Volume 6, 1981 Pages 363 369 http://topology.aubur.edu/tp/ A NOTE ON LEBESGUE SPACES by Sam B. Nadler, Jr. ad Thelma West Topology Proceedigs Web: http://topology.aubur.edu/tp/ Mail: Topology Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA E-mail: topolog@aubur.edu ISSN: 0146-4124 COPYRIGHT c by Topology Proceedigs. All rights reserved.
TOPOLOGY PROCEEDINGS Volume 6 1981 363 A NOTE ON LEBESGUE SPACES Sam B. Nadler, Jr. ad Thelma West L~t (X,d) be a metric space. A Lebesgue umber for a ope cover lj of X is a E > 0 such that for each poit p E X, the ope,ball Bd(p,E) = {x E X: d(p,x) < E} is cotaied i at least oe member of 0. A Lebesgue space (L-space)(is a metric space such that every ope cover of the space has a Lebesgue umber. It is kow that the L-spaces are precisely those metric spaces for which every cotiuous real-valued fuctio is uiformly cotiuous ([6, p. 112], [1, p. 12]). I particular, every compact metric space is a L-space. We will show that there are L-spaces which are ot eve locally compact. Furthermore, i Theorem 1 we characterize those metric spaces which cotai olocally compact L-subspaces. The proof of Theorem 1 shows how to costruct such subspaces i the most geeral possible settig. I Theorem 2 we show that ay L-space must belocally compact "at most poits. II I Theorem 3 we obtai a simple ecessary ad sufficiet coditio i order that a metric space be a locally compact L-space. This coditio determies the structure of all such spaces. We will briefly discuss Theorem 3 i r~latio to results i [4] ad [S]--see the Remark at the ed of the paper. Recall tha~ a metric space (X,d) is said to be totally bouded provided that for each E > 0, there exist fiitely may poits Pl,P2,,P of X such that X U{Bd(Pi,E): i = 1,2,,} [2, p. 22]. A subset A of a metric space
364 Nadler ad West (X,d) is said to be uifopmly isolated provided that there exists a C > 0 such that d(x,x') ~ c for all x,x' E A such that x ~ x' [5, p. 153]. Clearly, a metric space fails to be totally bouded if ad oly if it cotais a ifiite uiformly isolated subset. For use later o, let us ote the followig lemma. It follows easily from Theorem 1 of [1]. Lemma 1. A metric space (X,d) is a L-space if ad oly if the set L of all limit poits of X is compact ad for each ope subset U of X such that U.~ L, X - U is uifopmly isolated. The followig theorem is our characterizatio of those metric spaces which cotai olocally compact L-subspaces. Theorem 1. Let (M,D) be a metpic space. The: Every L-subspace of M is locally compact if ad oly if every poit of M has a totally-bouded eighbophood, i.e., if ad oly if the completio of M is locally compact. Ppoof. Assume that there is a poit p E M such that o eighborhood of p is totally bouded. The, as oted above, each eighborhood of p cotais-a ifiite uriiformly isolated subset. Let Xl be a ifiite uiformly isolated subset of the ball B o (p,2-1 ) such that p Xl. Assume iductively that we have chose ifiite subsets Xi of B O (p,2 -i ) - {p} for each i = 1,2,., ( < 00) such that Ui=IX i is uiformly isolated. Let o = if{d(p,x): x E ui=ix i } ad ote that c > O. Let E mi{c/2,2- - I }. Sice E > 0,
TOPOLOGY PROCEEDINGS Volume 6 1981 365 there is a ifiite uiformly isolated subset X 1 of + BD(p, ) such that p X + l. Note that u~:txi is uiformly isolated. Thus, we have iductively defied X for each = 1,2,--- such that, for each k = 1,2,---, uk X is ui=l formly isolated. Let X = {p} U (U:=lX ) ad let d deote the subspace metric for X obtaied from D. We see that p is the oly limit poit of X ad that if U is k ay ope subset of X such that p E U, the X - U c U=lX for some k < 00 ad, thus, X - U is uiformly isolated. Hece, by Lemma 1, (X,d) is a L-space. Furthermore, as is easy to see, (X,d) is ot locally compact at p. Therefore, we have proved half of Theorem 1. To prove the other half, assume that every poit of M has a totally bouded eighborhood ad let Y be a L-subspace of M. It follows easily from Lemma 1 that Y is complete (the fact that L-spaces are complete is also oted i [6, 8, p. 112]). Let y E Y. From our assumptio about M, there is a totally bouded eighborhood N(y) of y. We assume without loss of geerality that N(y) is a closed subset of M. The, N(y) Y is a closed eighborhood of y i Y. Sice Y is complete, N(y) Y is complete. Thus, sice N(y) Y is totally bouded, N(y) Y is compact [2, p. 22]. Hece; we have proved that each poit of Y has a compact eighborhood i Y. Therefore, Y is locally compact. This completes the proof of Theorem 1. For metric liear topological vector spaces, Theorem 1. becomes the followig result:
