REMOTE STORAGE ESSENTéAL FSXEL‘J POSNTS AND ALMGST CONTINUOUS FUNCWONS Thesis hat The Degree :3? P31. D. MICHIGAN: STATE UNWERSITY‘ Somashekhas‘ Amrifh Naimpally 1964 THESIS This is to certify that the thesis entitled 555977f'ict/ FiXecf PCM'LS /~7>z(/ ‘7 . I - ‘fl- ‘ ‘ f I714 057‘ (cnv‘niucug I'uncflcws presented by 307170 512924-1ka Hwy-"1-774 IVCU‘N/l/DCKH/ has been accepted towards fulfillment of the requirements for FDI’l’ Dc degree in [3761,14 9" n'ia‘IlKCS . I ajor profess 0-169 LIBRARY Michigan State University REMOTE STORAGE RS F PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE 20:: Blue 10/13 p/CIRC/DateDueForms_2013eindd - 09.5 ABSTRACT ESSENTIAL FIXED POINTS AND ALMOST CONTINUOUS FUNCTIONS by Somashekhar Amrith Naimpally The concept of an essential fixed point was first introduced by M. K. Fort, Jr. for single-valued continuous self-mappings on a compact metric space with the fixed point property. Schmidt extended the theory to a compact Hausdorff space with the fixed point property. Jiang Jia- he has extended the theory to upper semi-continuous multi- valued self-mappings on a compact metric space. In the first chapter of this thesis the theory of essential fixed points is further extended to upper semi-continuous multi- valued self-mappings on a compact Hausdorff space. Almost continuous functions were introduced by Stallings. In Chapter II two new tOpologies on the func- tion spaces of almost continuous functions are introduced. Then the theory of essential fixed points is extended to single-valued almost continuous self-mappings, first on a compact metric space and then on a compact Hausdorff space both having the fixed point property. In Chapter III consideration is restricted to real- valued almost continuous functions defined on a closed Somashekhar Amrith Naimpally interval. The chief results are: (1) almost continuity is equivalent to the connected graph property; (2) an almost continuous function is continuous if and only if the inverse image of each point is closed. Finally the relation- ship between almost continuous functions and other non- continuous functions such as neighborly functions and locally recurrent functions is investigated. In the last chapter Borsuk's concept of a retract is extended to that of an almost retract. It is shown that whereas almost retracts inherit the fixed point property from the original space, they do not always inherit such properties as local connectedness. ESSENTIAL FIXED POINTS AND ALMOST CONTINUOUS FUNCTIONS By Somashekhar Amrith Naimpally A THESIS Submitted to Michigan State University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1964 ACKNOWLEDGMENTS I am indebted to Professor J. G. Hocking for his helpful guidance during the preparation of this thesis. I wish to express my gratitude for his kind considera- tion and encouragement throughout my stay at Micnigan State University. I also wish to express my gratitude to Professor D. E. Sanderson for his helpful guidance during Summer 1963. Thanks are also due to Professors M. Edelstein and L. M. Kelly for-valuable suggestions and comments.- This thesis was written under partial support from NSF Contract GP-3l. I thank my wife Sudha for typing the preliminary draft of the thesis. ii DEDICATION To my mother and my wife for their patience and encouragement. iii CONTENTS Chapter Page I. ESSENTIAL FIXED POINTS OF MULTIVALUED FUNCTIONS ON A UNIFORM SPACE . . . . . 1 II. ESSENTIAL FIXED POINTS OF ALMOST CONTINUOUS FUNCTIONS. . . . . . . 7 III. PROPERTIES OF ALMOST CONTINUOUS FUNCTIONS. 19 IV. ALMOST RETRACTS . . . . . . . . . 31 BIBLIOGRAPHY. . . . . . . . . . . . . . 