FUNCTIONS with CLOSED GRAPHS • by F. JON LEITCH B.Sc
FUNCTIONS WITH CLOSED GRAPHS
• by
F. JON LEITCH
B.Sc. University of Guelph, 1968
THESIS SUBMITTED IN PARTIAL FULFILMENT
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ARTS
in the Department
of
MATHEMATICS .
We accept this thesis as conforming
to the required standard
The University of British Columbia
• March 1970 In presenting this thesis in partial fulfilment of the requirements
for- an advanced degree at the University of British Columbia, I agree
that the Library shall make it freely available for reference and
study* I further agree that permission., for extensive copying of this
thesis for scholarly purposes may be granted by the Head of my
Department or by his representatives. It is understood that copying
or publication of this thesis for financial gain shall not be allowed without ray written permission.
Department of
The University of British Columbia Vancouver 8, Canada Thesis Supervisor: J. V. Whittaker
ABSTRACT
This paper concerns itself mainly with those functions from one topological or metric.space to another that have closed graphs in the product space. Their relationship to closed, locally closed, compact, continuous and subcontinuous functions is studied in order to determine the relative strength of the closed graph condition. The paper collects and in some cases extends results found in papers by R. V. Fuller [2], P. E.
Long [7] P. Kostyrko'and T. Shalat [4], [5] and [6]. The main theorems deal with; l) the characterization of continuous functions in terms of subcontinuity and the closed graph property; 2) a proof that if f has a closed graph then f is the limit of a sequence of continuous functions; and
3) a study of the operations under which the class of functions with closed graphs is closed. Ill
Table of Contents page Chapter 0: Notation and Definitions 1
Chapter 1: Introduction of closed graph 5 • property; open functions with closed- graphs
Chapter 2: Subcontinuous functions; 10 relations between functions with various properties
Chapter 3: Characterization of continuous 17 functions; limit theorem; statements about domain and range spaces
Chapter 4: Points of irregularity; limit 29 numbers
Chapter 5: Closed and locally closed 35 functions; topologies determined by compact subsets
Chapter 6: Analysis of the class of functions. 43 with closed graph; product spaces. iv
Acknowledgements
I should like to thank Professor J.V. Whittaker for suggestion of the topic, his patient reading of the original draft and for his helpful criticisms. Also I would like to acknowledge the financial support of the
National Research Council of Canada. CHAPTER 0
This chapter is devoted to an explanation of the
notation to be used and some definitions.
Definitions: ,
FILTER
A class 5 of subsets of a nonempty set X is a
filter on X if the following conditions are satisfied:
1. if A,B e 3 then ADB e 3 '
2. if A e and BsA then B e 5.
3. 0 k *
A maximal filter on X is called an ultrafilter.
If is any filter on X and U. is an ultrafilter then
either $<^lLor and IX. are not comparable.
Maximal filters always exist since the class
Q = {ACX|Xq e A for a fixed Xq e X} is a principal (fixed)
ultrafilter. If X is an infinite set there exist non-
principal (free) ultrafilters.
TOPOLOGY
A class 3" of subsets of a set X is a topology for
X if the following conditions are satisfied:
1. If A,B e then ADB e 3\ 00 2. if A e ff n=l,2,... then u A e ff n n=i n 3. 0, X e if - 2 -
The • sets belonging to 3" .are called the open sets of X . If . A e 3" then the complement of A is said to be closed.
To avoid possible confusion we present our definition of, the four topologies mentioned in this paper.
FRECHET (T1): A topology 3" on a set X is T^ if and
only if for each x 4 y, x,y e X, there
exists M,N e 3" such that x e M, y | M
and y e N, x N .
HAUSDORFF.(T ) A topology J on a set X is T^ if and
only if for each x =j= y, x3y e X, there
exist M,,N e 3" such that x e M, y e N
and MON = 0
REGULAR: A topology 31 on a set X is regular if
and only if for each x | A, where A is
a closed subset of X , there exist open
M,N such that ACN, x e M and MDN = 0
NORMAL: A topology 3" on a set X is normal if
and only if for each pair of closed sets
A,BcX with ADB =' $ there are open,
disjoint sets M,N with the property that
.ACM and B<=N .
NEIGHBOURHOOD N
Let x e X and let 3" be a topology on X . Then - 3 -
N<=X is a ^-neighbourhood or simply a neighbourhood of x if
there is an 0 e J such that x e O5N
DIRECTED SETS AND NETS
A binary relation R on a set X is called a
preorder if it is reflexive and transitive: i.e.
1. (x,x) e R for all x e X
2. if (x,y) and (y,z) belong to R then
(x3z) e R .
A set X together with a definite preorder is called a preordered set.
A directed set D is a preordered set with the
additional property that for each a,b e D there exists a c e D such that a A net in a space X is a mapping some directed set D into X . We will denote nets either by {xola e A} or, if the directed set A need not be a mentioned,, simply by xQ . a A subnet of x will be denoted by x,- where a J Nb b is a member of the domain of x^ and N is the appropriate function from the domain of x^ to the domain of xa ' PRODUCT DIRECTION Let A and B be two ordered sets. In AxB we define the product direction as follows; (a-.,b-.) _< (ap,b?) - 4 - in A x B if and only if a1 CONVERGENCE x a e A A net C al } converges to a point x in a space X if and only if for each neighbourhood . N of x there exists an a„ e A such that if b>a.T , x, e N . N — N 3 b Throughout the remainder of the paper we will use the following notation: iff if and only if nbhd. neighbourhood X-A complement of A with respect to X A~ the closure of A Gf the graph of the function f X x Y the Cartesian product of X with Y x„ -• x the net xo converges to the point x a a Z+ the positive integers f:X -» Y the function f maps X into Y 3 there exists such that A' the derived set of A , i.e. the set of all accumulation (cluster) points of A . - 5 - CHAPTER 1 Following Fuller [2], we introduce the notion of a function having a closed graph in terms of lim sup (Ls) and lim inf (Li) of nets of sets. We also present a character• ization of open functions and give necessary and sufficient conditions for an open function to have a closed graph. Definition 1.1 A function f: X -* Y has a closed graph (relative