shall find an open set U ⊂
Sub A such that(7.2.1) P ∈ Gr ( k, A ) ∩ U ⊂ W .
First we observe that P = p k ( φ ) for some φ ∈ p − k ( W ), so there are u , . . . , u n ∈ C k and open sets U , . . . , U n ⊂ A such that(7.2.2) φ ∈ n \ i =1 S ( u i , U i ) ⊂ p − k ( W ) . We may assume without loss of generality that for some m ≤ n the vectors u , . . . , u m form a basis of h u , . . . , u n i . Then we are within the conditions ofLemma 7.1 with S = { φ ( u ) , . . . , φ ( u n ) } ,φ ( u i ) ∈ U i ( i = 1 , . . . , n ) ,e j = φ ( u j ) ( j = 1 , . . . , m ) . U ′ j be as in Lemma 7.1. Then P ∈ T mj =1 e U ′ j , so let us define U = m \ j =1 e U ′ j , and let us show that the inclusion in (7.2.1) holds:(7.2.3) Gr ( k, A ) ∩ U ⊂ W Let Q ∈ Gr ( k, A ) ∩ U . Then Q = p k ( ψ ) for some ψ ∈ V ( k, A ). For each j = 1 , . . . , m we have Q ∈ e U ′ j , so there is w j ∈ C k such that ψ ( w j ) ∈ U ′ j .The open sets U ′ j are linearly independent, so the vectors w j are linearlyindependent, too. Hence, there is g ∈ GL ( k, C ) such that g ( u j ) = w j for all j = 1 , . . . , m , and Q = Im( ψ ◦ g ) = p k ( ψ ◦ g ) . Now, for each i = 1 , . . . , n we can write u i = P mj =1 a ij u j , with a ij ∈ C . Then ψ ( g ( u i )) = m X j =1 a ij ψ ( g ( u j )) = m X j =1 a ij ψ ( w j ) ∈ m X j =1 a ij U ′ j ⊂ U i , and thus ψ ◦ g ∈ S ( u i , U i ). Since, by (7.2.2), we have n \ i =1 S ( u i , U i ) ⊂ p − k ( W ) , we obtain Q = p k ( ψ ◦ g ) ∈ W , showing that (7.2.3) holds. Corollary 7.3
For any k ∈ { , . . . , n } we have a homeomorphism Max k C n ∼ = Gr ( n − k, C n ) . A new basis for the lower Vietoris topology.
Let us provide a descrip-tion of the lower Vietoris topology in terms of the Grassmannian topologies.In order to do this, first we need a finer subbasis for the lower Vietoris topol-ogy:
Lemma 7.4
Let A be a Hausdorff vector space. Then the topology of Sub A has as a basis the collection of finite intersections T ki =1 e U i where U , . . . , U k ⊂ A are linearly independent open sets. roof. Given open sets U , . . . , U n ⊂ A let P ∈ T i e U i . We want to findlinearly independent open sets U ′ , . . . , U ′ m such that P ∈ T mj =1 e U ′ j ⊂ T ni =1 e U i .For each i let e i ∈ P ∩ U i . We may assume without loss of generality thatthere is some m ≤ n such that e , . . . , e m form a basis of h e , . . . , e n i . Thenwe are within the conditions of Lemma 7.1 with S = { e , . . . , e n } , so let U ′ , . . . , U ′ m be as in Lemma 7.1. Then we have P ∈ m \ j =1 e U ′ j , and we need to show that(7.4.1) m \ j =1 e U ′ j ⊂ n \ i =1 e U i . Let Q ∈ T mj =1 e U ′ j . For each j = 1 , . . . , m let u j ∈ Q ∩ U ′ j . By Lemma 7.1there are a ij ∈ C such that, for each i = 1 , . . . , n we have m X j =1 a ij u j ∈ m X j =1 a ij U ′ j ⊂ U i . Since we also have P mj =1 a ij u j ∈ Q , it follows that Q ∈ e U i for all i . Thereforewe have Q ∈ T ni =1 e U i , which proves (7.4.1).Now we shall use the notation ↑ W , for W any subset of Sub A , to denotethe upper-closure of W in the inclusion order of Sub A : ↑ W = { V ∈ Sub A | ∃ V ′ ∈ W V ′ ⊂ V } . Theorem 7.5
