Conditions of angelic type in function spaces

Pub . Mat . UAB Vol . 30 Ns 1 Maig 1986 CONDITIONS OF ANGELIC TYPE IN FUNCTION SPACES by Miguel Barceló  and Miguel A . Canela ABSTRACT : This paper deals with a class of topological spaces in which a-compactness and compactness coincide and the tightness cf a compact subset is less or equal than a, a being an infinite cardinal number . This class is a natural extension of the class of strictly angelic spaces, introduced by W . Govaerts . Sufficient conditions are given for a space of continuous functions to belong to this class, and some results on locally convexspaces are cbtained as an application . 1 . Introduction - The aim of this paper is to give a description uf a class of topological spaces in which a-compactness and compactness coincide, a being an infinite cardinal number, which will remain fixed throughoutthis paper . This class is a-productive (closed by products of families of card .iriality <a) . The class described will be denoted by Aa , where the "A" stands for, angelic . Indeed, for a = a 0 , our class coincides with the class of strictly angelic spaces introduced by W . Govaerts [2] . These spaces are a subclass of the angelic spaces studied by Pryce in [5] . The properties of the class of angelic spaces can be found in In.Section 2 we recall some topological notions which will be used throughout this work . In Section 3 we introduce the class A n , proving some 'properties . The main result is the stability for products of cardi nality < a, and the key for this result is a theorem cf V .I . Malyhin [41 . In Section 4 we give sufficient conditions for a space of continuous functions to belong to A , , endowed with the topology cf pointwise convergence . In Section 5 we restrict ourselves to locally convex spaces, obtaining with the tools given in the previousSections some results appeared in a former paper by M . Valdivia [6] . 2 . Some topological - notions .- a will denote a fixed infinitecardinal number . A subset S of a topological space X (all the spaces involved are Hausdorff) is acompact every net (xcontained in S, with  III < a has a cluster , point x E S .  S  is  relatively  acompact  if every  net  (x i ) iEI'  with  III <a  has a cluster  point  xEX .  Every  (relatively)  compact  subset  is  (relatively) a-compact, but not conversely . A counterexample can be obtained modifying the usual example of a sequentially compact space which is not compact [1, 1 .2(7)] . if If X is a topological space, the tightness of X, t(X),is the minimal cardinal number v with the following property : if S is a subset of X and xE cl(S)  is a closure point of S,  there is a subset  MC S  with 1M1 < v and  x E cl (M) . . The  density  character  of  X,  d (X), is  the  minimal cardinality of a . dense subset of X . The weight of X, w(X), is the minimal cardinality of a basis of open subsets for the topology of X . The weight  of  X  at  the  point  x E X,  w(X),  is  the minimal  cardinality of a x  > basis  of neighbourhoods of x . Clearly wx (X) < w(X),  but not conversely . - For there and cther cardinal functions of the General Topology, [3) can be used as a standard reference . 3 . The class A a .- The class A a will be the class of all topological spaces X satisfying the following conditions : (i) Every relatively a-compact of Xis relatively compact . (ii) Every compact subset cf X has tightness < a . (iii) If a subset SCX is ccmpact and d(S)< a, then w x (S) < a for every xES . We give next the properties of this class . 3 .1 . Proposition : The condition (ii) of the preceding definition can be replaced by : (ii)' If SCX is relatively compact and xEcl(S), there isanet (xi)iEI contained in S, with  III < a and lim x i = x, obtaining an equivalent definition . Proof :  (ii) '  implies  (ii) .  Conversely,  if t(S) < a  for a compact subset SC X,  we  take  MC S  with  x E cl (M)  and  1 M1 < a .  Then cl (M)  is  compact  and has density character < a, and, by (iii) the weightat x is < a . Therefo- If X is a topological space, the tightness of X, t(X),is the minimal cardinal number v with the following property : if S is a subset of X and  xE cl (S)  is  a  closure  point  of  S,  there  is  a  subset  MC S  with ¡MI < v and  x E cl(M) .  