to X x Y) iff [(x, f(x))|x e X} is closed in the product topology of X x Y . In terms of nets this becomes Gf is closed iff (x , f(x )) -« (p,q) in X x Y implies that q = f(p) • The following Lemma will be used a great deal in Chapter jj. Lemma 1.2 Let f:X -» Y be given. Then Gf is closed iff for each x e X and y e Y , where y 4 f(x) , there exist open sets • U and V , containing x and y respectively, such' that f (U)nv = 0 . Proof: If the condition holds and (x,y) G^ then U x V is an open set in X x Y such that (U x V)nG^ = 0 which Implies G~ is closed. - 6 - If Gf is closed and (x,y) Gf , then y ^ f(x) , so there exists a basic open set of the form U x V where U and V are open^ containing x and y respectively, and such that (U x V)nGf = 0 . Therefore f(U)HV = 0 . The topological convergence of a net of subsets of a topological space X may be defined in the same manner as the topological convergence of a sequence of sets. Definition 1.3 If C-An|n e D} is a net of subsets of X , then Li A (Ls An) is defined as the set of all x e X such that n v ' every nbhd of x intersects An for almost all (respectively for arbitrarily large) n . We say that a statement S on elements of a directed set D is fulfilled for: 1. almost all n e D if there is an n e D such o that S is fulfilled for all n>n — o 2. arbitrarily large n e D if the set of all n e D for which S if fulfilled is cofinal in . D . A net is said to be topologically convergent (to a set A) if Li A = Ls A (= A), in which case A = Lim A ' n n v '' n For further definitions and theorems on nets of sets see S. Mrowka [8]. The following lemma is interesting in its own right and is needed in the proof of the succeeding theorem. - 7 - Lemma 1.4 If f:X -* Y is a function, y ' is a net in Y , a, and p e Ls £~^(ya) > then there exists a net x^ in X with the property that x^ -* p and f(xNt)) is a subnet of Proof: Assume yJ is a net in Y and let p e Ls f ^(y-K ) a. a' Then for each nbhd U of p and each index a there is an f-1 u index N(a,U)>a such that (y]\r(a TJ)^ ^ ^ • Direct the nbhds of p downwards by containment (i.e. ^ o £ 2 ••') > let the pairs (a,U) have the product direction, and choose x 6 f_1 )f1U thus btainin the net x N(a,U ) (%(a,U ) n > ° S N(a,U ) x = y wilictl ls a in X , Now we have .f( jj(a TJ )) N(a U ) subnet x s nce r of ya and jj/a u ) ~* ^ i £° any nbhd U of p , xN(b U ) 6 ^n Proviaed only b>a and k>n . k ., Theorem 1.5 If f:X -* Y is a function the following are equivalent: a) f has a closed graph _1 _1 b) If ya - q in Y then Ls f (ya)5f (q) c) If y - q in Y then Li f-1(y )cf_1(q) a a Proof: Assume • a) holds and let y0 -• q in Y . a Let p e Ls f~^(ya) • By Lemma 1.4 there is a net x^ in X such that x^ -* p and f(xNfe) is a subnet of ya . - 8 - x f x p q and since G is Thus we have ( Ntl.> ( Nb^ "* ( -> ) f closed Q = f (p) 5 i.e. p e f""1(q) . Therefore b) is true. If b) holds then c) holds since for any net of sets E we have Li E dLs E a a— a Assume that c) holds and suppose (x f(x )) -» (p,q) . a. a Then y = f(x.) - q and p e Li f~'1"(y_) . Since by assumption cl cl ci _1 _1 _1 Li f (ya)£f (q) we have p e f (q) or f(p) = q and hence a) holds. Definition 1.6 If a function f:X -» Y is such that U open in X implies f(U) is open in Y then f is said to be an open function. Theorem 1.7 If. f:X -* Y is a function, the following are equivalent: a) f is open _1 _1 b) ya - q in Y implies f (q)cLi f (ya) -1 1 c) ya - q in Y , implies f (q)5Ls f (ya) Proof: Assume a) holds, let y q in Y and let cl p e f_1(q) . Suppose p d Li f_1(y ) . Then there exists a nbhd U of p such that frequently f_1(y )DU = Qf . Since cl U must contain an open set containing x we can take U to. be open. Now we have y is frequently outside f(U) , but a since f(U) is open and q e f(U) , f(U) must be a nbhd of q and this gives the desired contradiction. Thus b) holds. If b) holds then c) holds since L 1 _1 f- (q)5Li f- (ya)CLs f (ya) . Assume c) holds, and suppose f is not open. Then there is an open UcX and a net y in Y-f(U) such that — a 1 1 yo -» q for some q e f(U) . By assumption f~ (q)ciJs f~ (y„) . _1 _1 Let p e f (q)nu . Then U is a nbhd of p and f (ya)nu _1 is frequently nonempty since p e Ls f (ya) • But this means that y is frequently in f(U) a contradiction to the fact that y is a net in Y-f(U) . Hence a) holds, a Using Theorems 1.5 and 1.7 and the definition of Lim E where E is a net of sets we arrive at the following a a theorem. Theorem 1.8 The function f:X -» Y is open and has closed graph iff _1 _1 ya — q in Y implies Lim f (ya) = f (q) Proof: By the theorems cited above f is open and has closed graph iff yQ -• q in Y implies _1 1 -1 _1 • f (q)eLI f- (ya)CLs f (ya)cf (q) _1 1 _1 _1 Thus Li f (ya) = Ls f" (ya) = Lim f (ya) = f (q) and the theorem holds. - 10 - CHAPTER 2 .In this chapter we introduce suboontinuous functions. We then study some relations between functions with closed graphs and closed, compact and subcontinuous functions. Throughout the chapter we will denote by U(X,Y) the class of all functions f:X -» Y whose graphs, , are closed. The following theorem will lead us to the definition of subcontinuity (Fuller[2]) Theorem 2.1 x n e be a Let f e U(X,Y) and let £ nl sequence of points in X such that . x x , where x e X . Also x x let {f(xn)|xn e X} be compact in Y . Then f( n) - f( ) • Proof: x Suppose that f( n) does not converge to f(x) . Then there exists a nbhd N of f(x) with the property x N that f( n_) 4 for i = 1,2,3 ... (1). Since '£f(xn)} is compact in Y there is a sequence xn j = 1,2, ... for which (f(x )}°° converges to some q e Y . Since x f e U(X,Y) and (x , f( n )) - we must have 1J - 1J q = f(x) and hence (f(x )}c° -* f(x) . Therefore H 3=1 - 11 - {f(x )}°° converges to f(x) also, but this contradicts (1) i i-1 Hence f(x ) - f(x) . Definition 2.2 The function f:X -. Y is said to be subcontinuous iff xn p in X implies there is a subnet f(x^) of f(xa) which converges to a point q e Y . Similarly a function f:X -» Y is said to be inversley subcontinuous iff f(x ) -• q in Y implies there is a subnet a Xj^k of x^ which converges to some point p in. X . The concept of a subcontinuous function is thus a generalization-of a function whose range is compact (cf Theorem 2.1). It is also a generalization of a continuous function. Theorem 2.3 If f:X -» Y is a function and each point x in X has a nbhd U such that