Let A be a Hausdorff vector space. A basis for the lowerVietoris topology of Sub A consists of the collection of all the sets ↑ W wherefor some k ∈ N > the set W ⊂ Gr ( k, A ) is open in the Grassmanniantopology.Proof. Due to Theorem 7.2 and Lemma 7.4 it is enough to show that if U , . . . , U k ⊂ A are linearly independent open sets, and if we let W = Gr ( k, A ) ∩ k \ i =1 e U i , then ↑ W = T e U i . The inclusion ↑ W ⊂ T e U i is immediate because W ⊂ T e U i and the open sets of Sub A are upwards closed: ↑ W ⊂ ↑ (cid:0)T e U i (cid:1) = T e U i . Forthe other inclusion, consider P ∈ T e U i . Then there are linearly independentvectors v , . . . , v k ∈ A such that v i ∈ P ∩ U i for all i . Let V = h v , . . . , v k i .Then V ∈ W and V ⊂ P , and thus P ∈ ↑ W .36 Appendix
Lemma A.1
Let A be a vector space and n a positive integer. Let also V, P ⊂ A be linear subspaces such that V ∩ P = { } and dim( V ) = dim( A/P ) = n . Then A = V ⊕ P , and the set V ∩ ( a + P ) is a singleton.Proof. Let a , . . . , a n be a basis of V . Then a + P, . . . , a n + P is a basisof A/P , and there is an isomorphism ι : A/P → V defined by a i + P a i ,which gives us a projection ˆ P : A → A by composing the quotient a a + P with ι and the inclusion V → A : A → A/P ι → V → A .
Hence, we get ˆ P ( A ) = V and ker ˆ P = P (and thus A = V ⊕ P ), and therefore V ∩ ( a + P ) = { ˆ P ( a ) } . Lemma A.2
Let A be a vector space and n a positive integer. Let P, Q ⊂ A be linear subspaces such that dim( A/P ) = dim(
A/Q ) = n . Then there is an n -dimensional subspace V ⊂ A such that V ∩ P = V ∩ Q = { } .Proof. We begin by fixing an isomorphism φ : A → ( P ∩ Q ) ⊕ A/ ( P + Q ) ⊕ ( P + Q ) / ( P ∩ Q ) ∼ = ( P ∩ Q ) ⊕ A/ ( P + Q ) ⊕ P/ ( P ∩ Q ) ⊕ Q/ ( P ∩ Q ) . Notice that φ ( P ) ⊂ ( P ∩ Q ) ⊕ P/ ( P ∩ Q ) and φ ( Q ) ⊂ ( P ∩ Q ) ⊕ Q/ ( P ∩ Q ).Let j = dim A/ ( P + Q ). Then, since A/P ∼ = A/ ( P + Q ) ⊕ ( P + Q ) /P ∼ = A/ ( P + Q ) ⊕ Q/ ( P ∩ Q ), we have dim Q/ ( P ∩ Q ) = n − j and, similarly,dim P/ ( P ∩ Q ) = n − j . Now pick an ( n − j )-dimensional subspace W ⊂ P/ ( P ∩ Q ) ⊕ Q/ ( P ∩ Q ) such that W ∩ P/ ( P ∩ Q ) = W ∩ Q/ ( P ∩ Q ) = { } ,and let V = φ − (cid:0) W ⊕ A/ ( P + Q ) (cid:1) . Then V ∩ P = V ∩ Q = { } . References [1] J. Feldman and C. C. Moore,
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E-mail: [email protected] , [email protected]@math.tecnico.ulisboa.pt