Maihematical Notes, Voi 66, No. 1, 1999

A Topological Structure on a Set of Continuous Functions With Various Domains S. A . D r o z d o v s k i i

UDC 515.122.55

ABSTRACT. W e construct a topological space of continuous functions which generalizes the previously studied space of functions defined on dosed intervals. For the n e w space, metrizability properties are studie~ 'r1"e results can be applied in the topological theory of ordinary differentialequations. KEY WORDS: topological function space, continuous functions, uniform convergence, metrizability properties, separation axioms, local compactness, countability, ordinary differential equations, Vietoris neighborhood.

The new topological function space F(U) studied in the present paper is a generalization of the previously known spaces Cs(U), C ~ ( U ) , C~'(U), and C~(U) (see the definitions below), which are fundamental in topological theory of ordinary differential equations. The space C~(U) is described in detail in [1, 2], C+(U) and C~-(U) were used in [3], and C~(U) was considered in [4]. In these papers, one can also find various theoretical results obtained with the help of these spaces. The set F(U) is the union of the sets C~(U), C + ( U ) , C~-(U), and C f ( U ) , and the topologies induced on these sets by the topology of F(U) coincide with their own topologies. The space F*(U), considered at the end of the paper, generalizes a similar construction used in [4]. We study metrizability properties of F(U) and subsets of it. w

The space

F(U)

Let U be an open subset of the product R x M , where R is the set of real numbers and M is a metric separable locally compact space. We consider continuous functions from R to M that axe defined on various closed, half-open, and open intervals and whose graphs are subsets of U. In the following, by Ca(U), C~(U), and C~'(U) we denote the sets of all continuous functions of the form z: [tl,~2] -'+ M , z : [t0,t) -~ M , and z : (t, to] --+ M , respectively, whose graphs axe closed in U; here t m a y be infinite. By C~(U) we denote the set of all continuous functions z : (tl,t2) -+ M whose graphs axe closed in U. Next, we define F(U) to be the union of Ca(U), C~(U), C~-(U),and C~(U). We shall equip F(U) with a topology such that the corresponding convergence is a natural generalization of uniform convergence for functions defined on a closed interval. Let z 6 C , ( U ) . A sequence {zi : i < w} C F(U) converges to z if and only if there exists an i~ such that z~ E Us(U) for all i > i l , the family {z~ : il < i < w} is equicontinuous, and the following property holds for each dosed interval I C (~r(z)), where ~r(z) is the domain of z, and each open interval I D ~r(z): there exists a n i2 s u c h that I C_~r(zi) C I" for i > i2 and IIz~lx - z l i l l ~ 0 a s g - ~ co, where II II is the uniform norm. Let z E C~(U). A sequence {zi : i < w} converges to z if and only if there exists an i~ such that zi E Ca(U) U C ~ ( U ) for all i > il and the following property holds for each closed interval I C /~r(z)/ and each open interval I" ~ 7r(z): there exists an iz such that ! C_ 7r(zi) C I" for all i > i2, the family {zil~(;,)n(-oo,,up q : i2 < i < w} is equicontinuous, and II lz - zlzll 0 as i co. The convergence to elements of C~-(U) is defined in a similar way. It suffices to replace all symbols " + " by " - " mad the half-open interval ( - c o , sup/] by the half-open interval [inf I , +co) in the definition

of

c,+(u). Translated from Matema~icheskle Zame~hi, Vol. 66, No. I, pp. 76-88, July, 1999. Original artlde submitted July 5, 1996; revision submitted May 20, 1997.

62

0001-4346/1999/6612-0062522.00

(~2000 Kluwer Academic/Plenum Publishers

For z E C ~ ( U ) , a sequence {zi: i < w} of elements of F(U) is convergent to z if a n d only if for each closed interval I C ~r(z) there exists an il such t h a t I C_ 7r(zi) for all i > il a n d Iiz~lx - zixll -+ 0. Now let us describe a base of the topology on F(U) corresponding to the convergence thus defined. First, for each z E F ( U ) we introduce sets c , ( z ) , c+(z), and c;-(z). Let z E C,(U) a n d ~r(z) = [tl,t2]. T h e n !

I

c.(z) = {zli~; s;]: tl < tl < t2 < t~},

c+(~) = {=lt~,,,]: t~ < t < t~},

c:(~) = {~l{,,~2j : t~ < t < t~}.