f(U) Is contained in a compact subset of Y then f is subcontinuous. Proof: Let x be a net in X such that x -» x . a a Since U is a nbhd of x there is an aQ such that for all x a>aQ , xQ e U . Then f( a) e f(U) for all a>aQ . Since f(U) is contained in a compact subset of Y there is a sub• x f x net f( Nb) of ( a) that must converge to some q in Y . Hence f is subcontinuous. - 12 - The following theorem will show that if the range is embedded in a completely regular space then a subcontinuous function is very nearly such that it preserves compactness. Theorem 2.4 Let f:X - Y be a subcontinuous function and let Y be completely regular. Then for each compact subset K of X , f(K) is compact. Proof: Let KcX be compact and let {Z |a e A} be a net in f(K) . Let B be a uniformity for Y . (see Dugundji [1] pp 200, 201). Direct. B downwards by containment (cf Lemma 1.4) and let A x.B have the product order. For each (a,b) e A x B choose y^a b^ e b[Za]nf(K) and L X(a ^ e Knf~" (y^a • Since K is compact a subnet X(Mc, Nc) °^ x(a,b) converges. Since, f is subcontinuous a subnet y(Kd ^d) of y^a ^ converges to a point q in Y . Clearly q e ~fJKJ Now consider the net ZT,, which is a subnet of Kd Z& . We will show that Z^d -» q . Let b e B . By definition there is a symmetric b e B such that b^ o b""1" = b^ o b^crb . By the choice of y^a b^ we have that (y^Kd Ld)* ZKd) is eventually in b± . Since Ld) - q , (q, y(Kd? Ld)) is eventually in-,, b1 . Since b-^ o b£b we have (q, ZKd) is eventually in b^ and hence -» q . Consequently f.(K) is compact. - 13 - Assuming the range is embedded in a Hausdorff space, the next theorem states that if f e U(X,Y) then f handles compact sets with less success than continuous functions but its inverse does Just as well as the inverse of a continuous function. Theorem 2.5 Let f e U(X,Y). If K is a compact subset of X [Y] then f(K) tf_1(K)] is closed. Proof: Let K be a compact subset of X and assume that f(KJ Is not closed. Then there is a net ya in f(.K) such _1 that yQ - q and q e Y-f(K) . Pick Xq e K(1f (y ) . Since a a a K is compact there is a subnet xNb of x& that converges x x Xj( to some point x e' K . Thus we have ( JJDJ ^( Nb^ ~* ( l) * Since f e U(X,Y) , q = f(x) and hence q e f(K) , but this contradicts the choice of q . Therefore f(K) is closed. The second proof is completely analogous and will not be done. Definition 2.6 Let f:X - Y be a function. If f [f_1] takes compact subsets of X [Y] onto compact subsets of Y[X] then f is said to be compact-preserving [compact]. - Ik - Definition 2.7 If f [f-1] takes closed sets of X [Y] onto closed sets of Y [X] then f is said to be closed [continuous]. In the following theorem we characterize compact and compact preserving functions in terms of inverse sub- continuity and subcontinuity respectively. Theorem 2.8 Let f:X -» Y be a function. a) The function f is compact iff f | f_1(K):f~1(K) - K is inversley subcontinuous for every compact KcY . b) The function f is compact-preserving iff f |K:K -» f(K) is subcontinuous for all compact KcY . Proof: The proof of a) is analogous to b) so only b) is proved. Assume that KcX is compact and' that f is compact preservingc . Let x be a net in K such that x -* x , ° a a x e K . Since f(K) is also compact the net f(x ) must cl have a subnet f(xN^) that converges to some q e f(K) . Hence f|K:K -• f (K) ' is subcontinuous. Let f be subcontinuous. Let. KcX be compact and _1 let y be net in f(K). - Pick x e Knf (y ). Then x a i ~s a net in K and hence there is a subnet x™ of x . that NB a converges, to some x e K . Since f.|K;K - f(K) is subcontinuous we must have a subnet - 15 - x f(xMb) of f( Nb) = yNb converging to a point q e f(K) . x = ls a SUDne- we nave Since MNTD^ ^MNb k °^ ya that f(K) is compact. Corollary 2.9 a) If f:X -» Y is inversely subcontinuous and f~"!"(K) is closed for each compact KcY then • f is compact. b) If f:X Y is subcont inuous and f(K) is closed for each compact KcX then f is compa preserving. Proof: Again only b) is proved. In any case we need only note that (using the notation of Theorem 2.8) since f(K) is closed, the limit point q of the net y^^ must belong to f(K) . Corollary 2.10 If f:X - Y and f e U(X,Y) and f is sub- continuous .[inversely subcontinuous] then f is compact- preserving [compact] Proof: By Theorem 2.5, • f(K) is closed for every compact KcX and hence by Corollary 2.9 f • is compact-preserving. - 16 - Corollary 2.11 • Let f: X -• Y be a function. Then:. a) if Y is Hausdorff and f is both continuous and inversely subcontinuous then f is compact. b) If X is Hausdorff and f Is both closed and sub- continuous then f is compact preserving. Proof: We prove only b). Since X is Hausdorff every compact K<=X is closed and hence f(K) is closed for every compact KcX . By Corollary 2.93 f is compact-preserving. - 17 - CHAPTER 3 In this chapter, we give characterizations of con• tinuous and closed functions in terms of the closed graph and subcontinuity properties. We then show that if f is a function with a closed graph then f is the limit of a sequence of continuous functions and point out the dependency of the closed graph property on the range space. The following two theorems will show that the closed graph property complements the two subcontinuities in interest• ing ways. Theorem 3.1 If f:.X -» Y is a function then a sufficient condition for f to be. continuous is that f have a closed graph and be subcontinuous. If Y is Hausdorff then the condition is also necessary. Proof:. (Sufficiency): Let x& be a net in X such that x& -» x . x Suppose that f( a) does not converge to f(x) , i.e. f is not continuous at x . Then f(x ) has a subnet, f(x„„ ) . v a' ' K Nb no subnet of which converges to f(x) . Since f is sub- x x continuous some subnet ^( ]y[NC) °^ ^^ Nb^ converges to some point y e Y ... Hence we have (xmc» f(xMNc)) - (x,y) and f x f x since Gf is closed y = f(x) . Therefore ( M£jc) ~* ( ) which implies that the assumption was false. - 18 - (Necessity): If f is continuous then it Is obviously sub- continuous. If (xa,f(xa)) - (p,q) then x& - p and thus f(xa) -* f(p) since f is continuous. If Y is Hausdorff. q = f(p) and hence f has a closed graph. The following example shows that Theorem 3.1 (Necessity) is false if Y is only a space. Example 3.2 Let X = Y = Z+' and let both X and Y have the cofinite topology [i.e. a set is open