A Note on Peano Spaces
نویسنده
چکیده
In three-dimensional space set up a cylindrical coordinate system (r, $, z). The Hahn-Mazurkiewicz theorem characterizes Peano spaces (locally connected metric (compact) continua) as the continuous images of the closed unit interval I on the z-axis. In this note we obtain an extension theorem for Peano spaces (henceforth called Pspaces) based upon this characterization. We first define a dendrite L. To this end order the rationals in the interior of I into a sequence {(0, 0, r,-}. For each pair of positive integers i, j, let Lij denote the closed line segment joining (0, 0, r<) and (lA'+j, l/i+j, Ti). Let P_i,y, Loj denote line segments from (0, 0, 0) and (0, 0,1) parallel to and the same length as Lltj for every j. Then the dendrite L is defined to be the union of I and all the segments Li,j. Let a,-,,be the end point of Lij which is not on I. We shall refer to Oij as the free end of Lij. Denote by D the sequence {(0, 0, di)} consisting of the dyadic rational points interior to I enumerated in the usual way: <fi=l/2, <f2=l/22, d3 = 3/2i, • ■ ■ . The following lemma is then easily established.
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تاریخ انتشار 2010