If z 9 C+(U) a n d ~r(z) = [tl, t2), then c,(=) = {zbi,~al : tx < tl _ el < t~},

c~+(=) = {=1[,,,,i : t~ < t < t , } ,

~7(~) = ~ .

c : ( z ) = {zl[,,,~] : tl < t _< t , } ,

~,+(z) = o .

If z 9 CZ(U) a n d ~r(z) = (tl, t2], t h e n ~,(z) = {zli~isa] : h < t~ < t~ _< t~}, For z 9

C~(U) with

~r(z) = (t~ , t : ) , we set

, , : tl < t '1 <_t:' < t2}, e, Cz) = { z I[t,,t21

c,+(z) = c:(z) = 0.

In the following, we denote the graph of a n element z 9 F(U) by the same s y m b o l z. As was defined in [1], a base o/ neighborhoods of a point z 9 C,(U) consists of sets of the form G(z, Y) = V ~ = C,(U) n V, where V is a Vietoris neighborhood of z in exp(U). Let z 9 Cs+(U). We define an element of a base of neighborhoods of this point by

O ( z , v , v ) = y + = {~ 9

o.(u) u

c ~ + ( u ) : c.+(~) n r # o } ,

where V is a Vietoris neighborhood of some element y 9 c+(z). A neighborhood of a point z 9 CZ(U ) is obtained from G(z, y, V), z 9 C+(U), by replacing all symbols " + " by " - . " For z 9 C~(U), we define a neighborhood by the formula

a ( z , y , V ) = V + = {x 9 F ( U ) : c , ( x ) ~ g # 0 } , where V is a Vietoris neighborhood of y 9 c~(z). Note t h a t the subsets C,(U), C~(U) U C+(U) , and Cs(U) U C7 (U ) are open in F(U). Proposition

1. The space F(U) has a countable base.

P r o o f . T h e space exp(U) is known to have a countable base. Let B be a countable base of exp(U). Then one can readily see t h a t the family {V ~ , V + , V - , V + : V 9 B} is a countable base of F(U). [] w

Generalized metrizability

In the following, we need some facts a b o u t metrizability properties. D e f i n i t i o n . A topological space X is said to be metrizable in the eztended sense, or o-metrizable, by an o-metric 8 if there exists a nonnegative function 8: X x X --+ R + with the following properties: 1) 6 ( x , y ) = 0 if a n d only if z = y ; 2) a subset U C X is open if and only if for each x 9 U there exists an e > 0 such t h a t {y 9 X : *(x, y) < , } _c u . If, in addition to 1) and 2), one has 8(x, y) = 8(y, x) for all x, y 9 X , then the space X is said to be symmetrizable by the symmetric 6. If, in addition to 1) a n d 2), one has 6(x, y) _< 6(x, z) + g(z, y) for all z, y, z 9 X , t h e n the o-metric g is called a A - m e t r i c a n d the space X is said to be A-metrizable. Condition 2) is equivalent to the following condition: 2') a subset F C X is d o s e d ff and only if 6(x, F ) = inf{g(x, y ) : y 9 F } > 0 for every x 6 f . The following assertion is obvious. 63

Proposition 2. s

An o-metrizable topological space satisfies the separation aziom 7"1.

L e m m a I. Suppose that a topological space X is o-metrizable by an o-metric ~ such that the function y) = 6(y, z) is also an o-metric. Then X is symmetrizable. P r o o f . T h e obvious formula r(x, y) = m a x { a ( z , y ) ; $(y, z)} gives the desired symmetric on X .

[]

L e m m a 2. If a topological space X is o-metrizable by an o-metric 8, and moreover, the function r(z,y) =

inf {6(z,z~)+6(zx,zz)+...+~i(z,~,y)} {zl .... ,z.}gX,,~<~,

is an o-metric, then X is A-metrizable and r is the corresponding A . m e t r i c . P r o o f . We have

r(~, z) + r(~, V) =

inf

{~x.... ,~.}CX,n<~,

{a(~, u,) + ,~(ux, u2) + - - - + a(u., 2)}

inf

+ {~, ..... ~,,,}c_x,,,~<,o

{~(~,,,~) + ~(~x, ,,~) + - - - + ~(,,,,, ~,)}

>

inf

--

{ u x . . . . . u , , , V l .... , n , ~ } C _ X , n + m < o a

>

{a(z, u~) + a(u~, u=) + . - - + a ( u , , z)

+ a(,, ,,,) + a(,,~, ,,,) + . . . + a(,,.,, y)} inf {a(., u~) + a(,,~,,,=) + . . . + a(u., ~o)

-- {u, .... ,u,,,~,nI,...,~,.,}C_X,n+m<~

+ a(w, ~x) + ~(~, ~,) + - . . + ~ ( ~ , u)} = r(~, u). [] Corollary.