iff it is void or has finite complement]. Obviously the identity map is continuous but it does not have a closed graph, because the sequence {(n,n)|n e Z+] converges to every point (x,y) € X x Y . In every nbhd of a point (x,y) there is an element of the base of the topology for X x Y of the form G^ x G^ where G-^G^ are open subsets of Z+ and (x,y) e G^ x G^ .If m = max [njn Gx or n ^ Gg} then (k,k) e G1 x G for all k X m . Corollary 3.3 Let the function f:X -• Y be surjective. If U is an ultrafilter on X such that U x for some x e X then a sufficient condition for the image filter f(U) to converge to f(x) is that G^ be closed and f be sub- continuous. If Y is Hausdorff the condition is also necessa - 19 - Proof: (Sufficiency): By Theorem.3.1 f is continuous and hence if U - x then f(U) -» f(x) . (Necessity): As in previous theorem. Corollary 3-i+ If f:X - Y is a function and Y is compact and Hausdorff then f is continuous iff G~ is closed. f Proof: Suppose is closed. Since Y is compact f is subcontinuous. Hence by Theorem 3-l> f is continuous. Let f be continuous and suppose there is a net (xa> f(xa^ in Gf such that (xa> f(xa^ "* (P^) • THEN x -» p and by continuity f(x ) -* f(p) . Since Y is cl cl Hausdorff q = f(p) and is closed. The following example shows that the compactness condition on Y cannot be dropped. Example 3*5 Let X = (0,1] and Y = [0,+») . Define f by - Jl/x x e (0,») 1 10 x = 0 Then f is clearly not continuous but G^ is closed. The next Theorem from P.E. Long's paper [7]» gives a characterization of continuity in terms of the closed graph - 20 - property and conditions on the domain and range spaces. Theorem 3.6 . Let f:X -> Y be a function where Y is countably compact and X is first countable. If Gf is closed then f • is continuous. Proof: Suppose f is not continuous. Then there is an open VcY such that f_1(V) is not open in X . Therefore f-1(V) contains a point x e X such that x is a limit point of X - f-1(V) . Since X is first countable, there 1 exists a sequence {xn} , with x^ e X - f~ (V) , that converges to x . In the countably compact space Y , the set (f(xn)} has an accumulation point y | V . Then (x,y) ^ G^ , but (x,y) is a limit point of G^ since any open set in X x Y containing (x,y) clearly contains points.of the form (x,f(x)) . This contradicts the assumption that G^. is closed, hence f is continuous. The above example shows that this theorem is not true for Y locally compact and X compact. Theorem 3.1 showed the result of combining the closed graph.property with subcontinuity. If instead we use inverse subcontinuity we might suspect that f_1 will be continuous. The following theorem confirms this fact. - 21 - Theorem 3-6 If the function f:X -* Y has a closed graph and is inversely" subcontinuous then f is closed (i.e. f-1. is continuous). Proof: Let C be a closed subset of X and suppose that f(Cv ) is not closed. Then there exists a net x in C and a x a point q e Y - f(C) such that f( a) ~* Q • Since f is~ inverselyJ subcontinuous there is a subnet x,^ of x such Nb a that Xj^ -* p and since C is closed p e C . Hence we have (xNb> f(xNb)) "* (p>q) and q ^ f(p) since 0. e Y - f(C) and p e f(C) . This however, contradicts the assumption that G^ is closed. Definition 3.7 • A function f:X -• Y belongs to Baire class a if the set f-1(G) is a Borel set of additive class A of Borel sets in X for every open GcY . _1 In particular, f e B1(X3Y) if f -(G) is an Fct subset of X for each open G5Y and f e B (X3Y). if f-1(G) is open for each open GcY ; i.e. if f is continuous. The following theorem (from P. Kostyrko and T. Salat [5]) shows that if X and Y are metric spaces and f:X -» Y has a closed graph then f is the limit function of a sequence of continuous functions. - 22 - Theorem 3.8 Let (X,p) and (Y,a) be metric spaces where CO Y = U C with C compact n = 1,2,3,... . Then . ll II I • U(X,Y)cB1(X,Y) Proof:- Let f e U(X,Y) and let GcY be open. Define the set M = Gfn(X x G) and note that M e P (X x Y) , i.e. M ^ CO is an F set in X x Y . Therefore M = U M where CT n=l n • . CO each M is closed in X x Y . Since Y = U C and each n=l C is compact it could be assumed that C ,, for n ^ n— n+1 n = l,2.,3j... • Let ZQ = (x *y ) e X x Y and let p be a positive number. Let S(x ,p) be a spherical nbhd of + radius p -of Xq . By assumption there is an nQ € Z such e c n that yQ n > (and hence to Cn for n>. 0) Define Dn = Cn +n-l for- n = 1>2>~5>-- > and % = S(x ,p) x D . o . ^ & Let Mnp = MnnRp . Now each M is closed (since both Mn and Rp are) and bounded (since Rp is) in X x Y and Mn = U Mnp . Let ENP = (x e x|2 y e Y • • (x,y) e Mnp} and let E={xeX|3yeY- • (x,y) e M] . Then ro -1 E = U E^„ and E = f (G) . If we can now show that n,p=l np for any fixed n and p ,- ENP is closed in X , we will be done. Let x e E' [E'" is the derived set of E 1 . np np npJ - 2> - x a € A in Then there exists a net ( al ) ^np such that x -» x . From the definition of E this implies that a np ^ for each a e A there is a ya e Y such that (xa,ya) e Now the elements y all belong to a compact D and hence there is a subnet yNfe that converges to some point y e . Then we have (x^, YNFE) - (x,y) e Mnp since Mnp is closed. Hence x e E and therefore E is closed set. Thus we np np -1 have f (G) e F (X) , whence f e B1(XAY) The following example, like example 3-5 will show CO that the condition that Y = U C where the C are n=l n compact cannot be dropped. The example actually illustrates a function that has a closed graph but belongs to no Baire class. Example 3.8 Let X = [0,1] with the Euclidean metric and let Y be a set with cardinality, the continuum, and the trivial metric. Let f :X -» Y be bijective. Let {(x ,y ) |a e A] a a • d be a net in G> and assume that v (x ,y ) -»v (x 30,y ). Then f a^a' o o' xo -» and y„ - y^. . Since Y has the trivial metric CL o a o y = Jy .for almost all a e A 3 , i.e. there is an a e A . a o o such that Jy = y for all a>a . From y = f(K x ) it a Jo — o ^a a' a and follows that yQ = f(x&) for a>. 