Metrisability of Manifolds

Metrisabilit y of Manifolds Da vid Gauld ∗ † Ma y 10, 2018 Abstract Manifolds ha v e uses throughout and b ey ond Ma thematics and it is not surprising t hat top ologists ha v e exp ended a h uge effort in trying to understand them. In this article we are particularly in tereste d in the question: ‘when is a manifold metrisable?’ W e describ e man y conditions e quiv alen t to me trisabilit y . 1 In tro d u ction. By a manifold is mean t a connected, Hausdorff space whic h is lo cally homeomorphic to eu- clidean space (w e take o ur manifolds to ha v e no b o undary). No te that b ecause of connected- ness the dimension of the euclidean space is an in v arian t of the manifold (unless the manifold is empt y!); this is the di m ension of the manifo ld. A pair ( U, h ), where U ⊂ M is op en and h : U → R m is a homeomorphism, is called a c o or dinate chart . The follo wing notation is used. R denotes the real line with the usual (order) top ology while R n denotes the n th p ow er of R . B n consists of a ll p oin ts of R n at most 1 from the o rigin. The sets ω and ω 1 are, resp ectiv ely , the finite and coun table ordinals. Clearly eve ry manifold is T yc hono ff. Of course manifo lds share all of the lo cal prop erties of euclidean space, including lo cal compactness, lo cal connectedness , lo cal path o r arc con- nectedness , first countabilit y , lo cal second countabilit y , lo cal hereditary separabilit y , etc. As ev ery manifold is lo cally compact and Hausdorff, hence completely regular, it follows that ev ery manifold is uniformisable ([38, Prop osition 1 1.5]). The follow ing result sho ws that manifolds cannot b e to o big. Prop osition 1 L et M b e a (non-em p ty!) manifold. Then every c ountable subset of M is c ontaine d in an op en subset wh i c h is home om o rphic to euclide an sp ac e. Henc e eve ry two p oints of M may b e joi n e d by an ar c. Pro of . Supp ose that the dimension of M is m . Let S ⊂ M b e a coun table subset, say h S n i is suc h that S = ∪ n ≥ 1 S n , | S n | = n and S n ⊂ S n +1 . By induction on n w e choose o p en V n ⊂ M and compact C n ⊂ M suc h that (i) S n ∪ C n − 1 ⊂ ˚ C n and (ii) ( V n , C n ) ≈ ( R m , B m ), where C 0 = ∅ . F or n = 1, S 1 is a singleton so V 1 ma y b e any appropriate neigh b ourho o d of that p oint while C 1 is a compact neigh b ourho o d c hosen to satisfy (ii) as well. ∗ Suppo rted in part by a Mar sden F und Award, UOA611, from the Roy al So ciety o f New Zea land. † The author thanks Ab dul Mohamad, W ang Shuquan and F r´ ed´ eric Mynard for useful comments. 1 Supp ose tha t V n and C n ha v e b een constructed. Consider S = { x ∈ M / ∃ op en U ⊂ M with C n ∪ { x } ⊂ U ≈ R m } . S is op en. S is also closed, for supp ose that z ∈ ¯ S − S . Then we may ch o ose o p en O ⊂ M with O ≈ R m and z ∈ O . Cho ose x ∈ O ∩ S . T hen t here is op en U ⊂ M with C n ∪ { x } ⊂ U ≈ R m . W e may assume that O is small enough that O ∩ C n = ∅ . Using the euclidean space structure o f O w e ma y stretch U within O so as to include z but not uncov er an y of C n . Thu s z ∈ S . As M is connected and S 6 = ∅ w e mus t ha v e S = M . Th us t here is op en V n +1 ⊂ M with S n +1 ∪ C n = ( S n +1 − S n ) ∪ C n ⊂ V n +1 ≈ R m . Because S n +1 ∪ C n is compact we may find in V n +1 a compact subset C n +1 so that ( i) and (ii) hold with n replaced b y n + 1. Let U n = ˚ C n . Then U n is op en, U n ⊂ U n +1 and U n ≈ R m . Thu s by [8], U = ∪ n ≥ 1 U n is a lso op en with U ≈ R m . F urthermore S n ⊂ U n for eac h n so tha t S ⊂ U . There has b een considerable study of metrisable manifolds, esp ecially compact manifolds. In particular it is kno wn that there a re only t w o metrisable manifolds of dimension 1: the circle