37 iv CHAPTER I ESSENTIAL FIXED POINTS OF MULTIVALUED MAPPINGS ON A UNIFORM SPACE P Essential fixed points were first introduced by M. K. Fort, Jr. in [A] for single valued continuous self- mappings of a compact metric Space with the fixed point property. The central result in Fort's paper was that continuous self-mappings on such a Space can be approxi- mated arbitrarily closely by those which have all fixed points essential. Schmidt [13] extended the theory of essential fixed points to continuous self-mappings on a compact Hausdorff space with the fixed point property. Jiang Jia-he [7] considered upper semi-continuous multi- valued self-mappings on a compact metric space which need not have the fixed point property. In this chapter we extend the theory to the case of essential fixed points of upper semi-continuous multivalued mappings on a compact Hausdorff space. We use the techniques of Schmidt and Jiang Jia-he. We use the standard terminology of Hocking and Young [6] and Kelley [9]. Let (X,17) be a compact Hausdorff space with topology 1:. (X need not have the fixed point property.) Let B(X) be the family of all open symmetric neighborhoods of the diagonal in the product space X x X. Then B(X) is a base for the uniformity‘LL of (X,’U). Let C(X) be the space of all nonempty compact subsets of X. For each U in Mlet NW)=¥LW)uMmQXCM) KCUMW,KW:WM}. iM(U)} , with U in U: is a base forpthe uniformity {P of C(X). C(X) is a compact Hausdorff space with unifor- mity GD. Let f : Y --* C(X) be a mapping of Y into C(X), (f is a multi-valued mapping on Y to X). Let'VT‘be a uniformity for Y. Definition 1.1 f is upper semi-continuous at y in Y if and only if for each U in B(X), there is a W in'VW'such that for all (y.y') in W. ny”) c: UIf(y)] - Definition 1.2 f is lgwer semi-continuous at y in Y if and only if for each U in B(X) there is a w in'wv'euch that for all (y,y') in w, f(y) CL u[f(y')] . f is continuous if and only if it is upper semi-continuous and lower semi-continu- ous. Let D be a directed set and n be in D. It is well known that f is upper semi-continuous at y in Y if and only if yn ——->y, xn ____5x, n in D, xn in f(yn), implies x is in f(y). Let S(Y,C(X)) be the space of all upper semi- continuous mappings on Y to C(X). For each P in GP let 3 W(P) = &(f.a) ‘ f.s in S(Y.C(X)). (f(y), s(y)) in P for all y in Y.§. {w(1>) E , with P in (I) is a base for the uniformity JV of S(Y, c(x)). Theorem 1.1 S(Y,C(X)) is complete. 23223, This is proved by showing that S(Y,C(X)) is closed in the Space of all functions on Y to C(X). Let f be a limit point of S(Y,C(X)). For each W [M(U)] there is a g in S(Y,C(X)) such that (f(y),g(y)) belongs to M(V) for all y in Y where we may assume that VoVOVCZ U. This implies that S(Y)CV [f(y)] and f(y)CV [S(yH- Now there exists a W in'Mfsuch that for all (y,y') in W, s(y')<:: VIs(yI]. Now f(y”) CZ ‘Jhs(y')] CZ VoV [S(Y)] C VoVoV [f(y)] CZ U [f(y)]. This shows that f is an element of S(Y,C(X)). Definition 1.3 , Let f be in S(X,C(X)) and x be in X. Then x is called a fixedpoint of f if and only if x is in f(x). let {S CZ S(X,C(X)) be the subspace of all f in S(X,C(X)) which have at least one fixed point. Theorem 1.2 ‘S is complete. Proof. Let {fr}; be a Cauchy net in S . Since S(X,C(X)) is complete, there is an f in S(X,C(X)) such that Li fn ___> f. Since each fr1 is in S, there exist xn in X such that each xn belongs to fn (xn). Since X is a compact space there exists a convergent subnet of 1’an and we can then consider the corresponding subnet of' ifplg. So there is no loss of generality in assuming that the net ixn E itself is convergent. So let xn _; x . Since f1.1 __.> f there exists a subnet {xnmg of S’xn} such that for any U in QA', we can choose'xnm in f (xnm) such that (Ehm, xnm) belongs to U. Then 3% ——7 x and since f is upper semi- continuous at x it follows that x belongs to f(x). This shows that f is an element of S, i.e. IS is complete. Q.E.D. Definition 1.4 For f in‘S let Fifi = the set of all fixed points Of f. Then F is a function on‘S to C(X). Theorem 1.; F is in s('§,c(x)). £1922. Clearly for r in “s’, F(F) is in c(x). Let fn ——)f, x1.1 ___9 x where xn is in F(fn). Choose Ynm in f(xnm) such that (inm, xnm) is in U for given U in QL>, Ynm _; x. Since f is upper semi-continuous at x, x is in F(f). Q.E.D. Definition 1.5 x in F(f) is essential if and only if for each U in B(X), there is an N in\)/ such that whenever (f,g) is in N, x is in U [F(g)]. Definition 1.6 f in'S is an essential fixed point map if and only if all fixed points of f are essential. 5 Theorem 1.4 f in'S is an essential fixed point map if and only if F : S __..