0 - since f is injective the net x "is stationary for a>a . From x -• x it a — o a o follows that x„ = x for a>a . Hence (x ,y ) e G. and a o — o v o o' f therefore G^ is closed. - 2h - Now let E be any non-Borel set. in X and let G = f(E)cY . G is open since in Y all sets are open. Since f is single valued E = f'^G) Hence f belongs to no Bore'l class. The previous theorem was later generalized as follows (Kostyrko [6]) . Theorem 3.9 Let X be a normal topological space and let f e U(X,E^) where E^ = (-<», + «)• with the Euclidean metric, Then there is a sequence of functions fn e B (X^E^)- , x x n = 1,2,3,... > such that |fn(x)| Proof: Let Pn = Gfn(X x [-n,n]) n = 1,2,... . Fn is closed in X x' E-^ and its projection Xn to the set X is If also closed. (Xn = [x e X|3 y e (x,y) e Fn) . X 4 0 then the function g = f|1 X is continuous on X n °n > n . n x since it has a closed graph (namely F^) and | gn( )I all x e Xn . (cf Corollary J>.k). Since X is a normal space there is a continuous extension f of the function •r ft gn onto the whole space X such that |f (x)| x e X . If ., X^ = 0 put fn(x) s 0 . The equality f(x) = lim f (x) for X e X follows n^co n - 25 - x n e z+ is from the fact that the sequence of sets f nl ) CO increasing and X = U X „ If x e X then there is an n=l n . 4- n e Z such that for all n>n 9 , x e X and hence o — o n fn(x) = f(x) . We now. consider the dependency of the closed graph condition on the space.in which the range Is embedded. Note that Theorem 3.1 and the following corollaries require that the range space be Hausdorff. In general this condition cannot be dropped, for if i:X -» X is the identity map on X , then G^ is closed iff X is Hausdorff. Now let f:X -» Y be a function which has a closed graph.and is not continuous. Let Y* be a compactification of Y . Then f :X - Y* is. subcontinuous since Y* is compact If f had a closed graph it would be continuous (by Theorem 3. However continuity does.not depend upon the embedding space so that f:X -»-Y would be continuous, contrary to assumption. This shows that if f e U(X,Y) but f £ BQ(X,Y) then f 4 U(X,Y*) . The following theorems (from P.E. Long's paper [7]) show that if we know a function has a closed graph then we can make some statements about the domain and range spaces f . Theorem 3.1Q-, Let f:X -» Y be a function with G^ closed. Then f(X) is Tx . - 26 - Proof: Let y and w be distinct points in f(X) . Then there exists an x e X such that f(x) = w . Thus (x,y) | G^ , so. by Lemma 1.2 there exist open sets U and V containing x and y respectively., such that f(U)nv = 0 Therefore w | V and Y is T^ . Theorem 3-H Let f:X -* Y be any open surjection with G^ closed. Then Y is T^ . Proof: Let y and w be distinct points in Y . Then there are distinct points x and z in X such that f(x) = y and f(z) = w . Since (x,w) ^ G^ and G^ is clos there exist open sets U and V containing x and w respectively, such that f(U)nV = 0 ; but f(U) is open and contains y . Consequently Y is T^ . Theorem 3.12 Let f:X - Y be injective v/ith Gf closed. Then • X is Tx . Proof: Let x and z be distinct points in X . Then f(x) ^ f(z)* so there exist open sets U and V containing x and f(z.), respectively, such that f(U)nv = 0 . Thus z ^ U , implying X is T, . - 27 - Theorem 3.13 If f:X - Y is bijective with Gf closed, then both X and Y are T^ . Proof: Theorems 3.10 and 3.12 Theorem 3.14 Let f:X -» Y be infective and subcontinuous with closed graph. Then X is T^ . Proof: By Theorem 3.1 f is continuous, hence is an open surJection from f(X) onto X . Furthermore G^ is isomorphic to G^-l, hence f-1 has a closed graph in Y x X . By Theorem 3.11, X is Tg . Theorem 3.15 Let f :X -» Y be a homeomorphism of X onto Y having G^. closed. Then both X and Y are Tg . Proof: Theorems 3-H and 3.14 Theorem 3.16 Let. f:X -• Y be infective, open, connected and have G^ closed. Then if X is locally connected, X is m 2 ° - 28 - Proof: Let x and z be distinct points of X . Then f(x) ^ f(z) , so there exists an open connected set U containing x and an open set V containing f(z) such that f(U)nv = 0 . Since f(U) is open, z | U . For otherwise UU{z] is connected, so that f(Uu{z]) = f(U)u(f(z)} is connected, since f is connected. This is an impossibility. - 29 - CHAPTER 4 In this chapter we review and extend some results obtained by Kostyrko and Salat in [4], Spaces X and Y. will be Hausdorff unless otherwise stated. Definition 4.1 Let f:X -» Y be a function. Define a e X to be a point of irregularity of F if there exists a sequence x in X with x -» a 3 , x 4 a ; such that the sequence n. n n ' ^ {f(x )} is not compact in Y „ We will denote by the points of irregularity of f and by D^ the points of discontinuity of f . We note that for any,function f, N^cp^ • Theorem 4.2 If f e U(X,Y) then Nf = Df . Proof: Since for any function •N^cD^ we need only to show the reverse, inclusion holds for f e U(X,Y). Assume on the contrary that D^^W^. . Let x e D^ be such that x | N^. . Since x e D^ s x cannot be an isolated point of X . Let (x |n e Z+} be a sequence of points of X such that x - x and x 4 x . Since x k N„ the sequenc e ff(vx )) n n . i ^ n is compact in Y . From Theorem 2.1 it follows that f(xn) - f(x), which implies that x \ Df . Hence DfcN^ and the theorem Is proved. - 30 - Corollary 4.3 If f e U(X3Y) then f is continuous iff Nf = 0 . Theorem .4.4 Let f e U(X,Y) and assume that Y is locally compact, i.e. that each point y e Y .has a nbhd whose closure is compact in Y , then D^, is closed. Proof: Since fe U(X,Y) we have Df = Nf . Assume that 3\V is not closed. Then there exists a net 1 fx 1 la e A} in f . a • •* such that x -• x and x e X - Nf , i.e. x is a point of continuity of f . Since x e X - N. and xo -* x ' we have i a f(x ) - f(x) . Let K be a nbhd of f (x) such that K is compact. By continuity there is a nbhd V of x such that f (V)cKcK . Since x& -» x there is an aQ e A such that for all a>a •, x& e V and hence f (*a) e f(V) . Since x e N„ for all a e A then for each a i + a e A there is a sequence {xa n.|n e Z ] of points in X such that x -» x and {f(x „)|n e Z+l Is not compact . a, n a a, n in Y . i.e. there is no subsequence of {f(x „)|n e Z+} that converges in Y . - (l) Now V is a nbhd of x for all a>a . and since a — o x -» x for all a e A , therefore for a>a there exists a,n a . — o an n e Z+ such that for all n>n , x„ e V . Hence for each a — a J a,n a>a the subsequence (f(x x )ln>1 n , n e Z } is a subset — o ^ ^ . a.n' — a * •* - 31 - a a of f(V)dK . Therefore for each >. 