S 1 , where S n = { ( x 0 , . . . , x n ) ∈ R n +1 / x 2 0 + · · · + x 2 n = 1 } , and the real line R itself. In dimension 2 the compact manifolds ha v e also b een classified, this time in to t w o sequences: the orien table manifo lds, whic h consist of the 2 -sphere S 2 with n handles ( n ∈ ω ) sewn on, and the non-orien table manifolds, whic h consist o f the 2-sphere with n cross-caps ( n ∈ ω − { 0 } ) sewn o n. See [24, Chapter 14 and App endix B], fo r example. Despite considerable progress in the study of compact manifolds in higher dimensions there has b een no classification ev en of compact ma nifolds of dimension 3. Indeed, the 3-dimensional P oincar ´ e conjecture has only no w apparently b een resolv ed after ab out 1 00 y ears. The o riginal conjecture, [51], differs slightly from that p o sed b elo w and was found b y Poincar ´ e to b e false. P oincar ´ e’s counterex ample w as published in [52], where the following ve rsion w as also p osed. The conjecture say s that if a compact manifold M of dimension 3 is suc h that ev ery con tin uous function S 1 → M extends to a contin uous function B 2 → M , where B 2 = { ( x, y ) ∈ R 2 : x 2 + y 2 = 1 } , then M is ho meomorphic t o S 3 . The analogue of this conjecture in dimensions higher than 3 is kno wn to b e true, [21, Coro llary 7.1B] in dimension 4 and [44, Prop o sition B, p109] in dimension 5 and higher. In con trast to the compact situation, where it is kno wn that there ar e only countably man y manifolds [14], in the nonmetrisable case there are 2 ℵ 1 manifolds, ev en of dimension 2 ([47], p 669). Ho w ev er, there a re only tw o nonmetrisable manifolds o f dimension 1, the simpler b eing the open long r ay , L + , [9]. T o ease the des cription of L + w e firstly giv e a w ay of constructing the p ositiv e real n um b ers fr om t he non-negativ e in tegers and copies of the unit interv al. Bet w een an y in teger and it s successor w e insert a cop y of the op en unit inte rv al. More precisely , let R + = ω × [0 , 1 ) − { (0 , 0) } , order R + b y the lexicographic order f rom the na tural o rders on ω and [0 , 1) a nd then top ologise R + b y the order top ology . T o get L + w e do the same t hing but replace ω by ω 1 . More precisely , let L + = ω 1 × [0 , 1 ) − { (0 , 0) } , order L + b y the lexicographic order from the natural orders on ω 1 and [0 , 1) and then top ologise L + b y the order top o logy . The o ther nonmetrisable manifold of dimension 1 is the long line , whic h is obtained by joining together tw o copies of L + at their (0,0) ends in m uc h the same w ay a s one ma y reconstruct the real line R b y joining together t w o copies of (0 , ∞ ) at their 0 ends, thinking of one cop y as giving the p ositiv e reals and the o ther the negative reals. More precisely , let L b e the disjoint union of tw o copies of L + (call them L + and L − resp ectiv ely , with ordering < + and < − resp ectiv ely) as w ell as a single p oin t, which we denote b y 0, o rder L b y x < y when x < + y in L + , when x = 0 and y ∈ L + , when x ∈ L − and y ∈ L + or y = 0 a nd when x, y ∈ L − and y < − x , and top ologise L b y the order top olo gy . 2 The surv ey articles [47] and [49] are go o d sources o f information ab out nonmetrisable man- ifolds. A significan t question in top o logy is that of deciding when a top olog ical space is metrisable, there b eing man y criteria whic h ha v e now b een deve lop ed to answ er the question. P erhaps the most natural is the following: a top ological space is metrisable if and o nly if it is paracompact, Hausdorff and lo cally metrisable, see [58] and [37, Theorem 2.68]. Note t hat manifolds are alw a ys Hausdorff and lo cally metrisable so this criterion giv es a criterion for the