FINE TOPOLOGY ON FUNCTION SPACES

This paper studies the topological properties of two kinds of fine topologies on the space C(X,Y) of all continuous functions from X into Y.

used to obtain certain kinds of embeddings into infinite-dimensional manifolds.
This oaper studies the topological pro_erties of two kinds of "fine topologies" on C(X,Y).In order to avoid pathologies, all soaces will be Tychonoff soaces.

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The symbol ]R will denote the real line with the usual tooology, and lR will denote the positive real line.Also C(X, ]9) and C(X, ]R +) will be abbreviated as C(X) and + C (X). Finally let denote the set of natural numbers.I. UNIFORM TOPOLOGIES.
Whenever a space Y has a comoatible uniform structure on it, this induces a uniform structure on C(X,Y).If 6 is a diagonal uniformity on Y, then for each Dsg, define {( )c(x,) for every xX, (f(x),g(x))D}.

(i.I)
The family { Ds} is a base for a diagonal uniformity on C(X,Y).Denote the resulting topological space by C6(X,Y).On the other hand, if u is a covering uniformity on Y, then for each Cs, let ={(f,g) s C(X,Y) 2 for every xsX, there exists a UC with (f(x),g(x))sU2}. (1.2) The family { -Cs} is also a base for a diagonal uniformity on C(X,Y).Denote this spaee by C (X,Y).
There is a natural way of passing from a diagonal uniformity 6 to a covering uniformity U, so that U6 U and 6 6 (cf.Willard [5] section 36).It can be easily verified that Cu,(X,Y) =C(X,Y) and C6u(X,Y) C(X,Y).Therefore a uniform structure on Y may be considered either as a diagonal uniformity or a covering uniformity, and the resulting uniform structures on C(X,Y) will generate the same topology, called the uniform topology.Let stand for the fine (covering) uniformity on Y. Whenever Y is paracompact, has as a base the family of all open covers of Y.The topology on C (X,Y) will be called the fine uniform topology.
If u is any comnatible uniformity on Y, then the relationships between the various topologies discussed above are given by c (x ) _< Ck(X ) _< C (X,) _< C(X,Y) where the inequality means the space on the right is finer than that on the left.
Each compatible bounded metric p on Y induces the supremum metric p on C(X,Y), defined by p (f,g) sup{p(f(x),g(x)) xsX}.The resulting topological space will be deDoted by Cp (X,Y).A base for C (X,Y) consists of the metric balls + {B (f,e) faC(X,Y) and ea 19 }.If u is the uniformity on Y generated by p then c (x,z) c (x,).PROPOSITION i.i.If Y is metrizable, then C (X,Y) has as a base {B (f,e)" eM(Y), feC(X,Y) and ea19+}.
PROOF.To see that B (f,e) is a neighborhood of f in C (X,Y), define the open cover {B(y,e/3) yaY} of Y, and let ga[[f].Then for each xaX, there is a yaY with (f(x), g(x)) a B (y, e/B).Therefore each (f(x),g(x))<2e/3, so that p(f,g)s2e/3<e.This establishes that gab (f,e), and it follows that [f]cB (f.e).P >* * P On the hand, let a and feC(X,Y).Let i 2 > be a normal sequence of open covers of Y so that UI refines , and let p be the metric defined by this sequence (see Willard [5], P. 167, for this construction).It follows from the construction of + p that there is an edR so that B (f,e)c f].p Let ube any comDatible uniformity on Y.If X is compact then u on C(X,Y) is the same as the uniformity of uniform convergence on comnact sets, which is known to generate the comDact-open topology (cf.Willard [5], section 43).The converse is in fact also true for most Y. PROPOSITION 1.2.If Y contains a nontrivial path, then for any compatible uniformity u on Y C (X,Y) =Ck(X,Y) if and only if X is comDact.
PROOF.Let "I +Y be a continuous function from the closed unit interval into Y such that (0) / @(i).Let y =(0), let z =(i], and let f be the constant mare from X to y. Define [l={Y\{y}, Y\{z}}, which is an open cover of Y.To see that au, let Da6 with zD[y].Choose a symmetric Ea6 with EoEcD.It follows that {E[p] paY} refines 8, so that Suppose that X is not compact.To see that [f] is not a neighborhood of f in Ck(X,Y) let W [AI,V 1 ]n...n [An,Vn] be any basic open subset of Ck(X,Y) which contains f, where each [Ai,Vi] {feC(X,Y) f(Ai)cVi} and the A. are comDact and the 1 V. are open.If A A 1 u u A then there is some xeX\A.Define g:Au{x} Y 1 n by g(a) y for aeA and g(x) z. Since I) is arcwise connected, then g extends to some geC(X,Y)._Then eW\6[f], which establishes that C (X,Y) is finer than Ck(X Y)  In order to show that C0[f] U If], let geC0[f] and xeX.Then there exists an i such that f(x)eDxl [f(xi) ].Let UeC 0 with (f(x), g(x))eU 2. There is some Uiexl such that UcUi, so that (f(x),g(x))eU.