>C(X) is continuous at f. Prggf. Let f in'S be an essential fixed point map. For U in B(X) let v be in B(X) such that v o vc: U. Since F(f) is compact there is a finite subset Z;xl , x2 , . . .xn} of F(f) such that F(f)(: LJQZl V [x1]. Since Xi in F(f) is essential for each i=1 , 2 , ... n , there is an N in .N such that for (f,g) in N, g inAS’, x1 is in V [F(g)], i = 1 , 2 , . . . n. Therefore, F(f) C U121 V[xi] CUigl V o V [F(g)] C U121 U [F(g)] which shows that F is lower semi- continuous at f. But F is upper semi-continuous by theorem 1.3. It follows that F is continuous at f. Next let F be continuous at f. Let U be in B(X) and x be in F(f). Since F is lower semi-continuous, there is an N in VN” such that for g in‘S, (f,g) in N implies 5110 c: v [F(g)], i.e. x is in v [F(g)], which means that x is essential for f. This shows that f is an essential fixed point map. Q.E.D. Theorem 1.5 If f in‘S has a single fixed point then this fixed point is essential. £3293. Let i p} = F(f). For any U in B(X) there is an N in \A/‘such that whenever g is in‘S , (f,g) is in N, F(g) c: v [F(f)] = v [ p ] which means that p is in v [F(g)]. Therefore, p is essential. Q.E.D. 6 Theorem 1.6 Let f be in S and U, V be in B(X) such that V 0 VC. U. Then there is an N in Wsuch that if g is in "s’ and (r , g) is in N then v [F(g)] C U [F(f)]. 2322:. Since F is upper semi-continuous at f, there is an N in VN'such that if g is in‘S, (f,g) is in N then F(s) C V [F(f)]- If p is in V [F(g)], then p is in V [q] for some q in F(g) and q is in V [r] for some r in F(f). Therefore, p is in v o v [r]C U [r], that is v [F(g)]C U[F(f)]. Q.E.D. CHAPTER II ESSENTIAL FIXED POINTS OF ALMOST CONTINUOUS FUNCTIONS In this chapter we extend the theory of essential fixed points to almost continuous self—mappings on com- pact metric and compact Hausdorff spaces respectively. Stallings first introduced almost continuous functions in [14] and proved that an almost continuous self—mapping of a Hausdorff space with the fixed point property has a fixed point. In order to consider the theory of essential fixed points for almost continuous functions, we introduce two new topologies on function spaces of these functions. We shall show that these topologies agree with the usual topologies for function spaces of continuous functions. We then prove that the theorems of Fort [4] and Schmidt [13] hold when the functions under consideration are almost continuous. Let X and Y be tOpOlogical spaces. Let f:X ___) Y be a function. Let the graph of f be denoted by F(f) == £(x,f(x))| xinXE C XXY. Let X x Y be assigned the usual product topology. The following definition is due to Stallings [14]. Definition 2.1 f :IX.___; Y is almost continuous if and only if for each open set N in x x Y containing rI(r), there is a continuous function g : XZ____5pY'such that [7(a) CZ No Definition 2.2 X has the fixed point prOperty if and only if every continuous function f on X to X has a fixed point, i.e. there is a p in X such that f(p) = p. The following proposition due to Stallings [14] shows the existence of fixed points for almost continuous functions. Proposition 2.1 Let X be a Hausdorff space with the fixed point property. Then every almost continuous func- tion f on X to X has a fixed point. We not investigate the "essential" character of the fixed points of almost continuous functions. The main result is that if X has the fixed point property and f is an almost continuous function on X to X then f can be ap- proximated, in a certain sense, by an essential fixed point map. We first study metric spaces (up to Theorem 2.4) and then uniform spaces. SO let X and Y be compact metric spaces with metrics d , d' respectively and let X x Y be assigned the product metric D((x1 . Y1): (X2 . y2)) = d(xl . x2) + d'(y1 . y2)- Let H be the Hausdorff metric (see [9] page 131) on the hyperspace of all non-empty closed subsets of X X Y. 9 We introduce a metric Q in YX. This was used by Kuratowski [10] in discussing continuous functions defined on X as well as on subsets of X. Definition 2.3 For f and g in YX we define Q1351 = H (TITO. W >. Clearly (YX, Q') is a pseudometric space. We make it into a metric space by agreeing that f §_g if and only if rrrf7 = ‘Tfi‘tg7 for f, g in Yx and then passing to the quotient space with respect to this relation. Let S(X,Y) be the set of all continuous functions on X to Y and for f, g in S(X,Y) define Q1(f,g) = Sispéx d'(f(p),g(p)) where d‘ is the metric for Y. It is easily [seen that Q1 is a metric on S(X,Y). The next theorem relates the metrics g and Q1 . Theorem 2.1 The metrics Q and Q1 are equivalent for s(x,y). £3993. Let 1,70 be arbitrary and let U(f, a) = ‘Lg |f,g in s(x,y) such that Q(f,g) < a; , U1(f, e. ) ={g ‘f,g in S(X,Y) such that Qt(f,g) 4 L; . It is easily seen that U1 (f, i ) (:2 U(f, z ). We now show that there is a 5 >0 such that U(f, 5 ) C. Ul(f, i ). Since f is continuous on a compact set X, f is uniformly con- tinuous. Therefore, there exists a 8>o which we can, with- out any loss of generality, assume to be < t/3, such that for all x, y in X d(x,y) < 8 implies d'(f(x),f(y))< i . 