O n metric spaces aimed at their subspaces

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I : PRACE MATEMATYCZNE X (1966)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Serio I: COMMENTATIONES MATHEMATICAE X (1966)

W. H

(Warszawa)

O n metric spaces aimed at their subspaces

We say that an ordered pair of points x, у of Euclidean space is aimed at the point z of the same space, if x, y, z are collinear and у is between x and 0. We can represent this situation saying that the light ray from ^x through у will pass by For this it is necessary and sufficient that the equality

(1)

(

, y) + Q{y, z) =

(

,

) holds.

In the case of any metric space M, it is possible that for given points x, у elf there exists no set A с M containing x, у and isometric to the straight line, but still there exists a point zeM satisfying (1). Also in this case we can say that x , у are aimed at г. We can also consider the deviation of aiming of the pair (x, y) at z; the number

rciin[e(aV 2/)-f£(y, z )- g( x, z) , g{y,

) +

{

, z) — g{y, z)]

is a measure of precision of aiming expressed in terms of the theory of metric spaces. When x and у are aimed at z, this number is zero.

According to the Definition 1 (see below) the space M is aimed at its subspace X if every pair of points of M is aimed, with arbitrary small deviation, at some point of X. It appears that a class of all spaces M con­

taining X and aimed at X, coincides with certain other class. To every point p e M there corresponds a real-valued function f p, viz . f p(x) — @{p, x), defined on the space X. We shall prove that Ж is aimed at X if and only if the mapping p - ± f p of M into the space of real functions on X is iso­

metric imbedding. This mapping will be called natural.

Among the superspaces of A , aimed at X, there exists a maximal space isometric with some space Aim X, contained in the space of all real functions defined on X. The space Aim X is complete and an ab­

solute retract.

D

1. A metric space M is aimed at its subspace X , if for any points p, qeM and any e > 0 there exists such a point xe X that

б(Р, 2) + e(g, «) < Q(p, x ) + e or

Q{ ^P) Jr Q ( P ^ ) < Qiqy^Xe-

This alternative of inequalities is equivalent to a single inequality:

\

(P, oc)-Q(q, x)\+e > Q{p, q).

We formulate some properties of the introduced notion:

( i ) I f l c Y c I c f and M is aimed at X, then M is aimed at Y and N is aimed at X.

(ii) If a space Ж is aimed at X , then the completion of M is also aimed at X.

(iii) If Ж = T cz M and M is aimed at Y, then M is aimed at X.

The properties mentioned above show that it is possible, without loss of generality, to consider only complete metric spaces.

In the case when X is compact Definition 1 can be simplified as follows:

The space M is aimed at its compact subspace X , if for every p , qcM there exists x e X such that

Q(P>

Q(Pi x ) or

Q(bP)+Q(P, x) = Q(q, x).

De f i n i t i o n

2. A mapping

, Y being metric spaces) such that

6{f(x ),f(y)) < Q(x , У) for x, y €X is called a metric mapping (1).

Let now A im X denote, for a space X , the set of all real functions defined on X , having the property that

■ suple(®, y ) - f ( y ) \ = /( ® ) for any x e X .

y e X

In the set Aim A the uniform metric is defined by the formula 6(f, g) = sup }f(x)—g(x) |, / , g t M m X .

x t X

Let us remark that Q { f , g ) < oo even if / , g e A i mX are unbounded (if X is unbounded). Indeed

e (f ,9) = wp\f{ y)- g( y) \ <sup(|e(a?, y ) - f ( y ) \ + \Q(x, y) -g{y)\)

V t X y e X

< f i x ) + g ( x ) < XeX.

(J) These mappings are natural morphisms in the category of metric spaces and, hence, the name “ metric” is the most reasonable.

On metric spaces aimed at their subspaces 97

It is easy to see that f e A i m X if and only if f i x) + fi y) > e(®,

) > f ( ® ) - f ( y ) for every x, ye X.

All the functions of A im X are metric mappings. Indeed, let us sup­

pose that f e A i m X , x, y e X and, e.g. f(x) > f ( y ). Then f i x ) - f { y ) < f(x) - I

{

, y) - f ( x ) I <

(

, y).

Therefore \f{x)—f(y) | <

(

, y).

If У is a superspace of X, i.e., if 1 с Г, then there exists a natural metric mapping / of the space У into Aim X given by

f(y) = / tfeAim X, where f y(x) = g(x, y), y e Y , xeX.

We shall show that f y e A im X , for y e Y . Indeed sup \

{

, z)—f y(z) I = sup|e(a?, z)-g( z, y) | < g(x, y)

ZeX ZeX

= fy{x) = \Q(x, x) -f y{x)\ <sup|Q{ x, z) ~f y(z)\,

ZeX

and therefore sup \

{

,

) ~ f y(z) | — f v{x), y e Y , x e X .

ZeX

It remains to verify that the mapping / : У -> A im X is metric:

Qifv'Jv") = sup \fV’{ x)—fy’'{x) I = sup \

{

, y' ) - Q{ x, у") I < s u p e (y ', y")

xe X XeX x e X

= Q{y’,y")-

Obviously, if Z => Y => X and f : Aim У is the natural mapping th en /| y: У ^ A i m X is also the natural mapping.

Th e o r e m 1 .

A space Y is aimed at its subspace X if and only if the natural mapping / : У -> A im X is an isometric imbedding.

P r o o f. Let У be aimed at X. By Definition

for any y ' , y " e Y we have

mp\Q{ y' , x) - Q( y" , x) \ = Q{y',y")

XeX

and further Q{fV',fy") = Q{y',yf')', hence / : У -> A im X is an isometric imbedding.

