Ernest Michael and theory of continuous selections

ERNEST MICHAEL A ND THEOR Y OF C ONTINUOUS SELECTIONS Du ˇ san Repov ˇ s and P a vel V. Semenov T o fol low the thoughts of a gr e at man i s the most inter esting scienc e. A. S. Pushkin 1. In t ro duction F or a large n um b er of those w orking in top ology , functional analysis, multiv alued analy- sis, appro ximatio n theory , conv ex geometry , mathematical economics, con trol theory , and sev eral other areas, the y ear 1956 has alw ays b een str o ngly connected with t he publication b y Ernest Mic hael of tw o fundamen tal pap ers on contin uous selections which app eared in the Annals of Mathemati cs [4] [ 5]. With sufficien t precision that y ear marked the b eginning of the theory of con tinu ous selections of m ul t iv alued mappings. In the last fift y y ears t he approac h to m ulti v alued mappings and their selections, set forth by Michael [ 4] [5] , has well established itself i n con temp ora ry mathematics. Moreo ver, it has become an indisp ensable t o ol for man y mathematicians w orking in v astly differen t areas. Clearly , the principal reason for this is the naturality of the concept of selecti on. In fact, man y mathematical assertions can b e reduced to using the linguistic rev ersal “ ∀ x ∈ X ∃ y ∈ Y . . . ”. Ho wev er, as so on as we sp eak of the v alidity of assertions of the t yp e ∀ x ∈ X ∃ y ∈ Y P ( x, y ) it is natural to asso ciate to every x a nonempt y set of a ll those y for which P ( x, y ) is true. In this w ay we obtain a multiv alued map which can b e in terpreted a s a mapping, which asso ciates to every initial data x ∈ X of some problem P a nonempt y set of soluti ons of this problem F : x 7→ { y ∈ Y : P ( x, y ) } , F : X → Y . The question of the existence of selections in suc h a setting turns out to b e the q uestion ab out the uniq ue c hoice of the solution o f the problem under given initial conditions. Differen t t yp es of selections are considered in differen t mathematical categori es. 2000 M athematics Subje ct Classific ation . Pri mary: 54C60, 54C65, 41A65. Secondary: 54C55, 54C20. Key wor ds and phr a s es. Multiv alued mapping, upp er semic on tinuous , low e r semicontinuous , conv ex- v alued, con ti n uous selecti on, appro ximation, Vietoris top ology , Banach space, F r´ echet space, hyperspace, Hausdorff distance. T yp eset by A M S -T E X 1 2 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEME NOV One could sa y that the key imp ortance of Mic hael’s theory is not so muc h in pro v iding a comprehensiv e solution o f diverse selection problems in the category of top ologi cal spaces and contin uous maps, but rather the immediate i nclusion of the obtained results in to the general con tex t of dev elo pmen t of top ology . In a remark able num b er of cases, results of Mic hael on solv ability of the selection problems turned out to b e the final answe rs, i.e. they pro v i ded condit i ons whic h turned out to b e necessary and sufficien t. Initially w e were planning to write a surv ey pap er, which would presen t the dev elopmen t of the theory in the last hal f of t he cen tury and its man y applications. H o wev er, already our first attempts at suc h a pro ject sho w ed that the volume of suc h a surv ey w ould inv aria bl y fill an entire b o ok, hence i t would b e i nappropriat e for t his sp ecial i ssue. After some delib erati on w e decided to l imit o urselves to a survey of o nl y the pap ers o f Mic hael on the theory of selections and t heir mutual relat i ons. F or analogous reasons we do not giv e an y precise references to man y developmen ts in the theory of selections – the n umber of pap ers i n this area is by now a ro und one thousand. A considerable n umber of facts on selections and theorems, whic h go b ey o nd the presen t pap er, can b e found in b o oks and surv eys [R1-R15] listed at the end of the pap er. 2. Bibliography P a p ers in scien tific journals usually end wit h the list of r eferences. In our opinion, is it most reasonable to b egin a surv ey dedicated to the w ork of a single p erson, o n one sp ecial topic, spanning o v er 50 years, with a complete li st of his pap ers on the sub ject. List of all p apers by E. M ichael on s elections 1. T op olo gies on sp ac es of subsets , T rans. Amer. Math. So c. 71 (1951), 152– 182. 2. Sele ction the o r ems for c ontinuous functions , Pro c. Int. Congr. M ath. 2 (1954), 241–242. 