0 the sequence- + + ff(^ v x a,n)ln>/1 — na 3 . n e ZJ } has a subsequenc^ e . v ff(a,n^'x 1 )|i e J Z ] x e that converges to a point in K . But £f( a n.)l^- x n e + an is a subsequence of £^( a r,)l Z } ^ hence (l) has been contradicted. Therefore ' = . is closed. In the following theorems Y will be the real line Y = E^ = (+ co) with the Euclidean metric. Definition 4.5 If f:X -•. E-^ is a function then f is unbounded at a point x o e X if there is a nbhd N of x o in which it is impossible to find a single real M>0 such that |f(x)| Definition 4.6 The subset D .of E^ 4s said to be of the first CO category if D = u D where each D is nowhere dense n=i •'' M in E^ (i.e. the closure of D^ contains no nonempty interval). If D is not of the first category we say D is of the second category. Theorem 4.7 Let f e U(X,E1) and assume that each nbhd K of x e X is a second category set. Then D^ is nowhere dense in X . - 32 - Proof: Since f e U(X,E^) we have Df = . By Theorem 4.4, is closed and hence 0^ = X - 'is open x e X and 3 M>0 and a nbhd K of x x o o °f - i ° • 3- x e K , |f (x) | It is sufficient to show that 0^, is dense. Assume on the contrary that 0f is not dense in X . Then there exists an x e X and a nbhd K of x such that CpfiK = $ . o o f Hence there does not exist a single real M>0 such that |f(x)|<_M for all x e K [although it may well be true for some x e K .]. Let Tn = [x e K| |f(x)| |v| (xfc , f(x )) - (z ,v) i Gf since' |f(z ) |>n and |v|_ Let I be a (finite or infinite) interval on E^ . Then for any f e U(I,E^) then the set is nowhere dense in I . Definition 4.9 Let f be a real valued function determined on a partially ordered set (X5<) and let x e X . We will call t a limit of f at x on the right [left] if there exists x n € x x a sequence [ nl Z } of points in X such that n> [x Denote by L [L ] the set of limit numbers of f at a point x on the right [left] and let L = L UL be the set of all limit numbers of f at x . Theorem 4.10 Let f e .U(XjE^) and pick x e D^ , then i) L n = } 4 0 and ii) Ln[ (-.,«)-f(x) ] = 0 . A x Proof: i) This follows from Theorem 4.2 and the fact that in this case Nf = [x e X |f(x)| is unbounded] . il) Assume the contrary and let t e L D [(-«,«>)-f(x) ]. Then t e L and -f^ f(x) . • Evidently there is a sequence [x In s Z"1"} in X such that x -» x and f(x ) - t . Now n J n v n we have (x , f(x )) e G„ and (x ., f(x ) - (x.t) k G_. v n* v n'' . f v n ^ ny v 5 * f - 34 _ This contradicts the fact that f e V(X3E^) 3 hence the theorem, is proved. \ CHAPTER 5 This chapter discusses conditions for closed and locally closed functions to have closed graphs.. We also look at spaces whose topologies are determined in some measure by their compact subsets. Having already discovered conditions under which continuous and open functions have closed graphs we proceed to find such conditions for a closed function. Example 5.1 Let X = (0, 1] c.E1', Y = {0,1} and.let f be the characteristic function of X mapping E^ into Y . Nov/ f is a closed function but f does not have a closed graph. We note that f does not have closed point inverses. Following Fuller [2] we now define a locally closed function. Definition 5.2 . • ,c We call a function f:X-»Y locally closed if for every nbhd. U of each x <= X there is a nbhd V of x such that V c U and f(V). is closed in Y . If the domain of a closed function f is regular then f is locally closed. Also if f:X-*Y is such that X is locally compact and regular and f takes compact sets onto closed sets then .f is locally closed. - 36 - Example 5.3 Let X be the reals with the discrete, topology and Y the reals with the usual (Euclidean) topology, and let i:X-»Y be the identity. Then i is locally closed and is in fact continuous, but it is not closed. Theorem 5.^ If f:X-*Y is a locally closed function then I ya -+ q in Y implies that Ls f "*"(y )cf~' "(q) [cf. Theorems 1.5 and 1.7] Proof: _ L Let ya - q in Y and p e Ls f " (ya) . Assume _____ that p ^ f (q) . By Lemma 1.4 there exists a net x ^ in X such that xMh-» p and f (xw>i) =. y^ is a subnet of 1 c ya . Now X - f ( l) is a nbhd of p and since f is locally closed there is a nbhd U of p with the properties: i) UcX - f-1(q) and ii) f (U) is closed in Y . Since xNb ~*p xNb is eveni;ua^y ^ an(^ hence f(x^) is eventually in f(U) . This means that f(x^) is eventually in the complement of a nbhd of q , namely Y-f(U) . This contradicts the fact that y,„ = f(x,T, ) -» q . Corollary 5*5 If~„ f:X -» Y is a locally closed function and has closed point inverses then f has a closed graph. - 37 - Proof: By assumption f_1(q) = f~"I"(q) for every q e Y . If y -» q in Y then Lsf-1(y )cf_1(q) which by Theorem 1.5 implies that f e U(X,Y). Corollary 5.6 If the function f:X -» Y is closed with closed point inverses and X Is regular then f has a closed graph. Corollary 5.7 If the function f:X -» Y is closed and subcontinuous with closed point inverses and X is regular, then f is continuous. Proof: Apply Corollary 5.6 then Theorem 3.1 Corollary 5.8 If the function f:X Y is locally closed and inversley'subcontinuous with closed point inverses then f is closed. Proof: Apply Corollary 5.5 then Theorem 3.4 Theorem 5.9 ^ If '.f:X-» Y is a function where X is regular - 38 - and locally compact, the following are equivalent. a) f maps compact sets onto closed sets and has closed point inverses. b) f is locally closed and has closed point inverses c) f has a closed graph. Proof: We have already commented that (a).implies (b) and by Corollary 5>5s (t>) implies (c). Assume then that Gf is closed. By Theorem 2.5 f maps compact sets onto closed sets. Also since points are compact, the same theorem yields that f has closed point inverses. We now define some topological spaces which are, in a sense, determined by their compact sets, and prove two theorems concerning these .spaces and compact preserving functions. •. . Definition.5.10 Let X be a topological space and p. e X . a) X is said to have property k^ at point p iff for each infinite AcX having p as an accumulation point., there is a compact subset K of AU{p] such that, p e K and p is an accumulation point of K . b) X has property k^ at point p iff for each set A having p as an accumulation point, there is a - 39 - subset B of A and a compact K_>BU{p] such that p is an accumulation point of B . c) X has property k^ at point p iff U is an- open set in X precisely whenever UDK is open in K for each compact set K in X . X is said to be. a k^ space if X has property k^ at each of its points. We have as relations between the k^ spaces that k-^ implies kg implies k_. . Theorem 