The  density  character  of  X,  d(X), is  the  minimal cardinality of a dense subset of X . The weight of X, w(X), is the minimal cardinality of a basis of open subsets for the topology of X . The weight of X 2.t the point xEX, wx(X), is the minimal cardinality of a basis  of neighbourhoods of x .  Clearly wx(X) < w(X),  but not conversely . - For there end cther cardinal functions of the General Topology, [3] can be used as a standard reference . 3 . The class A a .- The class A a will be the class of all topological spaces X satisfying the following conditions : (i) Every relatively a-compact of X is relatively compact . (ii) Every compact subset cf X has tightness < a . (iii)  If  a  subset S C X  is  compact and d(S)< a,  then w x (S)< a for  every XES . We give next the properties of this class . 3 .1 . Proposition : The condition (ii) of the preceding definition can be replaced by : (ii)' If SCX is relatively compact and xEcl(S), there isanet (xi)ic-I contained in S, with  III <a and . lim x i = x, obtaining an equivalentdefinition . Proof :  (ii) ' implies  (ii) .  Conversely,  if t(S) < a  for a compact subset SC X,  we  take  MC S  with  x E cl (M)  and  1 M1 < a .  Then  cl (M)  is  compact  and has density character < a, and, by (iii) the weight at xis <a . Therefo52 We have (i) . Suppose  that  S C U X j is  compact,  and  we  are going to show that ¡EI t(S)< a . We can suppose, without loss of generality, S =  ni(S) . ¡El According to a result of V .I . Malyhin [4, Theorem 4), the finite product of compact spaces with tightness <a has tightness < a . If x is a closure  point  of a subset D C S,  we can  find,  for each  J C 1  finite,  a subset MJC rr J (D) with 1MJ1 < a and iT J (x)E cl(MJ) . We can choose now M J CD with nj(MJ ) = MJ and x E cl(M) . We have (ii) . Finally,  if  S C U X 1 .  is  cortrpact  and d (S ) < a ,  then  d (ir i (S) ) <a,  and ¡El - thus n (S) has weight < a in every point . Keeping in mind the construci  - tion of a basis of neighbourhoodsfor the product topology, and recalling III< a, it is easy to see that 11 n 1 .(S) has weight < a at eveiE I ry point . Q .E.D . and 1 M J j<a . Actually, 4 . Spaces of continuous functions .- MU M J J has cardinality <a We are going to see that certain a .ssumptions on a topological space X imply that the space C(X) = C(X,1R) of continuousreal functions belongs to the class A a , endowed with the topology of pointwiseconvergen ce W X . Indeed, we obtain results analogous to those obtained by W . Govaerts [c]  for the case a =¡{ 0 . 4 .1 . Proposition :. Let X be a compact space . Then (C(X),w x ) belongs to Aa . It is well-known that every subset S C C(X), wX relatively countacompact, is wX -relatively compact and countable tightnéss (e .g .[1]) . we can, replacing if necessary X by a suitable quotient, Proof : bly To check (iii), suppose that S separates the points of X . If S it wX -compact and DCS is dense,  with  ID 1 < a,  X  admits a basis of uniformityof cardinality < a , and hence d(X) < a . Reversing the argument, we have a set of continuous real functions, of cardinality <a , on S, which separates the po .ints of S, and therefore w(S) <el . Q .E .U . 4 .2 .  Proposition :  Let X be a topological space with a dense subset D C X which is relatively a-compact . Then (C(X),w x ) belongs to Aa . Proof : Conditions (i) and (ii) can be obtained as in 4 .1 from known results (see [1]) . To check (iii), we remark that, if S C C(X) is w X -cornpact, we can consider . the mapping 01 : X - C(S) defined by 0 1 (x)(f) _ = f(x),  which is continuous with respect to the topology w S .  Then 0 1 (D) is relatively a-compact in (C(S),w S ), and its closure FI is a compact subset of (C(S),w S ) . Now m2 : S 1 C(H), defined in an analogous way, is injective and continuous, and therefore a homeomorphism, and, using 4 .1, we are done . Q .E .D . 4 .3 . Proposition : Let X,Y be topological spaces . a)  If X  admits  a  dense  subset D C X such that  (C(D),w D )  is  in A,,,  then (C(X),W X ) is in A a . b)  If X  U  X i ,  where  (C(Xi),wx .)  