metrisabilit y of a manifold, viz that a manifold is metrisable if and only if it is paracompact. Man y metrisation criteria ha v e b een disco v ered for ma nifolds, as seen b y Theorem 2 b elo w, whic h lists criteria whic h require a t least some o f t he extra pro p erties p ossessed b y manifolds. Of course one mus t not b e surprised if conditions whic h in general top ological spaces are considerably w eak er than metrisabilit y a re actually equiv alen t to metrisability in the presence of the extra t op ological conditions whic h a lw ays hold fo r a manifold: suc h a condition is that of b eing nearly meta- Lindel¨ of, 10 in Theorem 2 b elow . Similarly one should no t b e surprised to find conditions whic h are normally strong er than metrisabilit y: suc h a condition is that M may b e em b edded in euclidean space, 35 in Theorem 2. Finally one ma y expect to find conditions whic h in a general to p ological space ha v e no immediate connection with metrisabilit y: such a condition is second countabilit y , 26 in Theorem 2. 2 Definiti o ns. In this section w e list nume rous definitions relev ant to the question o f metrisabilit y .. Definitions : Let X b e a top o logical space and F a family of subsets of X . Then: • X is submetrisable if there is a metric top ology on X whic h is con tained in the giv en top ology; • X is Polish if X is a separable, complete metric space; • X is p ar ac om p act (respectiv ely m etac omp act, p ar aLind e l¨ of and metaLindel¨ of ) if ev ery op en cov er U has a lo cally finite (resp ectiv ely p oint finite, lo cally coun ta ble, and p oin t coun table) op en refinemen t , ie there is another op en co v er V suc h that each mem b er of V is a subset o f some mem b er of U and each p oint of X has a neigh b ourho o d meeting only finitely (r espectiv ely lies in only finitely , has a neighbourho o d meeting only coun tably , and lies in only countably) many mem b ers of V ; • X is finitistic (resp ectiv ely str ongly finitistic ) if ev ery op en cov er of X has an op en refinemen t V and there is an in teger m suc h that eac h p o in t of X lies in (resp ectiv ely has a neighbourho o d whic h meets) at most m mem b ers of V (finitistic spaces hav e also b een called b ounde d ly metac o m p act and strongly finitistic spaces hav e also b een called b ounde d ly p a r ac omp act ); • X is str ongly p ar ac omp act if ev ery op en co v er U has a star-finite op en refinemen t V , ie for an y V ∈ V the set { W ∈ V / V ∩ W 6 = ∅ } is finite. If in additio n, giv en U , there is an in teger m suc h that { W ∈ V / V ∩ W 6 = ∅ } con tains at most m mem b ers then X is star finitistic ; • X is scr e enable (respective ly σ -metac o m p act a nd σ -p ar aLindel¨ of ) if ev ery op en cov er U has an op en refinemen t V which can b e decomp osed a s V = ∪ n ∈ ω V n suc h that eac h V n is disjoin t (resp ectiv ely p o in t finite and lo cally coun table); 3 • X is ( line arly) [ ω 1 -]Lindel¨ of if eve ry op en co v er (whic h is a c hain) [which has cardinalit y ω 1 ] has a coun table sub co v er; • X is (n e arly) [line arly ω 1 -]metaLindel¨ of if eve ry op en co v er U of X [with |U | = ω 1 ] has an op en refinemen t which is p oint-coun table (on a dense subset); • X is almost metaLindel¨ of if for ev ery op en co v er U there is a collection V of op en subsets of X suc h t hat eac h mem b er of V lies in some mem b er of U , that eac h p oin t of X lies in at most countably man y mem b ers o f V , and that X = S { V / V ∈ V } . • X is (str ongly) her e ditarily Lindel¨ of if ev ery subspace (of the coun tably infinite p ow er) of X is Lindel¨ of; • X is k -Lin d el¨ of prov ided ev ery op en k - co v er (ie ev ery compact subset of X lies in some mem b er of