3. Sele cte d sele ction the or ems , Ame r. Math. M on thly 63 (1956), 233–238. 4. Continuous sele ctions I , Ann. of Math. (2) 63 (1956), 361–382. 5. Continuous sele ctions II , Ann. of Math. (2) 64 (1956), 562–580. 6. Continuous sele ctions III , Ann. of Math. ( 2) 65 (1957), 375–390. 7. A the or em on semi-c ontinuous set value d functions , Duke M ath. J. 26 (1959), 647–652. 8. Dense families of c ontinuous sele ctions , F und. Math. 47 (1959), 174–178. 9. Par ac onvex sets , M ath. Scand. 7 (1959), 372–376. 10. Convex structur es and c ontinuous sele ctions , Canadian J. Math. 11 (1959), 556–575. 11. Continuous sele ctions in Banach sp ac es , Studia Math. Ser. Sp ec . (1963), 75–76. 12. A line ar mapping b etwe en function sp a c es , P roc. Amer. Math. So c. 15 (1964 ), 407–409. 13. Thr e e map ping the or ems , Pro c. Amer. Math. Soc . 15 (1964), 410–415. 14. A short pr o of of the A r ens-Eel ls emb e dding the or em , Proc. Amer. M ath. So c. 14 (1964), 415– 416. 15. A sele ction the or em , Pro c. Amer. Math. So c. 17 (1966), 1404–1406. 16. T op olo gic al wel l-or dering , Inv ent. Math. 6 (1968), 150 –158. (with R. Engelki ng and R. Heath) 17. A unifie d the or em on c ontinuous sele ctions , Pacific J. Math. 87 (1980 ), 187–188. (with C. Pixley) 18. Continuous sele ctions and finite-dimensional sets , Pacific J. Math. 87 (1980), 189–197. 19. Continuous sele ctions and c ountable sets , F und. Math. 1 11 (1981), 1–10. 20. A p ar ametrization the ore m , T op ology Appl. 21 (1985), 87–94. (with G. M¨ agerl and R. D. Mauldin) 21. A note on a sele ction the or em , P roc. Amer. Math. Soc. 99 (1987 ), 575–576. 22. Continuous sele ctions avoiding a set , T opology Appl. 28 (1988), 195–213. 23. A gener alization of a the or em on c ontinuous sele ctions , Pro c. Amer. M ath. So c. 105 (1989), 236–243 . ERNEST MICHAEL AND THEOR Y OF SELECTIONS 3 24. Some pr oblems , Op en Problems in T op ology , J. v an Mill and G. M. Reed, Edi t ors, North–Holland, Amsterdam, 1990, pp. 271–277. 25. Some r efinements of a sele ction the or em with 0 -dimensional domain , F und. M ath. 140 (1992), 279–287 . 26. Sele ction the o r ems with and without dimensional r estriction , Rece n t Developments of General T op ology and its Applications, In ternational Conference in M emory of F elix Hausdorff (1868– 1942), Math. Res. 67, Berli n, 1992. 27. R epr esenting sp ac es a s ima ges of 0 -dimensional sp ac es , T op ology Appl. 49 (199 3), 217–2 20. (with M. M. Choban) 28. A note on glob al and lo c al sele ctions , T op ology Proc . 18 (1993), 189–194. 29. A the ore m of Nep omnyash chii o n c ontinuous subset-sele ctions , T opology Appl. 14 2 (2004), 235– 244. 30. Continuous Sele ctions , E ncyclop edia of General T opology , c–8 (2004), 107– 109. W e ha ve selected the pap ers on selections [4], [5] and [7] to serv e as the basis of the classification of the en t ire list. Here is a reasonably precise diagram of relationship among the pap ers from the list: 4 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEME NOV Here the usual arro w means direct correlation and the dot t ed arro w means an implicit one. P a p ers [2], [11], [2 6 ], [2 8], [30] are not included in this dia g ram, since t hey are either short announcemen ts (or abstracts) on conferences or they are dev oted to p opularizatio n of the sub ject. 3. P ap ers from 1956 A considerable num b er of fundamen ta l mathematical pap ers can b e divided in to t w o t yp es. In suc h pap ers, as a rule, a significan t new theory is constructed or an imp ort an t problem is sol ved. This division is of course, conditi onal – on the one hand, i n constructions of new theori es one often encoun ters difficult problems, on the other hand a solution of a difficult problem often gives rise t o a developmen t o f a significan t new theory . The pap ers [4] and [5 ] are a clear cut example of suc h a divi sion. In [4] an essen tiall y new mathematical theory i s constructed, in the form of a branc hed tree, whic h unifies a l arge n um b er of sufficien tly differen t theorems. T o the contrary , i n [5] the principal result consists of t he proo f of a single hi ghly non trivi al theorem a nd a ll assertions and constructions in this pap er are devoted to the solution of this problem. Another, linguistic difference b et ween [4 ] and [5] is connected with the notion of con- v exit y: the form ulations of pract i cally al l theorems of [4] use the term con v ex , where to the con trary , this w ord is practically absen t from [5]. Finally , in [4] Leb esgue dimension is nev er used, while in [5] , there are dimension restrict i ons o n t he domains of the m ultiv alued maps ev erywhere. One can say , with sufficien t accuracy , that in [5] the finit e-dimensional, purely top o l ogi- cal analog ue of suc h a nontopolo gical notions as con vexit y and lo cal conv exity are presen ted and studied. Without an y doubt, the b est known assertion of [4] is Theorem 3 . 