is  in  Aa and  111< a,  then  (C (X),wX) iEI  1 is in Aa c) If 4) : Y - " X is continuous and surjective, and (C(Y),w y ) is in A a , then (C(X),w x ) is in Al . Proof : For a) Take the restriction map C(X) - C(D) and epply 3 .3 . For b), construct an injection C(X) - HC(X .) in a natural way and apply iEI 1 3 .4, followed by 3 .3 . For c), take the mapping 0* : C(X) - C(Y) defined by ~*(f) = fo~ and use 3 .3 . Q .E .D . 4 .4 . Corollary : Let X be a topological space with a family (X i ) iEI of relatively a -compact subsets  such  that  its  union  is  dense  and  ~11< a . Then (C(X),W X ) is in Aa . Finally, we have the following result, which is a natural extension of a theorem of D .H . Fremlin ([1,3 .5], [2, Proposition 9]) : 4 .5 . Proposition : Let X be a topological space, and Z a metric space, and suppose that (C(X),w x ) is in á . Then (C(X,Z),wx) is in A a . Proof : The argumentgiven in [2) for the countable case can be used . Q .E .U . 5 . Applications to locally convex spaces .- The results which have been stated here have a purely topological nature . Nevertheless, some particular cases have been proved by other methods, for instante for the case of a weak topology on a locally convex case . If X,Y are real locally convex spaces, the space L(X,Y) of continuous linear operators is a subspace of the space C(X,Y) of continuous functions,  closed with respect to wX .  The  topology induced by  w X  on L(X,Y) is usually called simple topology . From 4 .5 and ° .2, we obtain directly : 5 .1 . Proposition : Let X,Y be locally convex spaces, X being the union of a family (Di)iEI of relatively a-compact subsets, with 111< a, and Y metrizáble .  Then L(X,Y), endowed . with the simple topology, is in Al . We can consider, in particular, the case in which Y =IR and X is the dual E', endowed with the weak topology a(E',E) . We obtain thus the followingresult : 5 .2 . Corollary (M . Valdivia [6]) : Let E be a locally convex space,  E' being the union of a family (D i ) ¡ EI of a(E',E)-relatively a-compact subsets, with I II <a . Then every weakly (relatively) a-compactsubset of E is weakly (relatively) compact . NOTE : In [6], this result is stated under an extra assumption, the convexity of the D1 .'s, but this assumption is superfluous . Using 5 .1 and 3 .3, we have : 5 .3 . Corollary (M . Valdivia (6Í) : Let E be a vector space and T and T' two locally convex topologies on E, T finer than T' . Suppose that T' admits a zero-neighbourhood basis of cardinality < a , and denote by E' _ _ (E,T)'.If A C Eis a(E,E')-(relatively)a-compact, A is a(E,E')-(relatively) compact . FINAL NOTE : We have described a class of spaces in which compactness and a-compactness are the same, with good stability properties, which allows us to obtainthe results of this Section . It can be remarked that almost all is the same if we replace condition (iii) of the definition by the following  stronger condition :  If S CX  is compact and  d(S)<a,  then W(S) ~ a .  Nevertheless,  the  class  obtained  in  this  way  will .b e more restricted, because there are separable, first countable compact spaces which are not metrizable (e .g . the Helly compact) . REFERENCES 1 . K . Floret : Weakly compact sets . Springer, Berlin-Heidelberg-New York, 1980 . 2 .  W . Govaerts : A productive class of angelic spaces . J . London Math . Soc . 22 (1980), 355-364 . 3 .  I . Juhasz : Cardinal functions in Topology . Math . Centre Tracts 34, Amsterdam,1971 . 4 . V .I . Malyhin : The tightness and Suslin number in exp X and in a product of spaces . Soviet Math . Dokl . 13 (1972), 496=499 . 5 . J .D . Pryce : A device of R .J . Whitley's applied to pointwise compactness in spaces of continous functions . Proc . London Math . Soc . (3) 23 (1971), 532-546 . 6 . M . Valdivia : Some criteria for weak compactness . J . reine angew . Math . 225 (1972), 165-169 . Rebut el 15 d'ac tubne del 1985 MiguelBarceló EscolaTécnica Superior d'Enginyers Industrialsde Barcelona . Avinguda Diagonal Barcelona SPAIN . MiguelA . Canela Departament de Teoria de .Funcions, Universitat d Universitat de Barcelona Gran Via, 585 08007 Barcelona SPAIN .