the cov er) has a coun t able k -sub co v er; • X is (s tr on gly) her e ditarily sep ar able if ev ery subspace (of the coun tably infinite p o w er) of X is separable; • X is Hur ewicz if for eac h sequence h U n i of op en cov ers o f X there is a sequence hV n i suc h that V n is a finite subset of U n for each n ∈ ω and ∪ n ∈ ω V n co v ers X (not e the alternativ e definition of Hurewicz, [17 ]: X is Hur ewicz if for each sequence hU n i o f op en co vers of X there is a sequence hV n i suc h t hat V n is a finite subset of U n and f or each x ∈ X w e ha v e x ∈ ∪V n for all but finitely man y n ∈ ω . F or a manifold these tw o conditions are equiv alent.); • X is sele ctively sc r e enable , [1], if f or each sequence hU n i o f op en co ve rs of X there is a sequence h V n i suc h that V n is a family of pairwise disjoin t op en sets refining U n for eac h n ∈ ω and ∪ n ∈ ω V n co v ers X ; • X is hemic omp act if there is an increasing sequence h K n i of compact subsets of X suc h that for any compact K ⊂ X there is n suc h that K ⊂ K n ; • X is c osmic if t here is a coun table family C of closed subsets of X suc h that for eac h p oin t x ∈ X and eac h o p en set U containing x there is a set C ∈ C suc h that x ∈ C ⊂ U ; • X is an ℵ 0 -sp ac e ([33, page 4 93]) provided that it has a coun t able k -net w ork, i.e. a coun table collection N suc h that if K ⊂ U with K compact and U op en then K ⊂ N ⊂ U for some N ∈ N ; • X is an ℵ - s p ac e ([33, page 493]) provided that it has a σ -lo cally finite k -net w ork; • X has the Moving Off Pr op erty , [35], provided that ev ery family K of no n-empt y compact subsets of X large enough to con tain for each compact C ⊂ X a disjoin t K ∈ K has an infinite subfamily with a discrete op en expansion; • X is a q -s p ac e if eac h p oin t a dmits a sequence of neigh b ourho o ds Q n suc h that x n ∈ Q n implies that h x n i clusters; • X is F r´ echet or F r´ echet-Urysohn if whenev er x ∈ A there is a sequence h x n i in A that con v erges to x ; • X is a k -sp ac e if A is closed whenev er A ∩ K is closed for ev ery compact subset K ⊂ X ; • X is L a ˇ snev if it is the image of a metrisable space under a closed map; 4 • X is analytic if it is the contin uous image of a P olish space (equiv a len tly of the irratio nal n um b ers); • X is M 1 if it ha s a σ -closure preserving base (ie a base B suc h that there is a decomp osition B = ∪ ∞ n =1 B n where for eac h n a nd eac h F ⊂ B n w e ha v e ∪F = ∪{ ¯ F / F ∈ F } ); • X is str atifiable or M 3 if t here is a function G whic h assigns t o eac h n ∈ ω and closed set A ⊂ X an o p en set G ( n, A ) containing A suc h that A = ∩ n G ( n, A ) and if A ⊂ B then G ( n, A ) ⊂ G ( n, B ); • X is p erfe ctly no rm al if for ev ery pair A, B of disjoin t closed subsets of X there is a con tin uous function f : X → R suc h that f − 1 (0) = A and f − 1 (1) = B ; • X is mon otonic al ly n ormal if for eac h o p en U ⊂ X and each x ∈ U it is possible to c ho ose an op en set µ ( x, U ) suc h that x ∈ µ ( x, U ) ⊂ U and suc h that if µ ( x, U ) ∩ µ ( y , V ) 6 = ∅ then either x ∈ V or y ∈ U ; • X is e xtr emely normal if for each op en U ⊂ X and eac h x ∈ U it is p o ssible to c ho ose an op en set ν ( x, U ) suc h that x ∈ ν ( x, U ) ⊂ U and suc h that if ν ( x, U ) ∩ ν ( y , V ) 6 = ∅ and x 6 = y then either ν ( x, U ) ⊂ V or ν ( y , V ) ⊂ U ; • X is we akly normal if for ev ery pa ir A, B of disjoint closed subsets of X there is a con tin uous function f : X → S , for some separable metric space S , suc h that f ( A ) ∩ f ( B ) = ∅ ; • X is a Mo or e sp ac e if it is regular and ha s a deve lopmen t, ie a seq uence hU n i of op en co v ers suc h that for eac h x ∈ X the collection { st ( x, U n ) : n ∈ ω } f