Dispersion points and rational curves

Lipham, David

doi:10.1090/proc/14920

Dispersion points and rational curves
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Proc. Amer. Math. Soc. 148 (2020), 2671-2682 Request permission

Abstract:

We construct two connected plane sets which can be embedded into rational curves. The first is a biconnected set with a dispersion point. It answers a question of Joachim Grispolakis. The second is indecomposable. Both examples are completely metrizable.

References

Howard Cook, W. T. Ingram, and Andrew Lelek, A list of problems known as Houston problem book, Continua (Cincinnati, OH, 1994) Lecture Notes in Pure and Appl. Math., vol. 170, Dekker, New York, 1995, pp. 365–398. MR 1326857, DOI 10.2307/3618329
W. Dębski, Note on a problem by H. Cook, Houston J. Math. 17 (1991), no. 3, 439–441. MR 1126608
Jan J. Dijkstra and D. S. Lipham, On cohesive almost zero-dimensional spaces, preprint.
Ryszard Engelking, Dimension theory, North-Holland Mathematical Library, vol. 19, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author. MR 0482697
S. D. Iliadis, A note on compactifications of rim-scattered spaces, Rend. Circ. Mat. Palermo (2) Suppl. 29 (1992), 425–433. V International Meeting on Topology in Italy (Italian) (Lecce, 1990/Otranto, 1990). MR 1197185
S. D. Iliadis and E. D. Tymchatyn, Compactifications with minimum rim-types of rational spaces, Houston J. Math. 17 (1991), no. 3, 311–323. MR 1126596
F. Burton Jones, Wilder on connectedness, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977) Lecture Notes in Math., vol. 664, Springer, Berlin, 1978, pp. 1–6. MR 518401
B. Knaster and C. Kuratowski, Sur les ensembles connexes, Fundamenta Math. 2 (1921) 206–255.
E. Lamb, A Few of My Favorite Spaces: Cantor’s Leaky Tent, Scientific American, https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-cantor-s-leaky-tent/, June 20 2015.
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
David Lipham, Widely-connected sets in the bucket-handle continuum, Fund. Math. 240 (2018), no. 2, 161–174. MR 3720922, DOI 10.4064/fm378-3-2017
E. W. Miller, Concerning biconnected sets, Fundamenta Math. 29 (1937) 123–133.
Piotr Minc, Countable subsets of Suslinian continua, Fund. Math. 124 (1984), no. 2, 123–129. MR 774503, DOI 10.4064/fm-124-2-123-129
P. M. Swingle, Two types of connected sets, Bull. Amer. Math. Soc. 37 (1931), no. 4, 254–258. MR 1562132, DOI 10.1090/S0002-9904-1931-05144-2
E. D. Tymchatyn, Compactifications of rational spaces, Houston J. Math. 3 (1977), no. 1, 131–139. MR 428286
R. L. Wilder, Evolution of the topological concept of “connected”, Amer. Math. Monthly 85 (1978), no. 9, 720–726. MR 514036, DOI 10.2307/2321676