5*11 Let f:X -» Y be a compact function and let Y be a k-, space. A sufficient condition that f be closed is 3 that Gf be closed. If X is regular, T^ the condition is also necessary. Proof: . (Sufficiency): Let C be a closed subset of X . To show that f('C) is closed we need to prove that f(C)HK is closed for each compact Key . Let KcY be compact. Then f~"1"(K) is compact and hence so is cnf_1(K). By Theorem 2,5* f[Cnf_1(K)] is closed in Y . Since f[Cnf_1(K)] = f(.C)DK. we have that f(C)DK is closed in Y and hence in K for every compact KcY . Hence f is closed. - 40 - (Necessity): Since X is and f is compact, point inverses are closed. Thus by Corollary 5«6, f has a closed graph.. Using techniques similar to the predeeding theorem • we can arrive at another characterization of continuity. Theorem 5.12 Let f:X -» Y be an infective, compact-preserving function and X a space. A sufficient condition that f be continuous is that Q be closed. If Y is Tg then the condition'is also necessary. Proof: (Sufficiency): Let f e U(X,Y) and let CcY be closed. To show that . f~"*"(C) is closed it is necessary to show that f-1(C)riK Is closed for each compact KcX .Let KcX be compact, then so is f(K) . f (.Cnf(K)) is closed by Theorem 2.5. Since f is infective f-1[f(K)] = K so f-1(cnf(K)) = f_1(C)nK is closed in X and hence in K for each compact KcX . Hence f_1 is closed and therefore f is continuous. (Necessity): Let f be continuous and Y be Tg . Points are closed in Y and since f-"'" is closed f has closed point inverses.. Also f maps compact sets onto closed sets. Therefore by Theorem 5.9 C^ is closed. - 41 - Example 5.13 We will now construct a function which is compact and has closed graph but is not closed. Let X be an uncountable set. Let C-^ be the cocountable topology for X ; C-^ = {AcX|X-A is countable] \J0 . Let Cg be the discrete topology for X . The only compact sets in either (X, C-jO or (X,Cg) are the finite subsets of X . That this is true for (X,Cg) is obvious, to see that it is true for (X,C-^) let A. = {a-^ag,. .. } be a countable subset of X . Take as an open cover of A the collection [Mn|Mn = (X-A)U{an}} for n = 1,2,3,... - X-Mn = X-[(X-A)u{an)] - Afl(X-{an}) = A-{an) , so the Mn are legitimate open sets whose union contains A . Clearly no finite subcollection pf the M will cover A , so A is not compact. . Consider the identity map. i:(X,C2) -» -(X,Cj) . x x x The function i has closed graph, for let ( a? 1 ( a)) ~* ( ,y) • Then x& must eventually be constantly x in (X,C^) and so must i(x ) =xo . If i(x ) converged to y ^ x then we would have a contradiction to the fact that points are closed In (X,C-^) • The inverse of I carries compact (finite) sets of (X,^) onto compact (finite) sets of (X,Cg) and thus i is compact. Finally since C^pC-, , i is not closed. ( - k2 - Theorem.5.14 Let f :-X -» Y be a function that is compact preserving and has closed point inverses. Let X and Y be' T2 and let X have property kg at a point p . Then f is continuous at p . Proof: Suppose f is not continuous at p . Then there is a nbhd V of f(p) such that for each nbhd U of p there x v is a point xu e-U such that f( u) k . The collection x u A = C ul is a nbhd of p] has p as an accumulation point. Thus A has a subset B , for which there is.a compact K3BLl{p} .Also p is an accumulation point of B . Consider the function f JK:K -* Y . f |K is closed since Y is Tg . As (f|K)"1(y) = f_1(y)nK for all y e Y , f|K has closed point inverses. Finally, since K is.compact, Tg it is also regular and hence by Corollary 5.7 f|K is continuous. However if we pick a net x in BcK such that •a — f x s xa -* p then ( a) i never in the nbhd V of f(p) . This contradiction proves the theorem. - kj> - CHAPTER 6 This chapter is devoted to an analysis of the class U(X,Y) . We will follow Kostyro and Shalat in our effort to determine under what operations U(X,Y) is closed. Composition The following example will show that if f, g e U(X,Y) then it does not follow that fog e U(X,Y) . Example 6.1 Let X = [0,1] , Y - both with the Euclidean metric. Let f(x) = {1/x, * f [° >1] and let g(y) '= {^g J f °Q Then g is continuous on f(X) and f has a closed X graph. However g[f(x)] = {^/2 ^ ° 0 does not have a closed graph In XxY . We now give some conditions for the closedness of graphs of a composition of functions. Theorem 6.2 Let X,Y,Z be topological spaces. Let f:X -» Z be continuous and let g:y -» Z have a closed graph. Then g.f e U(X,Y) . Proof: Let G^ „ be the graph of g°f and let (x ,z ) be a net In G „ such that v(x , z/ ) -»v (x5 .z ) . Then got a-a o o' - 44 - x -> Xq and by the continuity contition, ^( a) ~* ^(-^o^ ' Hence (f(K K ox ),zo ) e Gg . Hence zo = <=*\p;(f( \ xo' )') and therefore G ,. is closed. cr o i Theorem 6.3 Let X.Y,Z be topological spaces such that f(X) is compact and Z is' Tg . Let f e U(X,Y) and let g:Y -• Z be continuous, then g°f e U(X,Y) Proof: Since f(X) is compact, f is subcontinuous. Hence by Theorem 3.1 that f is continuous. Since Z, is T^ the same , theorem yields that g has a closed -graph. Hence by Theorem 6.2 g°f e U(X,Y) ... Addition and Multiplication We shall show that, in general, U(X,Y) is not closed with respect to either addition or multiplication. . Example 6.4 . Let X = [0,1] and Y = , both with the Euclidean metric. Define functions f:X -> Y and g:X -. Y as follows: f(x) - *t ° -g(x) - {"^ * t % Then f and g both have closed graphs but (f+g)(x) = ^ - 0 does not have a closed graph. Since if x e X , x -* 0 , a a x 4 0 for any a then . (f+g)(x ).= 0 for every a and hence (xa,(f+g)(xa)) - (0,0) fGf+g . . - 45 - Example 6.5 Let X = [0,1] and Y = (-«>,+») and define /x g x) X f and g as follows: f(x) = {J * t °Q > < = for all x e X . Then f(x)-g(x) = |J * = 0 xsrhich does' not have a closed graph in XxY . Although f(x)«g(x) need not belong to U(X,Y) for f,g e U(X,Y) it is true that U(X,Y) is closed with respect to multiplication by a constant if Y = (-«>,+<») . Theorem 6.6 If f e U(X3E1) and c e E± then cf e U(X,E-L) Proof: For c = 0 the theorem is true, so assume that c e E1 , c 4 0 and f e U(X,Y) . Suppose that cf k U(X,E1) . Then there is a net (x ,y ) e G „ (where y = cf(x )) such that (v x 5J,y / ) -* (x ,,Jy ) Y I G » . i.e. y ^ cf(x ) and hence a a ^ o o cf * o. o' f(v x ) ^ y / . But we then have that v (x,f( v x )) -»K (xiJ ,y y ) YA o ^o/c a* a'' o o/c' which contradicts the fact that G^ is closed. Hence c-f e U(X,EX) . Maximum- and Minimum If f,g e e(X,Y) it does not follow that either max (f,g) or min(f,g) need belong to U(X,Y) . Example 6.7 Let X = [0,1] and Y = (-»,+») . Define f and g - 46 - 1//x X 1 and x 0 for a11 as follows: f(x) = (t -1 x f- o°'0 "' s( ) = x e X . 