PEANO COMPACTIFICATIONS AND PROPERTY $ METRIC SPACES

Let (X,d) denote a locally connected, connected separable metric space. We say the X is S-metrizable provided there is a topologically equivalent metric ρ on X such that (X,ρ) has Property S, i.e. for any ϵ>0, X is the union of finitely many connected sets of ρ-diameter less than ϵ. It is well-known that S-metrizable spaces are locally connected and that if ρ is a Property S metric for X, then the usual metric completion (X˜,ρ˜) of (X,ρ) is a compact, locally connected, connected metric space, i.e. (X˜,ρ˜) is a Peano compactification of (X,ρ). There are easily constructed examples of locally connected connected metric spaces which fail to be S-metrizable, however the author does not know of a non-S-metrizable space (X,d) which has a Peano compactification. In this paper we conjecture that: If (P,ρ) a Peano compactification of (X,ρ|X), X must be S-metrizable. Several (new) necessary and sufficient for a space to be S-metrizable are given, together with an example of non-S-metrizable space which fails to have a Peano compactification.

X, then the usual metric completion (,0) of (X,o) is a compact, locally connected, connected metric space, i.e. (,) is a Peano compactification of (X,o).There are easily constructed examples of locally connected connected metric spaces which fail to be S-metrizable, however the author does not know of a non-S-metrizable space (X,d) which has a Peano compactification.In this paper we conjecture that: If (P,0) a Peano compactification of (X,01X), X must be S-metrizable.Several (new) necessary and sufficient for a space to be S-metrizable are given, together with an example of non-S-metrizable space which fails to have a Peano compactification.Throughout this note let (X,d) denote a locally connected, connected separable metric space.We say that X is S-metrizable provided there is a topologically equivalent metric 0 on X such that (X,0) has Property S, i.e. for any e > 0, X is the union of finitely many connected sets of 0-dlameter less than e.It is well-known that S-metrizable spaces are locally connected and that if 0 is a Property S metric for X, then the usual metric completion (,) of (X,0) is a compact, locally connected, connected metric space, i.e. ,0) is a Peano compactification of (X,0) [8,p.154].
There are easily constructed examples of locally connected, connected metric spaces which fail to be S-metrizable, however the author does not know of a non- S-metri'zable space (X,d) which has a Peano compactification.We therefore ask: QUESTION I.If (P,p) is a Peano compactification of (X,01X) must X be S- me tr izab le ? 2.

DEFINITIONS AND BASIC RESULTS
A space Z is an extension of a space Y if Y is a dense subspace of Z.If Z is an extension of Y, we say that Y is locally connected, in Z if Z has a basis consisting of regions (that is, open connected sets) whose intersections with Y are region& in Y. Z is a perfect extension of Y if Z is an extension of Y and whenever a closed subset H of Y separates two sets A, BcY in Y, the set cl H (the closure of H in Z) separates A, B in Z. [6] z For completeness we include the following: THEOREM 2.1 [6].Let Z be an extension of X.Then X is locally connected in Z if and only if Z is a perfect locally connected extension of X.
THEOREM 2.2 [6].Let (X,d) be a metric space.Then X is S-metrizable if and only if X has a metrizable compactification Z in which it is locally con- nected.
THEOREM 2.3 [6].A topological space is S-metrizable if and only if it has a perfect locally connected metrizable compactification.
THEOREM 2.4 [6].Let X be a space having a perfect S-metrizable extension.
Then X is S-metrizable.
THEOREM 2.5 [5].Let X be a separable, locally connected, connected rim compact metric space.Then X is S-metrizable.
3. RELATED RESULTS AND QUESTIONS.
THEOREM 3.1.Let (P,d) be a Peano space and let X be a dense, locally con- nected, connected subset of P. Then there exists a G6-subset Y of P containing X such that X is locally connected in Y (as an extension of X).
PROOF.Let n be a positive integer and define Z Y E P: if U is an open n connected subset of P containing y and 8(U)<2 -n, then UX is not connected}.
(Here 6(U) denotes the d-diameter of U).We first assert that Z is closed.For suppose Yl' Y2' is a sequence in Zn which converges to y E (P\Zn).Since -n yZn, there exists an open connected subset U of P containing y and 8(U)<2 and IINZ t and this is a contradiction.Hence Z is closed.is a notonically increasing sequence and if for each iI, Yi=PZi Y Y. is a connected G6-subset of P which contains X.
i= I i We now assert that X is locally connected in Y, as an extension of X.For -n let >0 and let yY.Then there exists a positive integer n so that >2 2-n and since yZ there exists an open connected subset U of P with 6(U)< and n such that UX is connected.This implies that W--intyclyU is an open connected subset of Y. Thus Y has a basis, consisting of regions whose intersection with X is connected.This completes the proof.COROLLARY 3.1.1.Every dense, locally connected, connected G6-subset of Peano continuum is S-metrizable if and only if dense, locally connected, con- nected subset of a Peano continuum is S-metrlzable.
Since every nested ntersection of countably many sets can be represented as an inverse limit space and since every Y. above is S-metrizable, by (2.5), we ask: QUESTION 2. If [Yi' fl,j' is an inverse limit sequence of S-metrizable spaces and continuous maps (blcontlnuous injections), must Y inv lira {Yi' flj' be S-metrizable?
Of course an affir=mtlve answer to Question 2 would yield an affirnmtlve answer to Question i. THEOREM 3.2.Let (X,d) be a locally connected, connected separable metric space, let X denote the Stone-ech compactlflcatlon of X.Then X is S-metrlz- able if and only if there exists a Peano compactiflcation P of X such that f, the continuous extension of the identity injection f:X P to X, is monotone.
PROOF.Recall that a map between compact Hausdorff spaces is monotone if every point inverse is connected.Suppose that (X,d) is S-metrizable, say p is an S-metrlc for X.By (2.3), there exists a Peano compactiflcation P of X and X is locally connected In P. Let f'X P be the continuous extension of the -I identity map f:X P to X.We need to show that for y E P, 8f (y) is connected.
But since P is a metric space and X is locally connected in P, there exists neighborhood basis for y in P, Ui}i= I such that for i E q, cl UI+I cU i and KEY WORDS AND PHRASES.Propy S metric, Peano space, compaification.1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.54D05, 54F25.i. INTRODUCTION.
Z N X--.For let xX and let V be an open connected subset n 2 "n. of X such that (clV)< Then U--int clV is open in P and contains x and (U) <2 -n Furthermore, UOX is connected since V=IIOX=clV and V is connected Thus xZ and Z nX=.@.