2 ′′ . Theorem 1. The f o l lowing pr op erties of a T 1 -sp ac e X ar e e quivalent: (a) X is p ar ac omp act; a nd (b) If Y i s a Bana c h sp ac e, then every lower s emic ontinuous (LSC) c arr i er ϕ : X → F c ( Y ) admits a singlevalue d c ontinuous sele ction. Here F c ( Y ) denotes the famil y of all nonempt y closed con v ex subsets of Y . Observ e that in [4] Michael origi nally used the term ”carrier” instead o f ” m ulti v alued mapping”. In mathematical practice the implication ( a ) ⇒ ( b ) has the widest application and i s in folklore k no wn as t he ”Con vex-V alued Selection Theorem”. The impli cation ( b ) ⇒ ( a ) give s a selection c haracterizatio n of paracompactness. The unus ual n umeration 3 . 2 ′′ for the theorem has a very simple explanation. In chapter 3 of [4] Mic hael started b y a citati on of Theorem 3 . 1 ( Urysohn, Dugundji, Hanner) and Theorem 3.2 (Dowk er) on the extensions of si nglevalue d mappings and then presen ted the sequences: Theorem 3.1, Theorem 3.1’ , Theorem 3. 1”, Theorem 3 .1”’ Theorem 3.2, Theorem 3.2’ , Theorem 3. 2” of t heir analogs for multivalue d mappings. T o b e mo re clear, let us unify Theorems 3 .1 (a,b,c b elow) and 3 . 1 ′ (a,d,e b elow) as fol lo ws: ERNEST MICHAEL AND THEOR Y OF SELECTIONS 5 Theorem 2. The f o l lowing pr op erties of a T 1 -sp ac e X ar e e quivalent: (a) X is normal; (b) The r e al line R i s an extensor for X ; (c) Ev e ry sep ar able Banach sp ac e is extensor for X ; (d) Eve ry LSC c arrier ϕ : X → C ( R ) admits a singlevalue d c ontinuous sele ction; and (e) If Y is a sep ar able Bana c h s p ac e, then every LSC c arrier ϕ : X → C ( Y ) admits a singlevalue d c onti nuous sele ction. Th us, pla ying with w ords, suc h a series shows that the selection theory in fact, extends the theory of extensors. Here C ( Y ) = { Z ∈ F c ( Y ) : Z i s compact or Z = Y } . As asserted b y Mich ael, Theorem 3 . 2 ′′ w as his very first selectio n theorem, the initial goal of which were generalizations of a theorem due to R. Bartle and L. Gra ves o n sections of linear con tin uous surjections b et w een Banach space s. In parti cular, Prop osition 7.2 of [4] states that such a section can b e c hosen in an ”almo st ” linear fashion (scalar homog enous) and with the p oin twise norm arbitra ri ly clo se to the ”mi nimal” of all p o ssible. Th us the remai ni ng Theorems 3 . 1 ′′ –3 . 2 ′ are selection c haracterizations of ot her prop- erties of t he domain of a conv ex-v alued mapping: normality , col lection wise normalit y , normality and countable paracompactness, and p erfect normality . Man y constructio ns and ideas from [4] l a ter b ecame the basis for subsequen t researc h. F or example, Lemma 5.2 in [4] was t he first result in finding p oi n twise dense families of selections. In comparison wi th [4], the pap er [ 5] original l y dealt only with the unique Theorem 1.2, the so-called ”Finite-dimensional selection theorem”: Theorem 3. L et X b e a p ar ac omp act sp ac e, A ⊂ X a c lose d subse t with dim X ( X \ A ) ≤ n + 1 , Y a c omplete metric s p ac e, F an e qui- LC n family of nonempty clos e d s ubsets of Y and ϕ : X → F an LSC map. Then every singlevalue d c ontinuous sele ction of ϕ | A c an b e extende d to a sing levalue d c ontinuous sele ction of ϕ | U , for some op en subset U ⊃ A . If additional ly every memb er of F is n -c onne cte d (bri e fly, C n ) then one c an take U = X . Without any doubt, this is one of the mo st complicated to p ological theorems, t he six– step pro of in [5] is cl early a mathematical mast erpiece. V arious efforts were made b y sev eral p eople in the la st 5 0 years to simpli fy this pro of (or “improv e” it) , including ourselves . Ho w ever, none of these versions turned out to b e shorter or simpler. In our opinion, none of them reac hed the clarity of exp osit i on in [5] . Tw o years lat er, in 1958, Dy er and H amstr¨ om