10 Now let g be in U(f, 5 ) and let p be an arbitrary point in X. Then there is a q in X such that d(p,g) + d'(g(p), f(q))<8- Now d'(f(p). 8(a)) 5 Man) + d'(S(P):f(Q)) + d(p.q) + d'(f(p):f(q)) ’4 2 5 + ‘/3<‘-- This means g is in Ul(f, 2 ). Therefore, Q and Q1 are equivalent for S(X,Y). Q.E.D. Let (04,?) be the space of all almost continuous functions on X to X where X is a compact metric space with the fixed point property. Let C(X2) denote the space of all non-empty closed subsets of X2== X x X with the Haus- dorff metric H. Let A = i(p,p) \ for p in ch x2 be the diagonal in X2. We modify the definition of essential fixed points to suit their study in relation to almost continuous functions. Definition 2.4 For f in ()4 we define ngz =1 r|[f5(\Z§. Clearly F is a function on we to C(Xe). Definition 2.5, p in F(f) is an essential point of f in J4 if and only if corresponding to each open set V in X2 containing p, there exists an open set W in I14 contain- ing f such that for all g in w , F(g) n V .é ¢' Theorem 2.2 F :fl ___) C(X2) is upper semi-continuous. Proof. Let g>o be arbitrary. Let 8 be the Hausdorff distance between [X and FIZf) - U(F(f),£. ) if "p" (ff 5; U(F(f), a ) and let 5 = 1 otherwise. If g is 11 in a? such that guys) <5 then F(g) C U(F(f), a ). Therefore, F is upper semi-continuous. Q.E.D. Theorem 2.3 If F(f) is a single point p then it is an essential point for f. Proof. By Theorem 2.2 for e>O there exists a 870 such that for all g in A such that Q(f,g)<5, F(g) c: U(p, a ). That is F(s) nU(p, a) :4 4). Therefore, p is essential for f. Q.E.D. Definition 2.6 f in 64 is an essential fixed point map if and only if all points of F(f) are essential for f. Theorem 2.4 F is lower semi-continuous at f in 64 if and only if f is an essential fixed point map. lggggf. Let F be lower semi-continuous at f in ufl . Then for all £7 0, there is a 57 0 such that for all g in A. Q(r.s)<8 implies an: U(F(s). a )- Let p be in F(f). Then p is in U(F(g), E ), implies F(g) (I) U(p, g ) 1‘ ¢ . Therefore, p is essential for f. Thus f is an essential fixed point map. Next let f be an essential fixed point map. Then every point in F(f) is essential for f. Let £7 0. For each p in F(f), there is a 8p > 0 such that for all g in ‘54 , Q(f,g) < 5p implies F(S) n U(p, t/2) a4 4D . Since F(f) is compact a finite number of U(p, 8/2) cover F(f). Let 5 be the minimum of all the corresponding 12 8p's- Now if masks then each p in F(f) is in U(F(g), g ). Therefore, F(f) c U(F(s). e. ). This implies F is lower semi-continuous at f. Q.E.D. Theorem 2:5 For f in é4 and 23> O, there is an essential fixed point map g in a" such that “(g) c U( Tfl—L a I. “£3292. Since f is in 54 , there is a continuous function h on X to X such that fl(h) C U( TIT—f7, i ). Since ['1 (h) is compact, there is a 5'7 0 such that U( ['1 (h), 5 ) C U(T‘TT—I, é. ). By Fort's theorem [4], there is a continuous function g on X to X, which is an essential fixed point map and Q1(g,h) <8 . Then clearly F(s) c: U( F(h). 5 I: U( DTT. 2 )- Q.E.D. It should be noted that we have actually proved that g can be chosen to be continuous. Next we consider the essential points of almost continuous functions on X to X where X is a compact Haus- dorff space with the fixed point property. Lat 54 be the space of all almost continuous functions on X to X and let S be the subspace of J4 containing only continuous functions. We introduce a topology '1: for 5Q by defining the Kuratowski closure operator for a subset B of .99 (see [9], p. 43). 