Every contractible fan is locally connected at its vertex
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- by Lex G. Oversteegen PDF
- Trans. Amer. Math. Soc. 260 (1980), 379-402 Request permission
Abstract:We prove that each contractible fan is locally connected at its vertex. It follows that every contractible fan is embeddable in the plane. This gives a solution to a problem raised by J. J. Charatonik and C. A. Eberhart.
- David P. Bellamy and Janusz J. Charatonik, The set function $T$ and contractibility of continua, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 1, 47–49 (English, with Russian summary). MR 500858
- Karol Borsuk, A countable broom which cannot be imbedded in the plane, Colloq. Math. 10 (1963), 233–236. MR 155300, DOI 10.4064/cm-10-2-233-236
- J. J. Charatonik and C. A. Eberhart, On contractible dendroids, Colloq. Math. 25 (1972), 89–98, 164. MR 309082, DOI 10.4064/cm-25-1-89-98
- J. J. Charatonik, Problems and remarks on contractibility of curves, General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976) Soc. Czechoslovak Mathematicians and Physicists, Prague, 1977, pp. 72–76. MR 0464197
- J. J. Charatonik and Z. Grabowski, Homotopically fixed arcs and the contractibility of dendroids, Fund. Math. 100 (1978), no. 3, 229–237. MR 509549, DOI 10.4064/fm-100-3-229-237 G. B. Graham, On contractible fans, Doctoral Dissertation, Univ. of California, Riverside, Calif., 1977.
- K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
- J. Mioduszewski, Everywhere oscillating functions, extension of the uniformization and homogeneity of the pseudo-arc, Fund. Math. 56 (1964), 131–155. MR 176452, DOI 10.4064/fm-56-2-131-155
- Lex G. Oversteegen, Noncontractibility of continua, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 9-10, 837–840 (English, with Russian summary). MR 518989
- Lex G. Oversteegen, Fans and embeddings in the plane, Pacific J. Math. 83 (1979), no. 2, 495–503. MR 557948 —, Properties of contractible fans, Doctoral dissertation, Wayne State Univ., Detroit, Mich., 1978.
- Lex G. Oversteegen, An uncountable collection of noncontractible fans, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 5, 385–389 (English, with Russian summary). MR 557407
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 260 (1980), 379-402
- MSC: Primary 54F20; Secondary 54F25
- DOI: https://doi.org/10.1090/S0002-9947-1980-0574786-1
- MathSciNet review: 574786