Characterizations of continua in which connected subsets are arcwise connected

Tymchatyn, E. D.

doi:10.1090/S0002-9947-1976-0423318-6

Characterizations of continua in which connected subsets are arcwise connected
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by E. D. Tymchatyn

Trans. Amer. Math. Soc. 222 (1976), 377-388

DOI: https://doi.org/10.1090/S0002-9947-1976-0423318-6

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Abstract:

The purpose of this paper is to give several characterizations of the continua in which all connected subsets are arcwise connected. The methods used are those developed by B. Knaster and K. Kuratowski, G. T. Whyburn and the author. These methods depend on Bernstein’s decomposition of a topologically complete metric space into totally imperfect sets and on Whyburn’s theory of local cutpoints. Some properties of connected sets in finitely Suslinian spaces are obtained. Two questions raised by the author are answered. Several partial results of Whyburn are obtained as corollaries of the main result.

References

James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
J. Grispolakis, A. Lelek, and E. D. Tymchatyn, Connected subsets of finitely Suslinian continua, Colloq. Math. 35 (1976), no. 2, 209–222. MR 407815
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
B. Knaster and C. Kuratowski, A connected and connected im kleinen point set which contains no perfect subset, Bull. Amer. Math. Soc. 33 (1927), no. 1, 106–109. MR 1561328, DOI 10.1090/S0002-9904-1927-04326-9
B. Knaster, A. Lelek, and Jan Mycielski, Sur les décompositions d’ensembles connexes, Colloq. Math. 6 (1958), 227–246 (French). MR 107850, DOI 10.4064/cm-6-1-227-246
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
A. Lelek, On the topology of curves. II, Fund. Math. 70 (1971), no. 2, 131–138. MR 283765, DOI 10.4064/fm-70-2-131-138
E. D. Tymchatyn, Continua in which all connected subsets are arcwise connected, Trans. Amer. Math. Soc. 205 (1975), 317–331. MR 365523, DOI 10.1090/S0002-9947-1975-0365523-2
Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095, DOI 10.1090/coll/028
G. T. Whyburn, Concerning points of continuous curves defined by certain im kleinen properties, Math. Ann. 102 (1930), no. 1, 313–336. MR 1512580, DOI 10.1007/BF01782349
Gordon T. Whyburn, On the Existence of Totally Imperfect and Punctiform Connected Subsets in a Given Continuum, Amer. J. Math. 55 (1933), no. 1-4, 146–152. MR 1506952, DOI 10.2307/2371118
G. T. Whyburn, Sets of local separating points of a continuum, Bull. Amer. Math. Soc. 39 (1933), no. 2, 97–100. MR 1562558, DOI 10.1090/S0002-9904-1933-05567-2
G. T. Whyburn, Local separating points of continua, Monatsh. Math. Phys. 36 (1929), no. 1, 305–314. MR 1549688, DOI 10.1007/BF02307620

On a problem of W. L. Ayres

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