On the other hand, if f is an isometric imbedding, then for any y', y " e Y and e > 0 there exists such an жсХ that

Q{y',y

= Qifv’Jv") < \fv'ix)~fv"ix)\+e

Qiy'i x)~Q{y

x)\+£- Therefore У is aimed at X , q.e.d. a ' A

Prace Matematyczne X.l 7

Obviously, every space X is aimed at itself, and any one-point ex­

tension of X (i.e., a superspace Y of space X such that Y \ X is a one-point set) is aimed at X. From above considerations we obtain

C

. The natural mapping of a one-point extension of X into Aim X, analogous to the natural mapping of X into Aim X , is an isometric imbedding.

In the sequel it will be convenient to consider a superspace of space X as an ordered pair (M , f ) consisting of a metric space M and an isome­

tric imbedding f : X -> M.

In particular, the pair (Aimж, c), e being the natural mapping of X into Aim X , is a superspace of a space X.

A superspace ( M , f ) of a space X will be called aimed at X , if and only if M is aimed at f { X ) in the sense of Definition 1.

T

2. (Aim X, c) is aimed at X .

P r o o f. The theorem states that A im X is aimed at c(X) in the sense of Definition 1.

Given any / , y eA im X and s > 0. Then there exists such x e X that Q ( f , 9 ) < I/ И — 0 (®)l+e

= sup I^(ж, y) -f (y) \~mp\Q(x, y ) - g ( y ) | + e

VeX ytX

= IQ (/, c M ) ~<P (а, c {x)) I + e, q.e.d.

Let (N , g) and (M , f ) be two superspaces of X. The isometric imbedding h: N -> M will be called an isomorphic imbedding of {N, g) into (31, f ) if the diagram

h N ---* M

\ s

g Y f is commutative, i.e. hg{x) = f ( x ) for x e X .

In particular, if h is an isometric mapping, spaces (N, g) and (M , f ) will be called isomorphic and h will be called an isomorphism. Obviously, superposition of isomorphic imbeddings is isomorphic imbedding.

T

3. The superspace (Aim X, c) is maximal among all super­

spaces (M , f ) aimed at X 1 i.e. every superspace of X can be isomorphically imbedded into it. Moreover, every isomorphic imbedding of (Aim X, c) into any (M, f) is an isomorphism onto.

P r o o f. If (M , f ) is aimed at X , then in virtue of Theorem 1, the natural mapping g: J f-> A im /(X ) is an isometric imbedding. Of course, the pair (Aimf { X ) , gf) is a superspace of X and g is an isomorphic imbed­

ding of superspace (M , f ) into (A im /(X ), gf).

On 7netric spaces aimed at their subspaces 99

Let cp be the function which associates with every function Ji in A im /(X ) the function ¥ in A im X defined by ¥ (x) = hf(x) for x e X.

Then cp is an isomorphism between (A im /(X ), gf) and (AimX, c) for (<pgf{x))(y) = (gf(x))(f(y)) = a(f{x)J{y)) = e i ^ y ) = И®))(#)- Hence we obtain the first part of the theorem.

Now, cpg: J T A i m X is an isomorphic imbedding of (Ji",/) into (Aim X, c). If there existed an isomorphic imbedding of (Aim X, c) into ( J i,/) which is not an isomorphism onto, then the superposition of this imbedding with cpg would be an isomorphic imbedding к: Aim X Aim X of the superspace (Aim X, c) into itself, not being an isomorphism. We thus get a contradiction since the function к, as an isomorphic imbedding of (Aim X, c) into itself, is an identity. Indeed, к must be an identity on c{X). Moreover, к is an isometric imbedding, and for /e A im X we have

f{x) = sup H y ,

) - f{y) I =

(

{

) J) = g(kc{x), kf)

УеХ

—

(

(

), kf) = (kf)(x) since kf eAimX.

Hence kf = f for any /e A im X . This concludes the proof of the theorem.

From the above we obtain as a corollary that the space A im X is com­

plete (this can also be verified directly).

C

. Let X be a compact space. Then every bounded subset of any superspace aimed at X is completely bounded.

This follows from Theorem 1 and Arzela’s theorem, since if X is compact, then A im X has the desired property.

T

4. For any X , A im X is an absolute metric retract (cf. [1]) i.e. if A im X is a subspace of a metric space Jf, then there exists a metric mapping (a metric retraction) r: M -> Aim X, which is the identity on Aim X.

P r o o f. Let с: X -> Aim X denote the natural mapping. The desi­

red retraction is defined as follows: r{p) —rp, where rp{x) — g{ p, c ( x)), pcM, XeX.

Indeed, for /e A im X , жеХ, we have

M®) = e(f> c (®)) = /(® );

hence r is identity on A im X .

Furthermore, r(p) = rpeA im X for pe M, since

sup \rv{ y) - Q( x, y) I =sup|e(p, с( у) ) - д( с( х) , c{y))\ <

{

)) = rp(x)

y e X y c X

= \Гр{я) — 9{я,х)\ <sup|rp(y)-e(a?,y)|.

yeX

Thus rp(x) = snplrp(y) —

{

, y) |.

v*x

Finally, r is a metric mapping: for p, qeM we have Q(r(p), r(q)) = mp\g(p, c(x))-Q(q, c(®))| < g(p, q),

X e X

References

[1] N. A r o n s z a j n and P. P a n i t c h p a k d i , Extension <

tinuous transformations and hyperconvex metric spaces, Pacific J.

pp. 406-439.

q.e.d.

f uniformly con- of Math. 6 (1956),