13 Definition 2.1 f is in B' (the set of limit points of B) if and only if for each open set Vc: X2 containing [1(f), there is a 804 f) in B such that F(S)C V. Let BC = B LJ B'. We shall now prove that the Operator '0' satisfies the conditions (a) to (d) of the Kauratowski closure operator ([9], page 43). in is in in If (a) (b) (0) (d) in B , 1 Clearly (to = ch . For each B c: 54 , obviously B :2 BC. We want to show that BCC = Bc for each BC354. Clearly B0 C BCC. Suppose f is in B00 but not in BC. Then for each open set V containing r1(f) there is a g in BC different from f such that [1(g) C:’ V, If g is in B‘ then clearly there is an h in B different from f such that [7(h) C: V. Therefore, f is in B'. If g is in B then f is in B'. In both cases f is in Be, a contradiction, therefore, BCC - BC. Let B1 and 32 be subsets cd'd4. we wish to show 0 c that Bl (J 32 a (El L.) B2)c. Let f be in BE LJ BC. Then either f is in B: or f is 2 BS. Suppose f is in BE. Then f is in E1 or Bi. If f then f is in Bl\;j B2. If f is in B' then f is 1 (Bl LJ B2)c. Therefore, Bi L) B; c: (Bl L) B2)c. Next let f be (131 L) B2)°. Then f is in B1 L) 32 or in (131 L) B2)'. f is in 31)») B2, then f is in El or B2, i.e. f is in 14 B: or BS, i.e. f is in BE L) 3;. If f is in (El LJ B2)' then clearly f is Bi or Bé. Therefore, '0' is the Kuratowski closure operator. It is easyto see that the tOpology 13 of J4 induced by 'c' is the same as the one generated by the basis con- sisting of sets of the form flV = {f l f in 64 such that U(f) C V E for V an open set in X2. Theorem 2.6 If X is a compact metric space then (fl ,6 ) is equivalent to (04 , T ). 2329:. Let f be an element of [99 and let V be an Open subset of X2 containing 'TTCF). Since _TTTTT' is compact there is a positive 5 such that U[ WT), 5 ]C. V. Now if g belongs to U(f,5 ) then Q( WT), W) x i , V = i(x,y) I y < xi each contain points of F(f) and together contain the whole of r7(f). This shows that F (f) is not connected and so by theorem 3.2 f is not almost continuous. The function f constructed above is unbounded, in fact 'P (f) is dense in [a,b] x R1. The question naturally arises as to whether there is a counter example with a bounded Darboux continuous function. Such a function can be easily constructed by modifying an example given by 24 Lebesque [11]. Let every x be written as a non-terminating decimal I . a1 a2 . . . an. . . . If the decimal . a1 a3 . a2n-l . . . is not periodic, set g(x) = 0; if it is periodic and the first period commences with a2n-l’ set g(x) = .a2n a2n + 2 a 2n + 4. . . . The function g takes on every value between 0 and 1 inclusive in every interval, no matter how small and O 5 g(x) g l for all x. Now if g(x) = x for any x in [0, l] we change it to another value different from x but still lying between 0 and 1. In this case g takes on every value between 0 and l countably many times in every non-degenerate subinterval of [0, 1] and so g is Darboux continuous. But g is not almost continuous as its graph [\(g) is not connected. The function f constructed above is Darboux contin- uous but P (f) is not connected. The question arises as to whether there is an example of a Darboux continuous function whose graph is totally disconnected. The answer is in the affirmative and this can be done by modifying the function f such that [7(f) has no points on the countable collection of straight lines {—y a‘r x I where V’ is rational g . Then given any two points P, Q of [1(f), there exists a straight line y =‘r x whose complement contains [1 (f) and such that P and Q lie in the disJoint open com- ponents of that complement. We now give a few examples of almost continuous func- tions to serve as illustrations. 25 Example 3.2 Let f(x) sin % (x £ 0) =O(X=O) where -l g x g 1. Clearly [1(f) is connected in [—l, l] X R1. Therefore, by theorem 3.2 f is almost continuous but clearly f is not continuous. Example 3.3 We now give an example of an unbounded almost continuous function. sin -% (x £ 0) 0) where -l e x < 1. f(x) = O ><|+—J (x= Example 3.4 Stallings [1M] asked the following question if f : X ___.;Y and g : Y ————> Z are almost continuous then under what conditions is gf : X -———e> Z. almost con- tinuous? Clearly if we restrict ourselves to real-valued almost continuous functions and X = [a, b] then by theorem 3.2, gf is almost continuous if and only if r7(gf) is con- nected. As