1 Then min (f,g) = -[^ °£t ^Q' ^ which does not . have a closed graph in XxY . Theorem 6.8 If f,g e U(XJ,E1) and both f and' g are subcontinuous then max (f,g) and min (f,g) belong to U(X5E1) . . Proof: By Theorem J>.1 both f and g are continuous. Hence both max (f,g)- and min (f,g) are continuous. Since E^ is Hausdorff then by Theorem J.l we have that both max (f,g) and min (f,g) belong to U(XaE1) . Absolute Value • Theorem 6.9 If f:X - E1 and f e U(X,E-L) then |f | e U(X3E1) . Proof: Let f e U(X,E ) and let (x ,y ) be a net in i. a a G|F| such that (x&,ya) (xQ,yo) . If f(x&) > 0 for infinitely many a e A , i.e. for a = a^ , •k = 1,2,3,..., then y = f X = f x f x = f x and hence ' aa ' (a a )!' -( a ) ~ ( J = 'l ( JI k k k ° ° ° f x 0 for (xQ,yo) e G|fj . If on the other hand ( a) < infinitely . - 47 - many a e A then y - |f(x ) | = - f(x - - f(x ) = -y k k k = |f(xQ)| and again (x.Q,yQ) e Gjfj . Therefore |f| e U(X,E1) Convergence Pointwise The following example will show that U(X,Y) need not be closed with respect to pointwise convergence. Example 6.10 n Let X = [0,1] and Y = E-^ and define fn(x) = x for n = 1,2,33... and x e X . Then f e U(X,E-L) for all n . However f(x) = lim f (x) does not belong to U(X,E^) n-*t» since f(x) = .{$> * f [°>^ Almost Uniform n € A sequence of functions {f"nl ^ } converges almost uniformly to f on a space X if [f |n e Z+] converges uniformly to f on all compact subsets of X . Theorem 6.11 If fn e U(X,Y) , n~l,2,3,... , where (x,p) and (Y,a) are. metric spaces and f -» f almost uniformly on X then lim f = f e ,U'(X,Y) . n-»ro Proof: Assume that ff In e Z+] converges to f almost n 2 D uniformly on X and that f e U(X,Y) . Let % be the metric - 48 - 2 2 2 for XXY , then # = p +a . Since f.tt U(X,Y) there x n exist elements ( n^yn) ^ &f such, that x ..(xn,y ) -» ( 0,yo) 4 . Then there exist positive numbers '&1 and 6g such that S((xQ.yo),'61)nS((xQ.f(xQ)), 62) = 0 1 As . fv (x ') _> fx (x ) there exists a k, such that for all ' n o' o' 1 x 6 and hence for a11 k k k >_ k1 , a(fk(xo), f( 0)) < 2 2_ i ' 2 •6((x0.f(x0)),(x0,fk(x0))) = ^a (f(x0)/,fk(x0)) < S2 . Therefore (xo,fk(xQ)) e S((xo,fk(xo)), 62) for all k > ^ . 2 Since f -» f almost uniformly on' X , then K' = [xo,x-jXgj...] is a compact subset of X . Then for every integer k there is an n^ (n1 CT(f (x )..f(x )) < 1/k for p = 0,1,2,... . k y L It follows from the above that If. p -> co .then >(((xo,yo), (xp,f (xQ))) < 1/k and hence there Is a k ' such that for all k>k„ (x ,fv (x1 )) 2 — 2 ^ o' nfc o ' e S((xo,yo),oi ) . Hence from 1 (x -f (x )) 4 S ((x , f (xQ) ),'p 6 k • for k_>kg which contradicts 2 . The following v/ill show that the converse of the preceeding theorem Is not necessarily true. Example 6.12 Let X = [0,1] and define f = -f1 / = 0 n Lx-1/n x e (0,1] for n - 1,2,3,... . Then lim f (x) = f(x) = 1 for n-+<= all x e X . Now f e U(X,E1) and f e U(X,E1) for all n- _ 49 - But since sup Ix^^-l | = +» it follows that convergence is xeX • . not almost uniform. • In what follows if [X |a e A), is any system of topological spaces, then X X will-, be the Cartesian product GeA a of the sets X and TT X will be the topological product a . aeA a supplied with the product topology. The following is a well known lemma for which no proof is given. :: .6.13 Let [X |a e A] be a system of topological spaces and. set X = TT x • The net {xb|b e B] of points in. X aeA a D x x a where x = i a|a e A} converges to x = ( al e A] iff for each aeA the net [x^ jb e B, x^ e X&} converges to x& . Theorem 6.14 Let '(XQ|a e A] be a system of topological spaces and on a space. X define functions f :X -» X for each ^ o a o a a e Xa . Let Gf • = {(x,f (x))|x e X } be closed sets in a .TT{X |s = 0,a ; a e A] . Then the function f:X -» XX s ° aeA a defined by f(x) = (f^(x),f2(x),...} has a graph Gf that is closed in X x(X X ) . aeA Proof": We shall show that G^ cz Gf . Let x e G^ , x = (v x .fx 1la e A]) for some x e X '. There is a net' o <• a ' ' o o - 50 - b b {x |b e B, x = (xj,{xj|a'e A})} of points in Gf - {x} such that xb -* x . Note that x° = f (xb) for all aeA a a.v oy By the lemma 6.13 a xb x for all aeA. Hence a a b b f v(x ) •* x for all aeA. Also x -• x . Therefore a o. a •• o o b b (x ,ff (x )|1 a e A}) - v (x bJx la e J A}) x = (x , ff (x )|a e A}) . ^o^a^o- , o a o ^ a o Since G„ is closed for all aPV, (x , ffv ( x )|a e J A}) f o^a o a = (x ,f(x )). Hence G„ is closed. v o v o I Theorem 6.15 Let [X la e A] be a system of topological spaces. Let X be a space and for each aeA define f :X -» X o ^ a o a "co be subcontinuous. Define f:X -» X X as m Theorem 6.14. o A a aeA Then f is subcontinuous. Proof: Suppose f Is not subcontinuous. This implies that there is a net fxb|b e B] in x such that fxb1 -» x e X o1 o L oJ o o but ..o subnet of {f (x ) |b e B] converges in x X . Since aeA a ° b x b a, for any fixed b e B 3 f(xQ) = Cfa( 0)l e A] this means that for at least one aeA, ff (xb)|b e B] has no convergent a o subnet in xa (by Lemma 6.13) . But this contradicts the fact that f is subcontinuous for each aeA. a Combining Theorems 3»1* 6.14, and 6.15 we see that If f is defined as in the two theorems above then f is continuous. - 51 - BIBLIOGRAPHY 1. Dugundj'i, J., Topology, Allyn and Bacon Inc., Boston, .1966. 2. Fuller, R.V., Relations Among Continuous and Various Non~Contifiuous Functions, Pacific Journal of Mathematics, Vol. 25, No. 3, (1968), PP 495-509. 3.. Husain, T.,- Almost Continuous Mapping, Roczniki- Polskiego Towarzystwa Matematycznego, X, (1966), pp 1-7. 4. Kostyrko, P. and Salat, T., o &VHK.UHax., rPA$b! KQToPblX Casopis Math., Vol. 89, (1964), pp 426-4J1. i 5 • Ci ^VHKUHflX. PPA$bl ^OTOPblX flBTTjaiOCfl .^AMK.HYh MHO^ECTBAMM JL ? , Acta fac. rer. natur. Univ. Comenian, Math 10, No. 3, (1965), PP 51-61}. 6. Kostyrko, P., A Note On Functions ¥ith Closed Graphs, Casopis Math., Vol. 9"4, (1969), PP 202-205. 7. Long, P.E. , Functions With Closed Graphs, Mat. Montbliy, .Oct. 1969, PP 9~30-932. 8. Mrowka, S., On The Convergence of Nets of Sets, Fund Math. 45 (T958), PP 237-2T6". :