applied this theorem to get the sufficient conditions for a regular map f to b e a trivial fibration. Suc h a condition turned out t o b e lo cal n -connectedn ess ( LC n ) of t he homeomorphisms group H ( M ) of the fib er M o f f . The problem w hen H ( M ) is LC n w as one of the cen tral in top ology ov er a p erio d of almost 20 years and serv ed as o ne of the k ey sources for the dev elopmen t o f infinite–dimensional top ology , as a separate part of to p ology . F or a first encoun ter wi t h t he theory of selectio ns, the papers [4] and [5] a re t o o difficult and to o v oluminous. On the other hand, the short note [3] q uic k l y tells the reader of the most p o pular metho d of selection theory – the metho d of outside approximation. The note consists of the pro of of the Conv ex–v alued and the the 0–dimensional selection Theorems. The last theorem is a part i cular case o f the Fi ni t e-dimensional theorem for n = 0. Theorem 4. If X is zer o-di mensional ( dim X = 0 ) a nd p ar ac omp act sp ac e, and i f Y is a c omplete metric s p ac e, then every LSC mapping ϕ : X → F ( Y ) admits a singlevalue d 6 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEME NOV c ontinuous sele ction. In spite of i ts relat iv e simplicity and cla ri t y of it s pro o f, the Zero-dimensional selection theorem has ma ny a pplications in selecti o n theory and ot her areas of mathemati cs. The last pap er of the series [4,5,6] dealt mainly with restrictions o n t he displacemen t of a closed subset A in X . F or example, as in the Borsuk pairs, when X = Z × [0; 1] and A = ( Z × 0) ∪ ( B × [0; 1] ) for an appropriate B ⊂ X . Al so the low er semicon tinuit y assumption in [6] was strengthened b y con ti n uity i n the corresp onding Hausdorff metric h ρ in ex p X . Here we repro duce a typical statemen t, Theorem 6. 1: Theorem 5. L et X b e a p ar ac omp act sp ac e with dim X ≤ n + 1 and A ⊂ X a we ak deformation r etr act of X . L et ( Y , ρ ) b e a c omplete metric sp ac e, F a uniformly- LC n family of nonempty close d subsets of ( Y , ρ ) and ϕ : X → F a c ontinuous map with r esp e ct to h ρ . Then every sele ction of ϕ | A c an b e extende d to a singlevalue d c ontinuous sele cti o n of ϕ . Surprisingly deep constructions and results of [6] hav e until no w had no real and clear applications. 4. P ap ers from 1959 W e b egin b y the first pap er of the series [7, 8,9,10]. If one com bines arbitrary paracom- pact domains, as i n the Con vex-v alued selection theorem, and arbit rary complete metri c ranges for closed-v alued mappings, as in the Zero-dimensional selection theorem, then of course, there is no hop e of o btaining a singlevalue d contin uous selection. It turned out that under t hose assumptions a sufficien t ly fine multivalue d selectio ns ex ist. It was rather an unexp ected and ”...curious result ab out semi-contin uous..., [7] ” selections. Belo w, 2 Y denotes t he family o f all nonempty subsets o f a set Y : Theorem 6 ( [7; Theorem 1.1] ). L et X b e a p ar ac omp act sp ac e, Y a metric sp ac e, and ϕ : X → 2 Y an LSC map with e ac h ϕ ( x ) c omplete. Then ther e exist ψ : X → 2 Y and θ : X → 2 Y such that: (a) ψ ( x ) ⊂ θ ( x ) ⊂ ϕ ( x ) for al l x ∈ X ; (b) ψ ( x ) and θ ( x ) ar e c omp act, for al l x ∈ X ; (c) ψ i s LSC; and (d) θ is USC (upp er semic ontinuous). It app ears that this was in principle, the v ery first theorem on multiv alued selecti ons. The pro of of t his Compact-v alued selection theorem is based on t he so called metho d of inner appro ximations. Roughly sp eaking, one can inscrib e in to eac h v alue ϕ ( x ) a tree wit h a coun table set of levels, with finite sets of vertices on each level so that each max i mal linearly ordered sequence of vertices will b e fundamen tal. Th us the sets ψ ( x ) and θ ( x ) are constructed as the sets of li mits of differen t kinds of suc h maximal paths in the tree. Shortly , ψ ( x ) and θ ( x ) are limits of certai n i nv erse (coun table) sp ectra in the complete metric space ϕ ( x ) . Beginning b y [7] multiv alued selections b ecame b y then a fully resp ected part of general selection theory . The comprehensiv e fundamen tal pap er [10] also had a n imp ortant impact on the dev elopmen t of selection theory . In this pap er t he a xiomatic theory of conv exity in metr i c spaces was presen ted. As far as we k no w, t his was al so one of the first pap ers o n axiomat ic con v exities. It serv