a special case the following function is almost continuous. l -l l h(x) = sin [ (sin ; ) ] (x i n7? ) =O(X=nlw )0 -l ‘E: x :5 1. Here h = f2 where f is the function of example 3.2. Jones ([8] p. 117) has constructed an example of a real-valued function f whose graph is connected but which 26 is discontinuous everywhere. By theorem 3.2 it follows that there exist almost continuous functions which are not Riemann integrable. Suppose {fr}; is a sequence of almost continuous functions and fr1 ___) f, then the question arises under what kind of convergence will f be almost continuous. The answer is obvious from the topology used in the function space 59 namely : for each open set U containing [7(f) there is a positive integer m such that P (fm) C U. Next we give a necessary and sufficient condition for an almost continuous function to be continuous. Theorem 3.4 A real-valued almost continuous function f defined on [a, b] is continuous if and only if for every 1 (x) is closed in [a, b]. real number x, the set f- ggggf. If f is continuous then f"1 (x) is closed for each real x. On the other hand if f is almost continuous then f is Darboux continuous by theorem 3.3. Let c be any point in [a,b] and let L be any arbitrary positive number. The sets U = f'1 [f(c) + i ] and V = f'1 [f(c) - t ] are disJoint and closed in [a, b] and c does not belong to either set. Therefore, there is a positive ‘5 such that the open interval (0 - 5 , c + 5 ) is disJoint from both U and V. Also f(x) does not equal f(c) + i or f(c) - t in (c -' S , c + S ). This shows that for all x in (C ' 8 : C + 5 ) f(x) lies between f(c) -i and f(c) +5, 27 for if not f(x) must equal f(c) - E or f(c) + i since f is Darboux continuous. This means that f is continuous. Q.E.D. Finally we discuss the relationships between almost continuous functions and other non-continuous functions. Definition 3.2 A real-valued function f of a real variable is neighborly at a real x if and only if for every positive g , there exists an open interval I such that for all y in I, |x-y| + [f(x) - f(y)] < z (Bledsoe [1]). Clearly if I contains x then f is continuous. We say that f is neighborly if f is neighborly at all real x. Example 3.1 shows that there exist non-continuous neighborly functions. Defintion 3.3 A real-valued function f of a real variable is locally'recurrent at x if and only if every deleted neighborhood of x , N(x), contains an element y such that f(y) = f(X) (Bush [3])- Definition 3.4 A real-valued function f of a real variable is almost locally recurrent at x if and only if there is a sequence {xni such that xn ———§ x and f(xn) ———+f(x). If in definition 3.4, xn ‘7' x(xn <1 x) for all n then we say that f is almost locally recurrent from the right (left) at x. Obviously a locally recurrent function is almost locally recurrent but the converse is not true (see example 3.6). 28 The following theorems are easy consequences of the definitions and so we omit the proofs. Theorem 3.5 If a real-valued function f of a real variable is neighborly at x then it is almost locally recur- rent at x. Theorem 3.6 If a real-valued function f on [a,b] is almost continuous then it is almost locally recurrent for all x in [a,b]. (Example 3.5 The Dirichlet function f(x) = 0 when x is rational, f(x) = 1 when x is irrational, shows that even a function which is locally recurrent everywhere need not be neighborly or almost continuous. H Example 3.6 Let f(x) = x sin , (x A l) l-x where O ‘3. x g 1. Here f is almost continuous, neighborly and almost locally recurrent but f is not locally recurrent at 1. Example 3.7 Let f(x) = O for 0 g; 3c 4; 1 f(x) - l for 1 <1 x g; 2 Clearly f is neighborly at all x in [0, 2] but since the graph of f is not connected, f is not almost continuous. So a neighborly function need not be almost continuous. Bledsoe [1] has shown that the points of discontinuity of a neighborly function form a set of the first category. 29 On the other hand, as remarked earlier, there exist almost continuous functions which are discontinuous everywhere. This means that an almost continuous function need not be neighborly. Now we prove a theorem which gives sufficient condi- tions for an almost continuous function to be neighborly. Theorem 3.7 If f is a real-valued almost continuous function on [a,b] and f has only one point of discontinuity then f is neighborly. 