Totally ordered subsets of Euclidean space

Journal

of Mathematical

Economics

23 (1994) 391-393.

North-Holland

Totally ordered subsets of Euclidean space Alan F. Beardon Department of Pure Mathematics, University of Cambridge, Cambridge, UK Submitted

May

1993, accepted

September

1993

We show that if a totally continuous, order-preserving

ordered, closed, connected subset of Euclidean space supports real-valued function, then it is homeomorphic to an interval.

Key words: Utility

Ordered

functions;

a

sets

JEL classification: C60

1. Introduction

A totally ordered topological space is continuously representable if it can be mapped by a continuous, order-preserving map into R. In a recent paper, Candeal and Indurain (1993) showed that a closed, convex, continuously representable subset of R” is homeomorphic to an interval. Now convexity is not a topological property, and the real issue here is to characterise those topological properties of a continuously representable subset of R” which ensure that it is a curve. The example X0= {(x,sin l/x)~ W:x>O}

u {(O,O)}

[ordered by the value x, and continuously represented by (x, y)~x] shows that a connected, continuously representable subset of R” need not be a curve. On the other hand, a connected compact, continuously representable topological space is a curve (for a continuous map of a compact space into a Hausdorff space is a homeomorphism). It is of interest to seek weaker hypotheses than compactness, and we prove the following result. Theorem I. Let X be a connected topological space and suppose that u: X + R is an injective, continuous map. Then u: X -u(X) is a homeomorphism if and only ij-X is Hausdorff and locally compact. Correspondence to: Alan Cambridge, Cambridge, UK.

F.

Beardon,

Department

of

0304~4068/94/$07.00 0 1994 Elsevier Science S.A. All rights SSDI 0304-4068(93)EOO74-6

Pure

reserved

Mathematics,

University

of

A.F. Beardon, Totally ordered subsets of Euclidean space

392

Of course, an equivalent formulation is that a connected, representable space X is homeomorphic to an interval if and Hausdorff and locally compact. To relate this to Theorem 4 in Induriin (1993), observe that a closed subset X of any manifold and locally compact, and so Theorem 1 has the following

continuously only if it is Candeal and is Hausdorff

Corollary. A closed, connected, continuously representable subset X of any manifold (and, in particular, of IV’) is homeomorphic to an interval.

A related problem has been considered by Eilenberg (1941) but, as far as I can see, neither of these results are an immediate consequence of his results. 2. A preliminary result We begin with a connected, topological space X that is mapped injectively and continuously by a map u onto a subset of R, and we denote the given topology by 5 As u(X) is a connected subset of R, it is an interval and so we can use the bijection u to transfer the order c on Iw to a total order < on X. Using this order, we endow X with the order topology 0, and u: (X, 0) + (u(X), 8) is an order-preserving homeomorphism, where d denotes the induced Euclidean topology on u(X). The fact that u: (X, Y) + R is both continuous and order-preserving guarantees that 0 c F. We need only prove that 9 c 0, for it will then follow that O=F, and hence that u:(X,F) + (u(X), B) is a homeomorphism. As usual, we write x5 y if and only if xi y or x = y, and we write (a,b)={xEX:a
[a,b]={xEX:asxlb},

with (- co, a), (- 00, a], (b, + 00) and [b, + co) being defined in the obvious way. It is clear that [a, b] is closed and connected in the O-topology (for it is homeomorphic to the real interval [u(a),u(b)]), but we need to know that it is both closed and connected in the F-topology. Lemma 1. Suppose that in the above circumstances, a and b are in X and that a< b. Then [a, b] is closed and connected in the F-topology. Proof. The interval [a, b] is closed in the F-topology because u(X) is an interval in R, and [a, b] is therefore the inverse image of the closed subinterval [u(a), u(b)] of R. To show that [a, b] is connected it is sufficient to show that the map

h(x)=

I

h(a) x h(b)

if if if

xE(--,a), x E [a, b], x ~(b, + oo),

A.F. Bear-don, Totally ordered subsets of Euclidean space

393

of X onto [a,b] is y-continuous. This, however, follows directly from the fact that if A is any (relatively) open subset of [a,b], then there is a r-open subset W of X with A = W n [a, b], and

h-‘(A)=

(-co,a)u Wu(b, +co) Wnkb) (-co,a)u(Wn(-co,b)) I (Wn(a,+oo))u(b,+co)

if if

aEA and bEA, a$A and b$A,

if if

SEA and b$A, a#A and beA,

3. The proof of Theorem 1 First, if u: X + u(X) is a homeomorphism then, as u(X) is a real interval, X is HausdorlI and locally compact. To prove the reverse implication, we need to show that y c 0, and it is sufficient to show that for any point x in a y-open set W, there is some Oopen set W, with {x} c W,, c W. As X is Hausdorff and locally compact, there is an open neighbourhood N of x whose compact closure lies in W [see Kelley (1968, p. 146)]. Suppose first that there is some Y in X with y
U(Y,)+ u(x),

we have u(Y)=u(x) which is false as u is injective. It follows that if there is some y with y