366 Nadler ad West Corlllary. A metric liear topological vector space is fiite dimesioal if ad ~ly if every subset which is a L-space is locally compact. Proof. The corollary follows immediately from Th~orem 1 above ad 7.8 of [3, p. 62]. The L-spaces costructed i the proof of Theorem 1 fail to be locally compact at oly oe poit. The questio arises as to whether there is a L-space which fails to be locally compact at every poit of some oempty ope subset. The followig result aswers this questio by showig that o such L-space exists. Theorem 2. If (X,d) is a L-space~ the X is locally compact at each poit of a dese ope subset of x. Proof. Let L = {x: x is a limit poit of X} ad let W = {x X: X is locally compact at x}. Note that X - W c L. Thus, sice L is compact (by Lemma 1), X - W ca ot cotai ay oempty ope subset of X. Hece, W is a dese subset of X. Clearly, W is a ope subset of X. This completes the proof of Theorem 2. We have the followig characterizatio of locally compact L-spaces: Theorem 3. A metric space (X,d) is a locally compact L-space if ad oly if X is the uio of a compact subspace ad a uiformly isolated subspace. Proof. Assume that (X,d) is a locally compact L-space. Let L deote the set of all limit poits of X. By Lemma 1,
TOPOLOGY PROCEEDINGS Volume 6 1981 367 L is compact. Thus, sice X is locally compact, L ca be covered by fiitely may ope subsets of X whose closures are compact. Hece, there is a ope subset U of X such that LeU ad such that the closure IT of U is compact. By Lemma 1, X - U is uiformly isolated. Therefore, writig X = IT u (X - U), we see that we have proved half of Theorem 3. Now, to prove the other half of Theorem 3, assume that (X,d) is a metric space such that X is the uio of a compact subspace C ad a uiformly isolated subspace E. We first show that X is locally compact. Note that E - C is a uiformly isolated ope subspace of X. Thus, for each poit x E E - C, we see that {x} is a ope subset of X. Hece, clearly, X is locally compact at each poit of E - C. To show that X is locally compact at each poit of C, it suffices (sice C is compact) to show that C is a ope subset of X. Suppose that C is ot a ope subset of X. The there is a sequece {x}~=l of poits of X - C such that {x}~=l coverges to a poit of C. Sice {x}~=l is a Cauchy sequece of poits of E, we have a cotradictio to the assumptio that E is uiformly isolated. Thus, C is a ope subspace of X. This completes the proof that X is locally compact. Next we show usig Lemma 1 that (X,d) is a L-space. Let L deote the set of all limit poits of X. It was show above that for each x E E - C, {x} is a ope subset of X. Hece, L ~ C. Thus, sice L is a closed subset of X ad sice C is compact, we have that L is compact. Let U be a ope subset of X such that U ~ L. We wish to show that X - Uis uiformly isolated (see Lemma 1). Note that
368 Nadler ad West (*) X - U = (C - U) U (E - U) Sice C - U is a closed subset of the compact set C, C - U is compact. Sice U ~ L, C - U cotais o limit poit of X. Thus, C - U must be fiite. Sice E - U is a subset of the uiformly isolated space E, E - U is uiformly isolated. It ow follows from (*) that X - U is uiformly isolated. Therefore, we ca ow use Lemma 1 to coclude that (X,d) is a L-space. This completes the proof of Theorem 3. Remark. Let E be a subset of a metric space (M,d). Cosider the followig two coditios o E: (1) every cotiuous real-valued fuctio o E is uiformly cotiuous; isolated. (2) E = E U E 2 where E is compact ad E 2 is uiformly l l Recall from the begiig of this paper that (1) is equiva let to E beig a L-space. (M,d) is locally compact. Assume that (1) holds ad that Sice E is complete [5, Thro. 1], E is a closed, therefore locally compact subspace of M. Hece, by Theorem 3 above, (2) holds. With o restrictio o (M,d), we see from Theorem 3 that (2) implies (1). I Theorem 4 of [5] it is show that (1) implies (2) if closed ad bouded subsets of (M,d) are compact. However, as we have just show, (1) is equivalet to (2) i more geeral spaces (M,d). Similar geeralizatios of some other results i [5] are also possible to obtai by usig our results. I coectio with this we ote that coditio (2), which occurs frequetly i results i [4] ad [5], is really equivalet to E beig a locally compact L-space (by our Theorem 3).
TOPOLOGY PROCEEDINGS Volume 6 1981 369 The authors are grateful to Professor Norma Levie for refereces [1] ad [6]. Refereces 1. M. Atsuji, Uiform cotiuity of cotiuous fuctios of metric spaces, Pac. J. Math. a (1958), 11-16. 2. N. Duford ad J. T. Schwartz, Liear operato~s: Part I, Itersciece Publishers, New York, 1964. 3. J. L. Kelley ad I. Namioka, Liear topological spaces, D. Va Nostrad Compay, Ic., Priceto, New Jersey, 1963. 4. N. Levie, Uiformly cotiuous liear sets, Amer. Math. Mothly 62 (1955), 579-580. 5. ad W. G. Sauders, Uiformly cotiuous sets i metric spaces, Amer. Math. Mothly 67 (1960), 153 156. 6. A. A. Moteiro ad M. M. Peixoto, Le ombre de Lebesgue et la cotiuite uiforme, Portuga1iae Mathematica 10 (1951), 105-113. West Virgiia Uiversity Morgatow, West Virgiia 26506