3322:. Let p in [a,b] be the point of discontinuity of f. At all other points f is continuous and so f is neighborly. By theorem 3.6 f is almost locally recurrent at p, therefore, corresponding to any positive 2 there is a q in [a,b] different from p such that [p - q]'<.E/4 and I f(p)-f(Q) ] <3 i/4. Since f is continuous at q, there is a positive 8 (less than i/4) such that for all y in [a,b] such that [y - q I <8, [f(y) - f(q)\<‘/4. Therefore, for all y in the open interval (q - 5 , q + 5 ), lp-yl + [f(p)-f(y)]S IP'CI] +]C1"Y] + [f(p) - f(q)] + [f(q) - f(y)](g. This shows that f is neighborly at p. Q.E.D. Corollary 3.8 It is easy to see that the conclusion of theorem 3.7 holds even if f has an infinite number of dis- continuities which have a finite number of limit points. 3O Corollary 3.9 An almost continuous function cannot have a removable discontinuity. Next we prove a partial converse of theorem 3.6. Theorem 3.10 Let f be a real-valued function continuous at all points except at p in the interior of [a,b] and let f be almost locally recurrent from the right and from the left at p. Then f is almost continuous. .grggf. The conclusion easily follows from the fact that the graph of f is connected but we give a direct proof below. Let V be any Open set containing r1(f). There exists a positive 2 such that the Open disc S((p,f(p)), i ) lies in V. From the given conditions we know that there are x1 , x2 in [a,b] such that xl <; p <1 x2 and (x1 f(xl)) and (x2, f(x2)) lie in S((p, f(p)), E, ). Let g be a function on [a,b] as follows. g(x) - f(x) for x in [a,b] - (xl , x2) (x-xl) f(x2) + (x2-x) f(xl) : for x in [xl,x2]. x2'xl Clearly g is continuous and P (g) C V which shows that f is almost continuous. Q.E.D. CHAPTER IV ALMOST RETRACTS Following Borsuk [2], we shall study almost retracts which are "retracts" under almost continuous functions. We shall show in this chapter that almost retracts inherit the fixed point property as do retracts but that they do not always inherit some other properties such as local connect- edness. We construct a non-locally connected set with the fixed point prOperty. The following propositions are due to Stallings [l4]. Pr0position 4.1 If f : X ——9Y is almost continuous and g : Y ___)z is continuous then gf : X——-> Z is almost continuous. Proposition 4.2 Let X be a compact Hausdorff Space, Y a Hausdorff space and Z any topological space. If f : X ————€>Y’is continuous and g : Y'————§>Z is almost contin- uous then gf : X —_) Z is almost continuous. We now give a counter example to show that theorem 3 page 91 of Kelley [9] cannot be extended to almost continu- ous functions. This answers negatively a natural conJecture. 31 32 Example 4.1 Define f : [0, 1] ————€>[ -l, l] x [ -l, 1] by setting f (0) = (o, o) f (x) (sin %-, cos %.) for x £ 0. If [7(f) denotes the graph of f, then [W(f) = §;(x, f(x)) \x 6 [o,1]§ C [0, 1] x [-1, 1] X [-l, 1]. Since (0, 0, o) is a point on r](f) and for x £ 2 l 1 defined by fog (x, x) = (sin — COS'E) is not almost Y X. continuous. This example shows that the composition of two almost continuous functions is not necessarily almost continuous. Definition 4.1 Let Y be a subset of a topological space X. Y is called an almost retract of X if and only if there is an almost continuous function f on X onto Y such that for all y in Y, f(y) = y. The following theorem states an equivalent condition (Cf. Borsuk [2] page 154). The proof is omitted as it is simple. Theorem 4.1 Y C; X is an almost retract of X if and only if there is an almost continuous function f on X onto Y such that for each x in X, f(f(x)) - f(x). 34 Theorem 4.2 If X is a Hausdorff space with the fixed point property and Y is an almost retract of X then Y has the fixed point property. £3293. Let f be an almost continuous function on X onto Y such that for all y in Y, f(y) = y. Let g be a con- tinuous function on Y to Y. ,By pr0p0sition 4.1, gf : XZ__—_:> Xiis almost continuous and by proposition 2.1, gf has a fixed point p in X, i.e. gf(p) = p. Clearly p is in Y and so f(p) = p. This implies g(p) p which means that g has a fixed point in Y. Therefore, Y has the fixed point property. Q.E.D. Example 4.3 Let X by the square [-1, l] X [-l, l]; and let Y = g(x, f(x)) I f(x) = sin % (x 74 O) and f(0) = O for x in [-l, l] g . Let A be the diagonal of X. The function g on X onto A defined by g(x, y) = (x, x) is continuous. The function h on X into X defined by h(x, y) = (x, f(y)) is almost continuous. Also since [5, is closed in X the function h 'L; : [5 .