Under the assumptions of Lemma 2, if ~ is a symmetric, then X is metrizable.

L e m m a 3 [5]. Suppose that a topological space X is A.metrizable by a A - m e t r i c 6. Moreover, let have the following property: for ea'ch point x E X and an arbitrary sequence {Yn : n < ca} C X such that lim,~_.~ ~(~, y,~) = O, one has l i m , _ , ~ ~(y,~, z) = 0. Then X is metrizable. The p r o o f is by verifying that r(z, y) = max{6(z, y) ; ~(y, z)} is a metric on X . L e m m a 4 [5]. The property of a topological space to be o-metrizable by a given o-metric ~ is inherited by closed subsets. P r o o f . Let Y be a closed subspace of a space X that is o-metrizable by art o-metric 6, and let F C Y. ff F is cloned in Y, then it is also closed in X . It follows that $(z, F) > 0 for all z E Y \ F C X \ F . ff F is not closed in Y , then it is not closed in X ; that is, there exists an z E X \ F such that 6(z, F) = 0 and hence 6(z, Y) = 0. Since Y is closed, it follows that z E Y. [] More detailed information about o-metrizability can be found in [5]. w

Metrizability properties of F(U)

Here we construct a series of o-metrics for various subsets of F ( U ) . Theorem

1. The spaces F+(U) = Cs(U) U Cs+(U) and F - ( U ) = Cs(U) U C : ( U ) are metrizable.

P r o o f . We shall explicitly construct a metric on F + ( U ) . The construction for F - ( U ) is similar. Since M is locally compact, it follows that the Aleksandrov one-point compactification U* = U U {ucr of U is well defined. The space U*, as well as M , has a countable base and hence is metrizable by some metric p. Accordingly, the space expc U* is metrizable by a metric H . We define a modified Hausdorff metric h on Cs(U) by setting

h(~, y) = m ~ { 9 ( ~ , y); g(y, ~)}, g(~, U) = max{ sup

64

inf g ( v , s);

sup

inf g(v, s);

sup

inf H ( ~ , , ) } .

vEc7(~) sEcT(y)

One can readily see t h a t h induces the Vietoris topology on C~(U). Let us prove the t h e o r e m for F+(U). The proof for F-(U) is similar. We define a s y m m e t r i c on F+(U) by the formula

~(~, v) =

inf ('-',s)eD(f|

A~(~, ~),

where D ( A ~ ) C c,(x) • c~(y) is the domain of A ~ . If x, y E (7~(U), t h e n D(f~v) = {x} x {y},

f~v(v, s) = f ~ ( s , v) = h(v, s).

If either z C C,(U) and y C C + ( U ) , or z, y e C+(U), then

D(f~v) = c+(x) x c+(y), D(fy~) = c+(y) x c+(x), h ~ ( ' , *) = h~(~, ~) = h(,, ,) + p(~(sup ~(~)), ~ ) + p(,(sup ~(,)), ~ ) . Since the g r a p h of z E F+(U) is closed in U, it follows that either z E C,(U) a n d p(z,uor > 0, or z E C+(U) a n d p(z(t), uo~) -+ 0 as t --+ super(z). It obviously follows t h a t g corresponds to the given topology. Let us prove t h a t the o-metric ~ satisfies the assumptions of L e m m a 2. Let z, y E F+(U) be arbitrary, and let {zl, z2, . . . , z,~} C F+(U) be a finite ordered set such t h a t zl = z , z,~ = y , a n d n > 2. For each pair zi, zi+l, 1 < i < n - 1, we use the definition of ~ to choose a pair (vi(i+l), v(i+~)i) E D ( f z , z,+t) of compact sets satisfying the inequality

IAizi+t (Ui(i+l), U(i+I)i) -- r

1

Zi+ 1)[ < ~ ( Z i , Zi+ 1).