orms a neighbourho o d basis at x ; • X has a r e gular G δ -diagonal if the diagonal ∆ is a r e gular G δ -subset of X 2 , ie there is a sequence h U n i of op en subsets of X 2 suc h that ∆ = ∩ U n = ∩ U n . • X has a quasi-r e gular G δ -diagonal if t here is a sequenc e h U n i of op en subsets of X 2 suc h that for each ( x, y ) ∈ X 2 − ∆ there is n with ( x, x ) ∈ U n but ( x, y ) / ∈ U n . • X has a G ∗ δ -diagonal if there is a sequenc e hG n i of op en co v ers of X suc h that for eac h x, y ∈ X with x 6 = y there is n with st( x, G n ) ⊂ X − { y } . • X has a q uasi - G ∗ δ -diagonal if there is a sequence h G n i of families of op en subsets of X suc h that for eac h x, y ∈ X with x 6 = y there is n with x ∈ st( x, G n ) ⊂ X − { y } . • X is θ -r efinab l e if eve ry o p en cov er can b e refined to an op en θ -co v er, ie a co v er U whic h can b e expressed as ∪ n ∈ ω U n where eac h U n co v ers X and for eac h x ∈ X there is n suc h that or d ( x, U n ) < ω ; • X is subp ar ac omp act if ev ery op en cov er has a σ -discrete closed refinemen t; • X has pr op erty pp , [42], pro vided that eac h o p en co v er U of X has an op en refinemen t V suc h that for each ch oice function f : V → X with f ( V ) ∈ V for each V ∈ V the set f ( V ) is closed and discrete in X ; • X has pr op erty (a) , [42], prov ided that for eac h o p en cov er U of X and eac h dense subset D ⊂ X there is a s ubset C ⊂ D suc h that C is c losed and discrete in X and st ( C , U ) = X ; 5 • X has a b ase of c ountable o r der , B , if whenev er C ⊂ B is a collection suc h that eac h mem b er of C con tains a part icular p oint p ∈ X and for eac h C ∈ C there is D ∈ C with D a prop er subset o f C then C is a lo cal base at p ; • X is pseudo c omplete pro vided that it ha s a sequence hB n i of π - b ases ( B ⊂ 2 X − { ∅ } is a π -b ase if ev ery non- empt y op en subset of X con tains some mem b er of B ) suc h that if B n ∈ B n and B n +1 ⊂ B n for eac h n , then T n ∈ ω B n 6 = ∅ ; • X has the c ountable chain c ondition (abbreviated c c c ) if ev ery pairwise disjoint f amily of op en subsets is coun table; • X is c ountably tight if for eac h A ⊂ X and each x ∈ ¯ A there is a coun table B ⊂ A for whic h x ∈ ¯ B ; • X is c ountably fan tight if whenev er x ∈ ∩ n ∈ ω A n there a re finite sets B n ⊂ A n suc h that x ∈ ∪ n ∈ ω B n ; • X is c ountably str ongly fan tight if whenev er x ∈ ∩ n ∈ ω A n there is a sequence h a n i suc h that a n ∈ A n for eac h n and x ∈ { a n / n ∈ ω } ; • X is se quential if for eac h A ⊂ X , the set A is closed whenev er for eac h sequence of p oin ts of A each limit p oint is also in A ; • X is we akly α -favour able if there is a winning strategy for play er α in the Banac h-Mazur game (defined b elo w); • X is str o n gly α -favour able if there is a stationary winning strategy for pla yer α in the Cho quet g ame (defined b elow); • f or eac h x ∈ X t he star of x in F is st ( x, F ) = ∪{ F ∈ F : x ∈ F } ; • X is Bair e prov ided that the in tersection of an y coun table collection of dense G δ subsets is dense; • X is V olterr a , [29], provide d that the inte rsection of any t w o dense G δ subsets is dense; • X is str ongly B air e pro vided that X is regular a nd there is a dense subset D ⊂ X suc h that β do es not ha v e a winning strat egy in the game G S ( D ) pla y ed on X . • F is p oint-star-op en if for each x ∈ X the set st ( x, F ) is op en. • The Banac h -Mazur gam e has tw o play ers α and β whose play alternates. Pla ye r β b egins b y c ho osing a non-empty op en subset of X . After that the play ers choose successiv e non- empt y o p en subsets of their opp onen t’s previous mov e. Player α wins iff the in tersection of the sets is non- empt y; otherwise player β win s . • The Cho quet game has t w o pla y ers α and β whose play alternates. Play er β b egins b y c ho osing a p oint in an op en subs et of X , sa y x 0 ∈ V 0 ⊂ X . After t hat the pla yers alternate with α c ho osing an op en set U n ⊂ X with x n ∈ U n ⊂ V n then β c ho oses a p oint x n +1 and an op en set V n +1 with x n +1 ∈ V n +1 ⊂ U n . Player α wins iff the intersec tion o f the sets is non-empt y; otherwise player β wins . 