Xl
The topology generated in Proposition 1.3 is in general not the compact-open topology.
The "comnleteness" of a function space can be a useful property for obtaining the existence of certain kinds of functions.If 0 is a complete metric on Y, then 0 is a complete metric on C(X,Y).On the other hand, C (X,Y) may not be metrizable.So comnlete metrizability of C (X,Y) is too much to exnect in general.But C (X,Y) does have "completeness" to the following extent.
THEOREH 1.4.If Y is completely metrizable, then C (X,Y) is a Baire space.
PROOF.Let 0 be a compatible bounded comnlete metric on Y. Also let {W n e} be a sequence of dense open subsets of C (X,Y), and let W be a nonempty n open subset of C (X,Y) Choose d I eM(Y) fl eC(X,Y) and 0< 1/2 so that v 1 (fl'e )cWInW.Define 0 max {0,d I}, so that B (fl'el)C B d (fl,e).Continue by Bd 1 1 1 en+l<l/2n+l induction so that at the n+l step, choose dn+lM(Y fn+IC(X,Y., and 0< such that Bdn+-fn+l n+l CWn+IoBo fn en/2 and define 0n+I max(0 n dn+l }" Now {f ne)is a Cauchy sequen n ce in C (X,Y), and therefore converges to some n feC (X,Y).Also for each ie, {f ne} will converge to f in C (X,Y), so that o n f is in the closure of BOi(fi,e/2 in Ci(X,Y).Therefore feBi(fi'ei Bdi(fi,ei WiW for every The conclusion of Theorem 1.4 cannot be strengthened to Cech-completeness, which can be seen from Theorem 4.1.Also the hypothesis cannot be changed to Y being compact. For examDle, if X is the closed unit interval I with the usual topology, and Y 12 with the order topology with respect to lexicographic ordering, then some functions in C (X,Y) have onen neighborhoods which can be written as countable unions of nowhere dense sets.

FINE TOPOLOGIES.
Throughout this section, (Y,0) will be a metric space.In this case there is a natural way to generate a topology on C(X,Y) which may be even finer than the fine uni fortuity topology.
From Propositions 1.3 and 2.1 it follows that if X is psuedocomact then Cf (X,Y) Cv(X,Y).But even if X is not Dsuedocompact,_ Cfo(X,Y) can still be reated to C (X,Y) as follows.
PROPOSITION 2.2.If X is paracompact, then C (X,Y)-<C (X,Y).v+ fp PROOF.Let dM(Y), let fC(X,Y), and let e ]R For each xX, there exists a (x) ]R such that B (f(x),(x))cBd(f(x),e/2). Let C =(Ua A) be a star- refinement of (f-l(BP(f(x),(x)/2)) xX).Let ( A) be a artition of unity p subordinated to C. For each A, let n be the cardinality of (A UnU), let xX be such that Ua f-l(B (f(x),(x))) and let m min((x)/2 U nU p ( / Then define Z((m /na) A}, which is a member of C (X).
The inequality in Proposition 2.2 is in general not an equality as indicated in the comment after Corollary 3.5.
Whenever 0 is complete then Cf(X, Y) is a Baire sace ([4], p.297).Instead of giving a proof of this here, a proof will be given in the next section that C (X) f0 is psuedo-complete, which is a property stronger than being a Baire space.
3. REAL-VALUED FUNCTIONS.For the rest of the paper, 0 will denote the usual metric on R bounded by i; that is, O(s,t) rain(l, s-tl}. + + LEMiA 3.1.Let f,gsC(X) and let @,@sC (X).Then the closure of B (f,) in + Cf0(X) is contained in B0(g,@) if and only if for each xsX the closure of B0(f(x),(x)) in R is contained in B (g(x),@(x)).+ PROOF.For the sufficiency, let hsB (f,) and let xX.Suppose h(x) were contained in the complement of the closure of B (f(x),(x)); call this set V. Then the set of functions taking x into V would be a neighborhood of h in Cf (X) which + misses B (f,).This contradiction shows that h(x) cl(B (f(x),(x))) pc B (g(x),O(x)).
PROOF.For each n, let C (X) (sC+(X) for all xsX, (x)<I/2n}, n + a (X) It n (B(f ) fC(X) and C+X/}.Each is base for Cf0 and define n n remains to show that if Bn n for each n with cl(Bn)CBn then {Bn:n} .If each Bn=B(fn, n) then by Lemma 3.1, for each ns and each xsX, cl(B(fn+l(X)'n+l(X)))cBp(fn(X)'n(X))" Since each Cn(X)<i/2n, then [(B (fn(X), (x)) n} (f(x)} for some f(x)s JR.This defines the function f, which n is the uniform limit of (f n}, and is hence continuous.Clearly n n as desired. The algebraic structure on ]R induces an algebraic structure on C(X).This structure interacts well with some topologies on C(X).For example_ C k(X) and C(X) are always locally convex linear topological spaces.On the other hand C (X) is only a topological group under addition while the scalar multiplication operation is not continuous for non-compact X.The space C (X) behaves much like C (X) in this f regard.It is.straightforward to show that Cfo(X) is a topological group under addition.As a result, Cfo(X) is homogeneous; and for many arguments it suffices to consider only basic neighborhoods of the zero function, f0" The next result establishes when f0 has a countable base.It is stated for (JR,o), but it is also true for any metric sace containing a nontrivial path.
PROPOSITION 3.3.If X is normal and f0 has a countable base in Cf0(X), then X is countably compact.
space Y, let M(Y) be the family of all comnatible bounded metrics on Y.The following fact gives a useful tool for working with the fine uniform tODology.
It follows from Proposition 1.2 that whenever X is compact, all compatible uniformities on Y generate the same topology on C(X,Y) the compact-open topology.Also when Y is compact, a compatible uniformity on Y generates a unique topology on C(X,Y) since there is only one compatible uniformity on Y but in this case the topology on C(X,Y) is not the compact-open topology, unless X is also compact.PROPOSITION 1.3.Let Y be a metrizable space.If X is psuedocompact, then all compatible uniformities on Y generate the same topology on C(X,Y).PROOF.Let U be an open cover of JR, and let feC(X,Y).For each xeX, let Now there exists an C0eu which refines each of