(3.1)

Let us consider two possible cases. 1) Suppose t h a t x, y E C~(U). Let i0 be the m i n i m u m number such t h a t zi E C~(U) for all i in the range i0 < i < n . If i0 = 1, t h e n

n--1 n--1 E ~(Zi' Z/+I)= E h(zi'Zi+l) ~ h(Zl' zn)= ~(x, y). i=l

i=1

Let i0 > 1; t h e n Zio_X C C+(U). If h(zio, y) < 89 uoo), then, with regard to the definition of h,

p(Vio(io_l) (sup 7r(V/o(io_l))), U,cr) ~_~p(y, ucr -- P(~io(io--1)(supql'(Vio(io_l))), y) 1

> p(y, u~) - h(zlo, y) > ~p(y, u~). It follows from inequality (3.1) that 5(zi,

quently,

2 ZI+I) ~> ~p(v(i+x)i(sup~r(v(i+a)i)), u~) for every i < n . Conse-

R--1

1

i=1 If h(zio , y) >_ 89

u~) , then i0 < n and

~--1 ~--1 1 Z ~(z,, Z,+l) > E ~(Zi' Zi+l)= E h(zi'zi+l) ~ h(zio'Zn) ~> 2P(Y' uc~). i=io i=io i=1 65

2) Suppose that at least one of the element~ z and y belongs to C+(U). Using the definition of h and inequality (3.1), one can readily verify thai the collection of all compact sets vi(i+x), vi(i-a) contains an d e m e n t wio that can be included in a chain {wi E cs(zi) : 1 < i < n} of compact sets such that ffYi ~ Vi(i--1) ["1 •i(i+l),

W 1 C v12 ,

zgtt C vtt(rt_l)

3 for 1 < i < n and h(wi,wi+x) < 5g(z~, zi+x) for 1 < i < n - 1. By the triangle inequality, ~--1

h(w,, ~,.) <_~ h(w,, ~,+,), i=1

p(~,~(sup ~(~,~)),,,~) < ~ h(w,, ~i+~) + p(W,o(sup ~(W,o)), u~), i=l rt--1

p(wn(suplr(wn)), uoo) < E h(wl, Wi+l) + p(wio(supTr(Wio)), uoo). i=1 3 Since h(~lJi,llJi+l) < 5~(z,i, 2:i+1) bald p(~Oio(Sllp?r(~Oio)),woo ) < 3*(Zio , Z/o+l ) b y (3.1), we o b t a i n n--1

*(~, v) = ~(zl, z.) < p(~l(sup.(~l)), woo) + p(~.(s~p.(~.)), wr162 h(~l, ~.) < 7~ Z * ( * i , zi+~). i=1

Thus we have proved that, for any given z E F+(U) and varying V E F+(U), if r(z, y) --+ O, then g(z, y) -+ 0, where r(z, y) is defined in Lemma 2. The converse is obvious. Thus r is an o-metric and the symmetric $ satisfies the assumptions of the corollary to Lemma 2, whence it follows that F+(U) is metrizable by the metric r. Theorem

[]

2. The space F(U) is A-metrizable.

Proof. To prove this, we construct the corresponding A-metric. Just as in the preceding theorem, g(z, y) =

inf

(~,~)~D(I~)

f , vCv, s),

DCf~v) C_c,(z) • csCy).

For x, y E C~(U), we have

D ( f ~ ) = {z} • {y},

f~v(v, s) = .fw(s, v) = h(v, s).

If z E C~+(U) and y E Cs(U), then

D(f~v) = c+(z) • c+(y), D ( f w ) = {y} x c+(z), fzu(v, s) = f w ( s , v) = h(v, s) + H(x \ v, {uoo}). The case in which z E C~-(U) and y E C,(U) is completely similar to the preceding; namely, 5(x, y) and 5(y, x) can be obtained from those defined above by replacing all occurrences of " + " by " - . " If z S C~(V) and V e C~(V), then

D(Av) = cs(z) x c,(y),

D(A~ ) = {y} x c,(z),

A~(., s) = h~(s, ~) = h(~, s) + H(~ \ ~, {Woo}). If z E C+(U) and y E C[(U), then D(f~v) = c+(z) x c-~(y), D ( f ~ ) = c-~(y) x c+(z), f,v(v, s) = f w ( s , v) = h(v, s) + g(~: \ v, {woo}) + H(y \ s, {uao}). 66

zI II

Z2 9

I

z~ 9

I

9

!

v

0

Z+

7

Z-

FIG. 1

Let z E C~(U) and y E C+(U). Then D ( f ~ ) = c,(z) x c,(y),

AN(v, s) = h(v, s) + H(z \ v, {uoo}),

D ( h . ) = c+(y) • c.(z),

Ax(s, v) = h(s, v) + g ( y \ s, {uo~}) + H(z \ v, {u~}).

