-connectedness in hereditarily locally connected spaces

Authors:
J. Grispolakis and E. D. Tymchatyn

Journal:
Trans. Amer. Math. Soc. **253** (1979), 303-315

MSC:
Primary 54D05

DOI:
https://doi.org/10.1090/S0002-9947-1979-0536949-2

MathSciNet review:
536949

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: B. Knaster, A. Lelek and J. Mycielski [Colloq. Math. **6** (1958), 227-246] 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), 87-96].

**1.**Schieffelin Claytor,*Topological immersion of Peanian continua in a spherical surface*, Ann. of Math. (2)**35**(1934), no. 4, 809–835. MR**1503198**, https://doi.org/10.2307/1968496**[1]**R. Engelking,*Outline of general topology*, Translated from the Polish by K. Sieklucki, North-Holland Publishing Co., Amsterdam; PWN-Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. MR**0230273****[2]**H. M. Gehman,*Concerning the subsets of aplane continuous curve*, Ann. of Math.**27**(1926), 29-46.**[3]**J. Grispolakis, A. Lelek, and E. D. Tymchatyn,*Connected subsets of finitely Suslinian continua*, Colloq. Math.**35**(1976), no. 2, 209–222. MR**0407815****[4]**J. Grispolakis and E. D. Tymchatyn,*Hereditarily 𝜎-connected continua*, Colloq. Math.**41**(1979), no. 1, 57–65. MR**550628****[5]**B. Knaster, A. Lelek, and Jan Mycielski,*Sur les décompositions d’ensembles connexes*, Colloq. Math.**6**(1958), 227–246 (French). MR**0107850****[6]**K. Kuratowski,*Topology. Vol. II*, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. MR**0259835****[7]**A. Lelek,*Ensembles 𝜎-connexes et le théorème de Gehman*, Fund. Math.**47**(1959), 265–276 (French). MR**0110087****[8]**A. Lelek,*Arcwise connected and locally arcwise connected sets*, Colloq. Math.**36**(1976), no. 1, 87–96. MR**0431106****[9]**A. Lelek and L. F. McAuley,*On hereditarily locally connected spaces and one-to-one continuous images of a line*, Colloq. Math.**17**(1967), 319–324. MR**0220251****[10]**T. Nishiura and E. D. Tymchatyn,*Hereditarily locally connected spaces*, Houston J. Math.**2**(1976), no. 4, 581–599. MR**0436072****[11]**G. Nöbeling,*Über regulär-eindimensionale Räume*, Math. Ann.**104**(1931), 81.**[12]**M. Shimrat,*Simply disconnectible sets*, Proc. London Math. Soc. (3)**9**(1959), 177–188. MR**0105070**, https://doi.org/10.1112/plms/s3-9.2.177**[13]**E. D. Tymchatyn,*Compactification of hereditarily locally connected spaces*, Canad. J. Math.**29**(1977), no. 6, 1223–1229. MR**0464186**, https://doi.org/10.4153/CJM-1977-122-5**[14]**E. D. Tymchatyn,*The Hahn-Mazurkiewicz theorem for finitely Suslinian continua*, General Topology and Appl.**7**(1977), no. 1, 123–127. MR**0431107****[15]**Gordon Thomas Whyburn,*Analytic Topology*, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. MR**0007095****[16]**G. T. Whyburn,*Concerning points of continuous curves defined by certain im kleinen properties*, Math. Ann.**102**(1930), no. 1, 313–336. MR**1512580**, https://doi.org/10.1007/BF01782349

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54D05

Retrieve articles in all journals with MSC: 54D05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0536949-2

Keywords:
Hereditarily locally connected spaces,
finitely Suslinian,
weakly -connected spaces,
-connected spaces

Article copyright:
© Copyright 1979
American Mathematical Society