ed as the starting p oint for ma n y inv estigati o ns in this directi o n. ERNEST MICHAEL AND THEOR Y OF SELECTIONS 7 Also, the metho d of inner approximations from [7] was c hanged and applied in [10] to con vex-v alued maps. Roughly sp eaking, at eac h level of a t ree ab o v e one can consider the barycen ter of all vertices at that level, with resp ect t o a suitable contin uous partition o f the unity of the domai n. In t his w ay it is p ossible to obtain a p oinwise conv ergen t sequence of singlev alued (discontin uous!) selections with degree of discon tin uit y uniformly t ending to zero. Therefore the limit gives the desired con ti n uous singlev alued selection. In our exp eri ence, w e ha ve encoun t ered sev eral times the si t uations when the simpler and more dir ect smo o thing metho d of outside appro ximations did not w ork, whereas the metho d o f inner approximations successfully solve d the problem at hand. Lo oking at the data on submission of the pap ers, o ne may p erhaps infer that [10] w as originally the source for [7 ]. Whereas Lemmas 5.1 and 5 . 2 and Theorem 3 . 1 ′′′ w ere pro ved in [4] for p erfectly normal domains and separable Ba nac h range spaces, a version w as obtained in [8] for metri c domains and any B a nac h ra nge spaces. The pro of w as based on t he replacemen t of the G δ -prop ert y for closed subsets of a p erfectly normal domain b y the A. Stone theorem on the exi st ence of σ -discrete closed basis in an y metric space. Not e also that Theorem 5 . 1 [8] on the one hand, used the ideas from the pro ofs in [6], and on the other hand w as t he basis for the later app earance of suc h noti ons as SEP and SNEP ( sele ction extension and sele ction neighb orho o d extensio n pr op erties ) [ 18]. While [ 10] estimates the relations and links b et w een con vex and metri c structures on the set, t he pap er [9] deals with t he degree of noncon v exit y of a closed subset P of a Banach space, endo wed with standard con v ex and metric structures. Simply put, imagine that we mo ve the endp oints of a segmen t of l ength 2 r o ver a set P . In this sit uat ion it is ve ry natural to lo ok for the distance b etw een the p oints of segmen t and t he set P . So if all suc h distances are less than or equal to α · r for some constan t α ∈ [0; 1), then the set P is p ar ac onvex i n dimension 1. By passing to triangl es, tetrahedra, and n -simplices, one obtains t he notion of a p ar ac onvex set. So, as was pro v ed i n [9], the statemen t of the Conv ex-v alued selection theorem [3, 4] holds whenev er o ne replaces the con vexit y assumption for the v alues ϕ ( x ) by their α -paraconv exity , for some common α ∈ [0; 1), for all x ∈ X . Moreo ver, the pro of lo oks as a double sequen tial “impro v emen t” pro cess of exactness of appro x i mation, o n the account of apply ing the Conv ex-v alued selection theorem. 5. P ap ers from 1964–1979 One of the main purposes of the series [12-15] was to ex a mine t he metrizability assump- tion for the range space i n the Co nv ex-v alued selection theorem. In the pap ers [12,13,14] impro v ements of the Arens-Eells em b edding theorem w ere prov ed and a selection theorem for mappings from metri c domains in to completely metrizable subsets o f lo cally con vex top ologi cal ve ctor (LCTV) spaces w as establ i shed. It w as sho wn in [ 1 5 ] that the statement holds for paracompact domains as well. Observ e that for LCTV spaces c ompletness is a delicate and i n g eneral, ”m ultiv alued” notion. Belo w, a LCTV space i s said t o b e c omplete if t he closed conv ex h ull of any compact subset is also a compact subset. Theorem 7 ( [15; Theorem 1.2] ). L et X b e a p ar ac omp act sp ac e and ( M , ρ ) a metric subset of a c omplete LCTV sp ac e E . L et ϕ : X → 2 M b e an LSC map s uch that every 8 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEME NOV ϕ ( x ) is ρ - c o mplete. Then ther e exis ts a c ontinuous s inglevalue d f : X → E such that f o r every x ∈ X , the value f ( x ) b elongs to the close d c onvex hul l of the set ϕ ( x ) . Note that one of the key ingredien ts of the pro of is the C o m pact-v alued selection the- orem. Next, if ϕ is con v ex-v alued and closed-v alued, then completness of the entire E can b e replaced b y completness of the closed spans of ϕ ( x ) , x ∈ X . Suc h a replacemen t can be also deriv ed from the Z ero-dimensional selectio n theorem and b y the