For the case in which z E Cf(U) and y E C;-(U), one obtains 5(z, y) by the same formulas with all symbols " + " replaced by " - . " If z, y E C ~ ( U ) , then

D ( f ~ ) = co(~) • co(y),

/ ~ ( v , ~) = h(v, ~) + H(~ \ v, {~**}) + H(y \ ~, {u**}).

The fact that 5 corresponds to the topology of F(U) follows by the argument already used for the ometric in the proof of Theorem 1. The only difference is that here the following assertion is important: for given z E F(U) \ Co(U) and varying v E co(z), v --+ z implies H ( x \ v, {uoo}) --+ 0 and vice versa. Let us prove that the o-metric 5 satisfies the assumptions of Lemma 2. Now suppose that z, y E F(U), { z l , . . . , z,,} C_ F(U), Zl = z, z,, = y, and n > 2. For each pair zi, zi+~, 1 < i < n - 1, of the neighboring elements, we choose a pair (vi(i+l), v(i+l)i) E D(fz, z,+~) of compact sets satisfying inequality (3.1). By descending induction, we construct compact sets wi(i+l), wi(i-1) E c~(zi), 1 < i < n , w n E e,(zl), and wn(,~-l) E co(z,~). We have w,~(,~-l) = v,,(,~-l). Suppose that we have already constructed w,~(,,_l),w(,~-l)n,... ,w(i+l)i, where w(i+l)i C_ v(i+l)i. Since h(vi(i+~),v(i+~)i) < 35(zi,zi+~) by (3.1), it follows from the definition of h that there exists a wi(i+l) C_vi(i+l) such that h(wi(i+l),W(i+l)i) < ]5(zi, zi+l). Let i > 1. Since

diam(zi \ Vi(i_l) ) < 2H(zi \ vi(d-1), {uoo}) < 35(zi-1, zi) (the second inequality follows from (3.1)), we see that there obviously exists a compact set wd(d_~) C_ vd(d-X) such that h(wi(i-1), wi(/+a)) < 35(zi-1, zi). Using these inequalities, we obtain

5(zl, z=) < h(w~, w=(~_~)) + g(z~ \ w ~ , {uoo}) + H(z= \ w=(=_~), { u ~ } )

i=2

36

3

1 ,~-1

+ 5 (~' ~) + ~5(~_~, ~) < ~ ~ 5(z,, ~+~). i=1

We have thereby proved that for any z , y E F(U), where z is fixed, r(z, y) -+ 0 implies 5 ( z , y ) -+ 0. The converse is obvious. Hence r(z, y) is an o-metric, and by Lemma 2, the space F(U) is A-metrizable by the A-metric r(z, y). The proof of the theorem is complete. [] Corollary.

F(U) is a Tt-space.

However, F(U) need not be Hausdorif in general. 67

Example

1 (see Fig. i). Suppose that M = R, U = R • R \ {(0, 0)}, and

z+eC+(u), .(z+) = [-1, 0), z+(t)=0,

z-(t)=0.

For each i, 1 ~ i < w, we take zi E C,(U), ~r(zl) = [ - 1 , 1], zi(t) ~ 1/i. The sequence zi obviously converges to both z + and z - . Hence F(U) is not Hausdorff. Under certain assumptions about subsets of F(U), these subsets possess stronger properties than F(U) itself. Theorem

3. A subset X C F(U) is metrizable if and only if it is regular.

P r o o f . We already know that F(U) has a countable base. By well-known theorems, a regular space with a countable base is normal and metrizable. [:3 The papers [1, 2] deal with properties of subsets Z C C~(U) corresponding to elementary properties of solution sets of ordinary differential equations. Let us list some of these properties: 1) if z E Z and I C 7r(z) is an interval, then zli E Z; 2) if zt, z2 E Z and Zl = z2 on 7r(zt) n ~r(z2), then z(t)

S zl(t) /

for

t e

for t E

also belongs to Z ; 3) (the compactness axiom) for any compact set K C U, the subset {z E Z : z C_ K } is compact. By RI(U) we denote the set of all spaces Z C Us(U) satisfying condition 1). By R(U) we denote the set of all spaces Z C_Cs(U) satisfying conditions 1) and 2). The set of spaces satisfying conditions 1) and 3) will be denoted by R~(U), and the set of spaces satisfying all conditions 1)-3) will be denoted by Re(U). We define the space Fz(U) corresponding to a subset Z C_ C,(U) as the set of all elements of F(U) whose graphs have the form [U.A]u, where ,4 is some set of graphs of elements of Z such that A is linearly ordered by inclusion, i.e., for any zl, z2 E A , either zl C_ z2, or z2 C zt. Remark.

