A new characterization oî Peano continua *For a metric space

ANNALES SOCIETATIS MATHEMATICAL POLONAE Series I : COMMENTATIONES MATHEMATICAE X Y I (1972) ROCZNIKI POLSKIEGO TOWAKZYSTWA M ATEMAT Y OZNE G O

Soria I : PRACE MATEMATYOZNE X V I (1972)

Ro b e r t B . Fr a s e r, Jr. (Halifax)

A new characterization oî Peano continua *

For a metric space (X , d), we set 8 {x, ô) — {ye X\ d ( x ,y ) < ô } and $[# , d] = {ye X\ d {x , y) < <3}.

Th e o r e m 1. Let (X ,d ) be compact and satisfy С 1$(ж , <5) = 8[x-, d]

for all X€ X tmd ô > 0. Then X is a Peano continuum.

P roo f. We show that each hall $[a?, S] is connected. If not, it can he written as A u B , x e A , В A 0 , A n B = 0 and A , В closed. Since d ( x ,B ) > 0, there exists (by conrpactness) an element be В such that d(x, b) — d(x, B). Then be 8 [x, d{x, B)], but b$ Cl$(a?, d{x, B)), contrary to assumption. Thus X is connected and locally connected. We now apply the theorem of Hahn and Mazurkiewicz characterizing Peano continua.

Th e o r e m 2. A space X is a Peano continuum iff it is compact and admits a metric d such that $[ж, й] = С1$(ж, d) for all x e X and ô > 0.

P roo f. The sufficiency follows from Theorem 1. The necessity follows from the fact that every Peano continuum admits a convex metric [2]

which then satisfies the required condition [1].

One might conjecture that some type of connectivity remains if some conditions are slightly weakened. However, consider the following

Ex a m p l e. Let

A = |(Г, в) e Щ r = 1 + y , 7t < 0 < coj and

B = [(s, a) e B'l \ s = 2 -j--- , 77 ^ a < co l.

I a J

Let a l( x , y ) denote arclength. Define d on A - f B by d(«i, a 2) al(Ui, a 2)

1 + i\\{al a 2) for at , a2 e A, d{b\, h ) a1(&!, b2)

for bx, Ь2е В 1 + a l^ i, b2)

* This research was supported by a Research Faculty Fellowship from L.S.U.

248 R. B. F r a s e r

and d(a, b) = r^-s for a = (r, B)eA and b = (s, o )e B . It is easy to see that (X , <5) is uniformly locally compact and satisfies $[ж, d] = СШ(ж, <5) for all oceX, ô > 0. But X is not even connected.

References

[1] N. A rte m ia d is , A remarie on metric spaces, Proc. Kon. Ned. Akad. van Wetensch.

A68 (1965), p. 315-318.

[2] R. H. B in g , Partitioning a set, Bull. Amer. Math. Soc. 58 (1952), p. 536-556.