tech nique of p oin t w i se in t egration (see [ R13]). In the joint pap er with Engelking and Heath [16], Mic hael in some sense returned to his first selection publication [1]. Namely , by using embedings into closed t op ologicall y w ell - ordered subspaces of the Bai r e space B ( m ), they pro ved ([16; Coroll ary 2]) that for a ny complete metric, zero-dimensional ( with resp ect to di m or Ind) space ( X , ρ ) there exists a singlev alued contin uous sele ctor f on the famil y F ( X ) of all nonempt y closed subsets of X . Here F ( X ) is endo wed wit h the Hausdorff top o l ogy , sa y τ ρ , and f : F ( X ) → X is a mapping with f ( A ) ∈ A for ev ery A ∈ F ( X ). The zero-dimensionality is the necessary restriction, b ecause there are no selectors for F ( R ) (see [16 ; Prop osi t ion 5 .1]). Note that formall y , a selector is simply a selection of the m ulti v alued ev aluation map- ping, whic h asso ciates to each A ∈ F ( X ) the same A , but as a subset of X . Ho w ever, historically the situation was reverse. In [ 1] Mic hael prop o sed a separation of the problem ab out the ex i stence of a selection g : Y → X for G : Y → 2 X in t o tw o separate problems: first, to c hec k that G is con tinu ous and second, to pro v e t hat there exi sts a selector on 2 X . Hence, the selecti o n problem w as ori ginally reduced to a certain selector problem. 6. P ap ers from 1980–1990 Pic k p oi nts x 1 , x 2 , ..., x n in the domai n X of a multiv alued mapping ϕ and arbi t rary select p oints f ( x i ) ∈ ϕ ( x i ), using the Axiom of c hoice. Th us w e find a partial selection of ϕ ov er the closed subset C = { x 1 , x 2 , ..., x n } ⊂ X . By replacing the v alues ϕ ( x i ) wi th the singletons { f ( x i ) } w e once aga i n obtai n an LSC ma pping, sa y ϕ C . If all assumptions of a selection theorem hold for t he new LSC mapping ϕ C , then suc h a mapping admits a selection, and hence ϕ also admits a selection. This simple observ ation shows that a n y restriction for the v alue of ϕ o ver a finit e subset C ⊂ X , like closedness, connectiv i t y , con v exit y , etc. are inessen tial for the existence of a con ti nuous selection of ϕ . But what can one sa y ab out such an omission for a n infinite C ⊂ X ? Clearly , C should b e a sufficien tl y ”small ”, ”disp ersed”, etc. subset of X . At the international congress o f mathematicians in V a ncouver in 1974 , Mic hael announced results for coun table C . Based on this, the following result was published in 1981 (see [19; Theorem 1.4]): Theorem 8. L et X b e a p ar ac omp act sp ac e, Y a Banach sp ac e, C ⊂ X a c ountable subset and ϕ : X → 2 Y an LSC map with close d and c onv ex values ϕ ( x ) for al l x / ∈ C . Then for every clos e d subset A ⊂ X , e ach sele cti o n of ϕ | A admits an e xtens ion which i s a sele ction of ϕ ( s hortly, ϕ has SEP). Briefly , ov er a countable subset of a domain w e can simply omit an y restriction for the v alues of LSC ϕ : X → 2 Y . A year b efore, in a join t pap er with Pixley [17], Mic hael prov ed that the conv exity assumption can b e omitted ov er any subset Z ⊂ X wi t h dim X Z = 0. ERNEST MICHAEL AND THEOR Y OF SELECTIONS 9 Roughly sp eaking, results of [1 7 ,18,19,23 , 25] are principally rela t ed to sev eral p ossibil i- ties for relax ing con vexit y in selection theorems and in particular, the closedness assump- tions for v alues of mu ltiv alued mappings. F or example, l et us men tion the following t wo results: Theorem 9 ( [19; Theorem 7.1 ] ). L et X b e a p ar ac omp act s p ac e, Y a Ba na c h sp ac e , C ⊂ X a c ountable subset, Z ⊂ X a subset with di m X Z ≤ 0 and ϕ : X → 2 Y an LSC map such that ϕ ( x ) i s close d f o r al l x / ∈ C and C l os ( ϕ ( x )) is c onvex, for al l x / ∈ Z . Then ϕ has SEP. Theorem 10 ( [18; Theorem 1.2] ). L et X b e a p ar ac omp act s p ac e, Y a Banach sp ac e, Z ⊂ X a subse t with dim X Z ≤ n + 1 and ϕ : X → F ( Y ) an LSC map s uch that and ϕ ( x ) is c onvex, for al l x / ∈ Z and the family { ϕ ( x ) : x ∈ Z } is uniformly e qui- LC n . Then ϕ has SNEP. If mor e over, ϕ ( x ) is n -c onne cte d for every x ∈ Z , then ϕ has SEP . Note that in [18] t he tech nique of the pro of i n [5] was rearranged in a more structured form, wit h exa ct extracting of the useful prop erties li ke SEP , SNEP and SAP ( s e le ction appr oximation pr op erty ). The join t pap er with Magerl and Mauldin [20 ] formally contains no ”selections” in the title or in t he statements of the main theorems (1.1 and 1. 