Obviously, Fz(U) n Cs(U) -- u{cs(z) : z E Fz(U)} = Z for Z E R~(U).

P r o p o s i t i o n 3. If Z E Ri~(U), then Fz(U) is closed in F(U). P r o o f . Since F ( U ) has a countable base, it is sufficient to prove that the limit of each sequence

{zk : k < w} C_ Fz(U) convergent in F(U) belongs to Fz(U). Let zk -+ z. It follows from the definition of convergence in F(U) that z -- Uj<,, y J , where yj E es(z),

Yi' C yf, for j' < j " , and each yj, j < w, is the limit of a sequence {z~ E cs(zk) : k < w}. Since Z E R~(U), we see that x~ E Z ; moreover, since U is locally compact, it follows that Yi E Z for each j . Consequently, z E Fz(U) by the definition of Fz(U). [] L e m m a 5. / f Z E Ri(u), then Fz(U) is a locally bicompact space in the following sense: Fz(U) has a base of open sets such that the closures in Fz(U) of all elements of this base are bicompact (but not necessarily Hausdorff) spaces. P r o o f . Since we have already shown that Fz(U) has a countable base and is closed in F ( U ) , it suffices to prove that there exists a base of open sets in F(U) such that each sequence of points of Fz(U) contained in an element of this base has a subsequence convergent to some point of F(U). It follows from the definition of the topology of F(U) and the local compactness of U that the family {V ~ V + , V - , V+: V is open and [V]u is compact} is a base of F(U). Let H be one of the elements V ~ V + , V - , V • of this base corresponding to some V, and let {zi : i < w} be a sequence contained in Fz(V) n g . For each i, we take an element v~ E c s ( z ~ ) M Y . For each j , 1 < j < w, let Wj be the e/j-neighborhood of the point u~r in V*, where e = p([Y], u ~ ) . 68

One can readily verify t h a t the sequence zi contains a subsequence of one of the following forms. Hence, without loss of generality, we shall assume that the sequence zi itself has one of these forms. 1) There exists a j0 such that zi C_ U \ Wjo for all i. Since U \ Wy0 is compact by the definition of U*, it follows that zl E Cs(U) ; by the above remark, we can assume that zi E Z for all i. Since Z E R~(U) and all zi lie in a compact set, it follows that the sequence zi contains a subsequence convergent to some

zEZ. 2) There exists a j0 such that zil(i,f,~(z,),sup~(~,)] C_ U\Wio for all i, but ziMWi ~ ~ for all i. Then, obviously, { z i : i < w} C_ C+(U). For each i and each j _< i, we take an element y~ E c+(zi) such that y~ f3 Wj # O and y~ C_ U \ W j + l . Since Z E R i ( u ) , we see from the remark following the definition of Fz(U) that y~ E Z . Using the fact that Z E R~(U) and the sets U \ Wi are compact, for each j , 1 < j < w, by induction we construct a subsequence {ySi(j,k) : k w} of the sequence {~/': i < ~} as follows. From the sequence y~ we choose a convergent subsequence Y~(1,k)" _

Suppose that the sequence yJi(j,~) has already been constructed. The sequence y~+l.(j+l,k) is a convergent subsequence of the sequence y ~ l ~ ) , i ( j , k) > j + 1. For each j , the sequence yJ~(j,k) converges to some element zi E Z 9 Next, p(zj(sup 7r(zj)), u ~ ) < - r , p(zi(inf~r(xj) ), u ~ ) >_ e, a n d z i C_ zj+~ . It follows that the diagonal sequence Yi(ji,i) (and hence the J subsequence zi(j,j) ) converges to the element z = U~__lzj E C+(U). 3) There exists a j0 such that zi][i~f,~(~),sup~(z,)) C_ U \ Wjo for all i, but zi fq Wi ~ ~ for all i. This ease is completely similar to the previous one. Then (zi : i < w} C_ C~-(U), and the sequence zl contains a subsequence convergent to an element of C~-(U). 4) The sequence z + ---- Zi[[ir,f,r(~,,),sup,r(z~)) has the same properties as the sequence zi in case 2), and the sequence z~- = zi[(i~f,~(~),sup~(~i)] has the same properties as the sequence zi in case 3). In the sequence z + , we take a subsequence z/~k) convergent to an element z + E C + ( U ) , and then in the sequence z~k ) we take a subsequence zffk(t)) convergent to an element z - E C~-(U). Obviously, the sequence zi(k(t)) converges to z = z + U z - E C~(U). The proof of the l e m m a is complete. [] C o r o l l a r y . If Z E R~(U) and Fz(U) is Hausdorff, then Fz(U) is locally compact. T h e o r e m 4. Suppose that Z E R i ( u ) .

