connectedness in hereditarily locally connected spaces
Authors:
J. Grispolakis and E. D. Tymchatyn
Journal:
Trans. Amer. Math. Soc. 253 (1979), 303315
MSC:
Primary 54D05
MathSciNet review:
536949
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Abstract: B. Knaster, A. Lelek and J. Mycielski [Colloq. Math. 6 (1958), 227246] had asked whether there exists a hereditarily locally connected planar set, which is the union of countably many disjoint arcs. They gave an example of a locally connected, connected planar set, which is the union of a countable sequence of disjoint arcs. Lelek proved in a paper in Fund. Math. in 1959, that connected subsets of planar hereditarily locally connected continua are weakly connected (i.e., they cannot be written as unions of countably many disjoint, closed connected subsets). In this paper we generalize the notion of finitely Suslinian to noncompact spaces. We prove that there is a class of spaces, which includes the class of planar hereditarily locally connected spaces and the finitely Suslinian spaces, and which are weakly connected, thus, answering the above question in the negative. We also prove that arcwise connected, hereditarily locally connected, planar spaces are locally arcwise connected. This answers in the affirmative a question of Lelek [Colloq. Math. 36 (1976), 8796].
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 S. Claytor, Topological immersion of Peanian continua in a spherical surface, Ann. of Math. 33 (1934), 809835. MR 1503198
 [1]
 R. Engelking, Outline of general topology, NorthHolland, Amsterdam, 1968. MR 0230273 (37:5836)
 [2]
 H. M. Gehman, Concerning the subsets of aplane continuous curve, Ann. of Math. 27 (1926), 2946.
 [3]
 J. Grispolakis, A. Lelek and E. D. Tymchatyn, Connected subsets of finitely Suslinian continua, Colloq. Math. 35 (1976), 209222. MR 0407815 (53:11585)
 [4]
 J. Grispolakis and E. D. Tymchatyn, On hereditarily connected continua, Colloq. Math. (to appear). MR 550628 (80i:54043)
 [5]
 B. Knaster, A. Lelek and J. Mycielski, Sur les décompositions d'ensembles connexes, Colloq. Math. 6 (1958), 227246. MR 0107850 (21:6572)
 [6]
 K. Kuratowski, Topology. Vol. II, Academic Press, New York, 1968. MR 0259835 (41:4467)
 [7]
 A. Lelek, Ensembles connexes et le théorème de Gehman, Fund. Math. 47 (1959), 265276. MR 0110087 (22:970)
 [8]
 , Arcwise connected and locally arcwise connected sets, Colloq. Math. 36 (1976), 8796. MR 0431106 (55:4108)
 [9]
 A. Lelek and L. F. McAuley, On hereditarily locally connected spaces and onetoone continuous images of a line, Colloq. Math. 17 (1967), 319324. MR 0220251 (36:3317)
 [10]
 T. Nishiura and E. D. Tymchatyn, Hereditarily locally connected spaces, Houston J. Math. 2 (1976), 581599. MR 0436072 (55:9023)
 [11]
 G. Nöbeling, Über reguläreindimensionale Räume, Math. Ann. 104 (1931), 81.
 [12]
 M. Shimrat, Simply disconnectible sets, Proc. London Math. Soc. 9 (1959), 177188. MR 0105070 (21:3816)
 [13]
 E. D. Tymchatyn, Compactifications of hereditarily locally connected spaces, Canad. J. Math. 29 (1977), 12231229. MR 0464186 (57:4121)
 [14]
 , The HahnMazurkiewicz theorem for finitely Suslinian continua, General Topology and Appl. 7 (1977), 123127. MR 0431107 (55:4109)
 [15]
 G. T. Whyburn, Analytic topology, Amer. Math. Soc., Providence, R. I., 1942. MR 0007095 (4:86b)
 [16]
 , Concerning points of continuous curves defined by certain im kleinen properties, Math. Ann. 102 (1930), 313336. MR 1512580
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905369492
PII:
S 00029947(1979)05369492
Keywords:
Hereditarily locally connected spaces,
finitely Suslinian,
weakly connected spaces,
connected spaces
Article copyright:
© Copyright 1979
American Mathematical Society