2 ). Nevertheless, the essence o f these theorems is con tai ned i n the selecti o n result. It is a classical fact t hat eac h metric compact X can b e represen ted as the ima ge of the Can tor set K under some con tinuous surjection h : K → X . Th eorem 5.1 of [ 2 0] states that if { X α } is a famil y of sub compacta of a metric space X whic h is con ti n uously parameterized b y α ∈ A with dim A = 0 then one can c ho ose a family of surjections h α : K → X α con ti nuously dep ending on the same parameter α ∈ A . Suc h parameterized v ersion of t he Alexandrov theorem i s in fact, deriv ed from the Zero-dimensional selection theorem. In general, the decade 1980–1990 w as marked b y Michael’s very di verse set of pap ers on selections, practically every one of whic h contained new ideas of high quality . F or one more ex a m pl e, t he Finit e-dimensional selection theorem from [5] was strengthened in [23] sim ult aneously in tw o directions. First, t he a ssumption that { ϕ ( x ) } x ∈ X is an equi- LC n family in Y w as replaced by the prop erty that fib ers {{ x } × { ϕ ( x ) } x ∈ X } constitutes an equi- LC n family i n X × Y . This answ ered the problem of Ei len b erg stat ed in 1956 (see the commen ts in [5]). Next, the closednes s assumption for ϕ ( x ) ⊂ Y can b e weak ened to the closedness of graph-fib ers { x } × ϕ ( x ) in some G δ -subset of X × Y . The k ey ingredien t of t he pro of w as a ”factori zation” construction. Briefly , it o ccurred that the LSC ma pping ϕ : X → Y with w eakened assumptions can b e represen ted as a comp osition ϕ = h ◦ ψ with singlev alued h : Z → Y a nd w i th ψ : X → Z sati sfying t he clas- sical assumptions of the Finite-dimensional selection theorem [ 5]. Hence the comp osition of a selection of ψ wi t h h gi v es the desired selection of ϕ . W e guess that t he idea of the app earance of the G δ -conditions w as a corollary of con- structions of selections, a voiding a countable set of obstructions, from the pap er [22] which app eared one ye ar earlier: Theorem 11 ( [22; Theorem 3.3] ). L et X b e a p ar ac omp act sp ac e, Y a Banach s p ac e and ϕ : X → F ( Y ) a LSC map with c onvex va lues. L et ψ i : X → F ( Y ) , i ∈ N b e c ontinuous, 10 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEME NOV Z i = { x ∈ X : ϕ ( x ) ∩ ψ i ( x ) 6 = ∅} and supp ose that dim X < dim ϕ ( x ) − dim ( conv ( ϕ ( x ) ∩ ψ i ( x ))) , for al l x ∈ Z i and i ∈ N . Then ϕ admits a sele ction f which a v o i ds every ψ i : f ( x ) / ∈ ψ i ( x ) . Briefly , in the v alues ϕ ( x ) t here i s sufficien t ”ro o m” t o av oid all sets ψ i ( x ). Based on [22, 23], Michael stated in ”Op en problems in t op ology , I” t he ” G δ ”-problem [24; Problem 396] : Do es the Con vex-v alued selection theorem remain true if ϕ maps X in t o some G δ -subset Y of a B anac h space B with con vex v alues whi ch are closed in Y ? In spite of n umerous cases wi th a ffirmative answ er this problem has in general a negative (as it w as conjected in [24] ) solution, for details see the pap er of Namiok a and Mic hael in this issue. 7. P ap ers from 1992 In general, al l pap ers [21,25, 27] are related to ”disp ersed”, ma inly to zero-dimensional, (in dim-sense) domains of m ultiv alued ma ppings. Briefly , in [25] results of [17; Theorem 1.1] and [19; Theorem 1.3] are unified and gen- eralized in the spiri t of [23] t o subsets C ⊂ X , which are unions of countable family of G δ -subsets C n of X and to a mappings ϕ , ha v ing SNEP at each C n . In the pap er written with Choban [ 27], the Compact-v alued selection Theorem 6 w a s derived from the Zero- dimensional one (Theorem 4). In fact, a para compact domain X was presen ted as the image h ( Z ) of some zero-dimensional paracompact space Z with respect to some appro- priate con tin uous (p erfect or inductiv ely op en) ma pping h : Z → X . Theorem 4 applied to the comp ositi on ϕ ◦ h gives a selection, say s : Z → Y . So, the comp osit ion s ◦ h − 1 will b e a desired m ultiv alued selection of ϕ : X → F ( Y ). The pair of pap ers [26,28] is related to ”the differences b et w een selection theorems whic h assume that the domain is finite-dimensional and those whic h do not”. More g enerall y , based on the Pixley coun terexample in [26] it was sho wn that a gen uine dimension-free analogue of the Finite dimensional selection theorem do es not exist or briefly , that there are no purely top ologi cal a nal ogs of conv exity . In comparison, in [ 28] a con v exity , or connectivit y , t yp e restrictions in the spirit of [10] for a mapping are presen ted and under suc h restrictions t he equiv alence is prov ed b et wee n the existence of global selections and the existence of selections lo cal ly . The pap er [2 9] o n con t in uum-v alued selections i s an elegan t simultaneous appli cation of the ”universalit y” idea from [27] and the one-dimensional selection theorem (sp ecial case n = 0 of Theorem 5). The k ey step can b e describ ed as follo ws. Due to a recent theorem of Pasynk o v, each paracompact domain X can b e represen ted in the form h ( Z ), for some p erfect, op en surjection h : Z → X with pathwise connected fib ers and for some paracompact space Z wit h dim Z ≤ 1. So, if the comp o sition ϕ ◦ h admits a selection, sa y s : Z → Y then t he comp osition s ◦ h − 1 will b e a con tinuu m-v alued selection of ϕ : X → F ( Y ). W e should men t ion the though tfulness, exactness, p erfectness of al l Mi chael’s pap ers. His laconic st yle of exp o si tion is p erfectly match ed with t he deepness of his results. In our opinion, A man o f few words but with great ideas could well serv e as a go o d description of his ch aracter. As a rule, all his pap ers are equipped with a considerable n um b er of ERNEST MICHAEL AND THEOR Y OF SELECTIONS 11 additional references, which were added at pro ofs, and whic h v ery precisely gi v e correct accen ts to the pap er needed for prop er understanding. In conclusion of this surv ey of Mic hael’ s results on selections we wish our jubilan t success ful realization of man y more pro jects. Ac kno wle dgements W e thank Jan v an Mill for commen ts and suggestions. The first a uthor w as supp ort ed b y the Slo v enian Researc h Agency gran t s No. P1-0292-01 0 1-04 and Bl -R U/05-07 -0 4 . The second author w as supp orted b y the RFBR grant No. 05-01-00993. Reference s [R1] J.-P . Aubin and A. Celli na, Differ ential Inclusions, Set-V alue d Maps A nd Viability The o ry, Grundl. der Math. Wi ss., vol. 26 4 , Springer - V erlag, Berl in, 1984. [R2] J.-P . Aubin and H. F rank owsk a, Set-V alue d A nalysis , Birkh¨ auser, Basel, 1990. [R3] G. Beer, T op olo gies on Close d and Convex Sets , Kluw er, D ordrech t, 1993. [R4] C. Bessaga and A. Pelczynski, Sele cte d T opics in Infinite-dimensional T op olo gy , Monogr. Math., vol. 58 , PWN, W arsza w, 1975 . [R5] Y u. G. Borisovic h, B. D. Gel man, A. D. Myshkis and V. V. Obuhovskij, Set-V alue d maps , Itogi Nauki T ehn. Mat. Anal. 19 (1993), 127– 230 (in Russian). [R6] F. Deutsch, A survey of metric sele ctions , Con temp. Math. 18 (1983), 49–7 1. [R7] J. Dugundji and A. Granas, Fixe d Point The ory, Monogr. Math., vol. 61 , P W N, W arsa w, 1982. [R8] L. Gorniewicz, T op olo gic al Fixe d Point The ory of Mu l tivalue d M appings, Mathematics and Its Applications, vol. 495 , Kl u wer, D ordre cht, 1999. [R9] J. v an Mi ll, Infinite-Dimensional T op olo gy: Pr er e quisites a nd Intr o du ction , North-Holland, Am- sterdam, 1989. [R10] J. v an Mi ll, The Infinite-Dimensional T op olo gy of F unction Sp ac es , North-Holland, Amsterdam, 2001. [R11] J.-I. Nagata, Mo d ern Gener al T op olo gy, 2 nd E d . , vol. 3 3 , North-Holland Math. Libr., Am ster- dam, 1985. [R12] T. P . Partha sarat y , Sele ction The or ems And Their App l ic ations. Lec t ure Notes Math., vol. 263 ,, Springer - V erlag, Be rlin, 1972. [R13] D. Rep o v ˇ s and P . V. Semenov, Continuous Sele ctions of Multivalue d Mappings , M athematics and Its Applicati ons 455 , Kl u wer, Dordrech t, 1998. [R14] D. Rep o v ˇ s and P . V. Semenov, Continuous Sele ctions of Multivalue d Mappings , Recent Progress in General T opology I I (M. Hu ˇ sek and J. v an Mill , Editors), Elsevie r, Amsterdam, 2002, pp. 423– 461. [R15] M. L. J. v an de V el, The ory of Convex Structur es , North-Holland, Amsterdam, 1993. In st i tu t e of M a t he matic s, P h ys ic s a nd M ec ha ni c s , a nd F acu l t y o f E d u c a t io n , U n ive r s it y o f L j ub l ja na , J ad r a ns ka 19 , P . O . B ox 2 9 6 4, Lj ub l ja na , S l ov en ia 1 0 0 1 E-mail addr ess : dusa n.repov s@guest. arnes.si D e p a r tm e nt o f M a th e ma ti c s , M os c ow C i t y P e dag o gi c al U niv er s it y, 2- nd S e l sko k ho z y a st - ve nn yi pr . 4, M o sc ow , Ru ss ia 12 9 2 26 E-mail addr ess : pave ls@orc. ru