Then Fz(U) is metrizabIe if and only if it is Hausdorff.

P r o o f . Here we again write out the metric explicitly. To prove the desired assertion, it suffices to show that if F z ( U ) is not metrizable, then it is not Hausdorff. Since Fz(U) is dosed in F(U) by Proposition 3, it follows from Lemma 4 that Fz(U) is o-metrizable by every o-metric by which so is F(U). Taking this into account, we use Lemma 3, which permits us to reduce the proof to establishing the following assertion. Suppose that y E Fz(U) , {yi : i < w} C_Fz(U), and g(y, yi) --~ 0 as i --~ co, where ~ is the o-metric constructed in the proof of Theorem 2, but limi--.oo J(yi, y) ~ O. Then some subsequence of the sequence yi converges to an element z ~ F z ( U ) , z r y , which contradicts the Hausdorff property. It follows from the definition of ~ that under the above conditions there exists a number y~ C_ y, y' E C+(V) U C~-(U), a n d a subsequence y~ C_ yi, y~ E C~(U), with the properties J(y',y~) ~ 0 and l i m i ~ ~(y~, y') r 0. Hence we can assume, without loss of generality, that y E C+(U) and yi E Cs(U). For each i, there exist si E c+(yi) and wi E c+(y) such that

h(wi, si) < 2~(y, yi),

g ( y \ wl, {uor

< 2~(y, yi).

Hence p(yi(supTr(si)), uoo) -+ 0 as i -+ co. It follows that for each ~ > 0 there exists a subsequence Yi(i) such that p(yi(i)(supTr(si(j))), uoo) < e / j . Since limi-~r $(yi, y) ~ 0 and lim~-,ooH(y~ \ si, {u~}) # 0, we see that for some ~ > 0 there is a subsequence y~(~) such that H(y~(~), {u~}) > e for all k. Without loss of generality, we shall assume that i = i(j) = i(k). Using the last two assertions, for each i we choose a ti E ~r(yi), ti > sup~r(si), such that p(yi(ti), uo~) > ~. Set zi = Yil[~p~(~,),t,]. 69

We have arrived at the situation described in case 3) in the proof of Lemma 5. By virtue of the relevant results, the sequence zi contains s subsequence zio,) convergent to some z e Fz(U) N U~-(U), z ~ y E Fz(U) f3 C+(U). T h e corresponding sequence yi(k) also converges to z. Thus, the space Fz(U) is metrizable by the metric r'(., y) = max{r(x, y); r(v, x)}, where r(x, V) is the A-metric constructed in the proof of Theorem 2. The proof of the theorem is complete. [] However, if Z does not satisfy the compactness axiom, then Theorem 4 fails even for the class R(U) regardless of whether Fz(U) is closed in F ( U ) , as shown by the following example.

-

~ r , ~ . x~ - . - - .

~ =

- -

-

i~,~----\..0

:

0

Zo

-

C

-

-_

:

C

Zl

Z2

FIG. 2 E x a m p l e 2 (Fig. 2). We set M = R and

u = R x R \ ({(1, 0)} u {(2k, 0): 1 < k <

~}),

[0, ~), ~0(0 ~ 0.

~o e c + , ( u ) ,

,~(zo) =

zi e C;(U),

7r(zi) -- ( 2 i , 2 i + 1],

For each i, 1 < i < w, let

zi(t) = O.

For each i >_ O, we define a sequence {z~ : 1 _< j < w} C C,(U) convergent only to z~ by the formula

~(~)

[~(~,)],

~(,)

1 3

For any i , j > 1, the segment joining the points z~(suplr(z~)) and zJ(infTr(z~)) is the graph of a function wij E Us(U). We define an element zij E Cs(U) as follows: i

9 ~jl<.,~)\~.(zo)u<.,))(O = ~,~J(~) + sin(2~r(i +

j

j)t).

