A Hahn-Mazurkiewicz Theorem for generalized Peano continua By R. AYALA, M. J. CHA Â VEZ and A. QUINTERO Abstract. S. Mazurkiewicz remarked after his proof of the celebrated Hahn- Mazurkiewicz Theorem ([5]) that any generalized Peano continuum is the continuous image of the half-line 0; 1 but the converse does not hold. Therefore continuous images of 0; 1 do not characterize generalized Peano continua. In this paper we use perfect maps and trees to obtain an analogue of the Hahn-Mazurkiewicz Theorem for generalized Peano continua in the spirit of the classical Hahn-Mazurkiewicz Theorem. 0. Introduction. The well known Hahn-Mazurkiewicz Theorem ([5]) establishes that a Hausdorff space P is a Peano continuum if and only if it is the image of the unit interval I 0; 1. This theorem was the culmination of the study started by Peanos celebrated example of a square-filling curve. When compactness is replaced by local compactness we still have an interesting class of topological spaces, namely, generalized Peano continua. These spaces were already considered by the founders of continuum theory. In fact, Mazurkiewicz in his seminal paper [5] showed that any generalized Peano continuum is the continuous image of the half- line 0; 1 but the converse does not hold. Therefore continuous images of 0; 1 do not characterize generalized Peano continua. In this paper we use perfect maps and trees to obtain an analogue of the Hahn-Mazurkiewicz Theorem for generalized Peano continua (Theorem 2.4). In spite of its simple nature this result seems to be new in the literature. We use this theorem to give two further results on generalized continua which extend two basic results on Peano continua (Corollaries 2.6 and 2.7). 1. Generalized Peano continua. In this paper we shall deal with the class of generalized Peano continua. We recall that a continuum X is a compact connected metrizable space. When compactness is replaced by local compactness the space X is called a generalized continuum. If in addition X is locally connected it is called a (generalized) Peano continuum. Hence, any (generalized) Peano continuum is arcwise connected by ([7]; 4.2.5). Moreover it follows from ([3]; 4.4 F.(c)) that any generalized continuum is separable and hence second countable and s-compact ( [3]; 4.1.16, and 3.8.c(b) ). The local compactness together with the s-compactness yield that X is a countable union [ 1 n1 K n of compact subsets K n 7 X with K n 7 int K n1 . Actually we can assume without loss of generality that each K n is connected and all the components of X  K n are unbounded. Indeed, each K n is contained in a finite Arch. Math. 71 (1998) 325 ± 330 0003-889X/98/040325-06 $ 2.70/0  Birkhäuser Verlag, Basel, 1998 Archiv der Mathematik Mathematics Subject Classification (1991): 54F15, 54E40.