6 • G ruenhage’s game G o K,L ( X ), [34], has, a t the n th stage, play er K c ho ose a compactum K n ⊂ X after which play er L c ho oses another compactum L n ⊂ X so that L n ∩ K i = ∅ for eac h i ≤ n . Player K wins if h L n i n ∈ ω has a discrete op en expansion, ie there is a sequence h U n i n ∈ ω of op en sets suc h that L n ⊂ U n and ∀ x ∈ X , ∃ U ⊂ M op en suc h that x ∈ U and U meets a t most one of the sets U n . • F o r a dense subset D ⊂ X the game G S ( D ) has tw o pla y ers α and β whose pla y alternates. Pla y er β b egins by c ho osing a non-empty op en subset V n of X . Aft er that the pla y ers c ho ose success iv e non-empt y op en subsets of their opp onen t’s previous mo v e, β c ho osing sets V n and α c ho osing sets U n . Player α wins iff the in tersection of the sets is non-empt y and each sequence h x n i , for whic h x n ∈ U n ∩ D , clusters in X ; otherwise player β wins . • F o r an ordinal k and families A and B of collections of subsets of a space X let G k c ( A , B ) b e t he game play ed as follow s, [6]: at the l th stage o f the game, l < k , Play er One c ho oses a mem b er A l ∈ A t hen Pla y er Two c ho oses a pairwise disjoin t family T l whic h refines A l . The pla y A 0 , T 0 , . . . , A l , T l , . . . l < k is w on by Pla y er Tw o pro vided that ∪ l 0 let W η,ζ ,r = { ( x, y ) ∈ S / ζ − r < y − η | x | < ζ + r and | x | < r } ∪ [ { 0 } × { η } × ( ζ − r , ζ + r )] . T op ologise M b y declaring U ⊂ M to b e op en if and o nly if U ∩ S is op en in S and fo r eac h (0 , η , ζ ) ∈ U ∩ ( M − S ) there is r > 0 so that W η,ζ ,r ⊂ U . Then M is a separable 2- manifold. There are ev en manifolds whic h ar e b oth norma l and separable but no t metrisable, [56]. W e need now some fa cts from Set Theory . The Contin uum Hyp othesis (CH), dating back to Can tor, states that an y subset of R either has the same cardinality as R or is countable. Martin’s Axiom (MA) can b e express ed in v arious forms, the most top o logical of whic h is the follo wing: in eve ry compact, ccc, Hausdorff space the interse ction of fewe r than 2 ℵ 0 dense op en sets is dense. Recall the Baire Category Theorem whic h states that if X is ˇ Cec h complete (ie X is a G δ -set in β X ; for example ev ery lo cally compact, Ha usdorff space or eve ry complete metric space) and { U n / n ∈ ω } is a collection of op en dense subsets of X then ∩ n ∈ ω U n is dense in X . F rom the Baire Categor y theorem it is immediate that CH ⇒ MA. Bo th CH and MA are indep enden t of the a xioms o f ZFC and otherwise of each other: thus there are mo dels of Set Theory satisfying ZFC in whic h CH (and hence MA) holds, mo dels in which MA holds but CH fails (denoted MA+ ¬ CH), and mo dels in whic h MA (and hence CH) fails. The question whether p erfect normalit y is equiv alen t to metrisability for a manifold is an old one, dating back to [2]. It w a s sho wn in [55] that under MA+ ¬ CH the t w o conditions are equiv alent. O n the other hand in [57] there is constructed an example of a p erfectly normal n on- metrisable manifold under CH. The same situation prev ails when w e consider strong hereditary separabilit y . In [41] it is sho wn that under MA+ ¬ CH ev ery strongly hereditarily separable space is Lindel¨ of . 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