We obtain the set

z = U 0
c,(z,) u

U

c,(~,j) c R(u).

l_
Let us prove that the corresponding space Fg(U) is Hausdorff. Since F(U) satisfies the first countability axiom, it suffices to prove that there is no sequence convergent to two points simulta.ueously. Moreover, it obviously suffices to consider sequences whose elements belong to Cs(U) f3 Fz(U). One can readily see that spaces like Fc.(=~j)(U) a n d Fe.(z~)(U) are Hausdorff. Hence it suffices to consider sequences of the form {Yk e c,(zi(i)j(k)) : h < w). 70

The structure of zij is such that for each infinite family X = {zlj : i E A, j E B} and each point

O
the set t x R N [UX]u \ UX necessarily contains an everywhere dense subset of {t} • I , where I is the unit interval. It follows that no sequence yk 6 c~(z~(k)j(k)) can converge to an element of C~(C • R). Here the compactness axiom is violated. By considering the few possible cases, one can readily verify that each sequence Yk 6 %(z~(k)j(k)) can have at most one limit. It is also easily seen that Z and F z ( U ) are closed in Cs(U) and F ( U ) , respectively. Suppose that F z ( U ) is metrizable by a metric p; then for any i _ 0 and k > 0 there exists a standard neighborhood G~ = G ( z i , v ~ , Vik) contained in the 1/k-neighborhood of z~. Since z iJ converges to zi as A(i,k) also j --+ oo for all i, it follows that for each k there exists a z_j(i,k) i E G ik . Moreover, each z,,~,~ 2 zi belongs to G~ and hence lies at a distance less than 1/k from zl. We set i(k) = j(O, k) and j(k) = j(i(k), k). The corresponding element zi(~)j(k) lies at a distance less than 1/k from z0 and zi(~). Consequently,

p(zo, zi(k)) _< p(zo, zi(k)./(k)) + P(Zi(k)j(k), zi(k)) < 2/k, that is, zi(k) --+ z0 as k --~ co, which obviously contradicts the topological structure of F(U). Hence F z is Hausdorff but nonmetrizable. w

T h e s p a c e F*(U)

We define the space F*(U) as the subspace {(t, z) : t e ~-(z), z e F(U)} of the topological product R x F(U). L e m m a 6. The space F*(U) is me'trlzable. To prove the lemma, we define the corresponding metric r*. Each element (t, z) induces two elements z + = zl[t,+co)n,~(~ ) e F+(U) and z t = zI(_~,tln,~(z) e F - ( U ) . Let (tl, Zl), (t2, z2) e F*(U). We set r*((tl, zl),

= ,+(zS, , z+

+ "-(zS,,

where r + and r - are the metrics on F+(U) and F - ( U ) , respectively, defined in the proof of Theorem 1. Obviously, (t, z) --+ (to, z0) if and only if z + --+ Z+oto and z~ --+ Zot o- . It follows that r* is an o-metric specifying the topology of F*(U). The symmetry and triangle axioms for r* can be verified trivially. Thus, F*(U) is metrizable by r*. C o r o l l a r y . If Z C_ F ( U ) is a subspace such that there ezists a to belonging to the domain of each z E Z , then Z is metrizable. P r o o f . The mapping f : Z -+ F*(U) given by f ( z ) = (to, z) is obviously a homeomorphic embedding. Hence Z , a s well as F * ( U ) , is metrizable. [] References I. V. V. Fedorchuk and V. V. Filippov, Genera/Topology. Basic Constructions [in Russian], Izd. Moskov. Univ., Moscow

(19ss). 2. V. V. Filippov, Solution Spaces of Ordinary D~erentia/Equations [in Russian], Izd. Moskov. Univ., Moscow (1993). 3. V. V. Fillppov, "TopoIogicM properties of solution spaces of ordinary differentialequations," DokL Ross. Akad. Nauk [Russian Acad. Sci. DokL Math.], 352, No. 6, 735-738 (1997). 4. J. Yorke, "Spaces of solutions," in: Lecture Notes in Operat. Res. and Math. Econom., Vol. 12 Springer-Verlag, Berlin (1969), pp. 383--403. 5. C. I. Nedev, "o-metrizable spaces," Trudy Mosk. Matem. Obshch., 24, 201-236 (1971). M. V. L O M O N O S O V M O S C O W

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