————;,Y is also almost continu- ous by proposition 3.2. By proposition 4.2 the function hg on X onto Y is almost continuous and for each p in Y, hg(p) = p. This shows that Y has the fixed point property. How- ever, Y is not locally connected and is not closed. In case of retracts the property of being closed and locally connected is preserved (see Borsuk [2] page 155). A retract of a retract of a set X is a retract of X but as the composition of two almost continuous functions 35 need not be almost continuous (example 4.2), we cannot extend this result to almost retracts. However, proposi- tions 4.1, 4.2 provide us with the following theorems the proofs of which we omit. Theorem 4.3 If Y is an almost retract of X and Z is a retract of Y then Z is an almost retract of X. Theorem 4.4 If Y is a retract of a compact Hausdorff space X and Z is an almost retract of Y then Z is an almost retract of X. Definition 4.2 Let X and Y be any topological spaces and let X1 be a subset of X. Let f be a continuous function on Xl to Y. We say that f admits an almost continuous exten- sion to X if and only if there is an almost continuous func- tion F on X to Y such that for all x in X, F(x) = f(x). Theorem 4,5 X1 is an almost retract of X if and only if every continuous function f on X1 to Y (Y arbitrary) admits an almost continuous extension to X. 23233, Let X1 be an almost retract of X, and let r : X ‘)X1 be the almost retraction. Then F = fr is an almost continuous function (proposi- tion 4.1) on X to Y and for all x in X1, F(x) = fr(x) = f(x), i.e. f admits an almost continuous extension to X. On the other hand if f : X1 ...—9 Y admits an almost continuous extension F : X -—-4> Y, then choosing Y = X1 and f the 36 identity map on X1 we find that F is an almost retraction of X onto X1. Q.E.D. Theorem 4.6 If B, a closed subset of a compact metric space A, is homeomorphic to an almost retract R of the Hilbert cube I” then B is an almost retract of A. £3223. The homeomorphism h on B onto R admits a continuous extension H on A to I (see Borsuk [2] page 158). By theorem 4.5 there is an almost continuous exten- sion of h'1 : H ——9B say f : I” ___)B. By proposition 4.2, fH is an almost continuous function on A onto B and fH is the identity function on B. This means that B is an almost retract of A. Q.E.D. Definition 4,3 R is called a metric absolute almost retract if and only if X is any metric space and R' is a closed subset of X that is homemorphic to R, then R' is an almost retract of X. The following theorem is obvious. Theorem 4.7 The property of being a metric absolute almost retract is invariant under a homeomorphism. 10. ll. 12. 13. 14. BIBLIOGRAPHY Bledsoe W. W. "Neighborly functions," Bull. A. M. Soc. 3, 114-115 (1952). Borsuk K. "Sur les retractes," Fund. Math. 17, 152- 170 (1931). Bush, K. A. "Locally recurrent functions,” American Math. Monthly, 69, 199—206 (1962). Fort, M. K., Jr. ”Essential and non-essential fixed points,” Am. J. Math., LXXII, 315-322 (1950). Halperin Israel. "Discontinuous functions with Darboux property,” Canad. Math. Bull. 2, 111- 118 (1959). Hocking and Young. TOPOLOGY, Reading, Mass. (1961). Jiang Jia-he. ”Essential fixed points of multivalued mappings," Scientia Sinica 11, 293-298 (1962). Jones, F. B. "Connected and disconnected plane sets and the functional equation f(x+ ) - f(x)+f(y)," 51111. An Mo So 48’ 115-120 (1942 o Kelley, J. L. GENERAL TOPOLOGY, Princeton, N. J. (1955). Kuratowski. "Sur 1'espace des fonctions partielles ," Annali Matematica, XL, 61-67 (1955). Lebesgue, H. Lecons sur l'integration (1928). Marcus, S. "On locallg recurrent functions,” Am. Math. MONthly, 70, 822- 26 (1963). Schmidt. Essential fixed points. Ph.D. thesis (1962), Iowa State University. Stallings. ”Fixed point theorems for connectivity maps,‘ Fund. Math ., XLVII, 249-263 (1959). 37 346 IHIIIIWIHIIW ] 7 6 9 1 3 0 3 9 2 [HIIIWIIH