Countable connected spaces
Author:
Gary Glenn Miller
Journal:
Proc. Amer. Math. Soc. 26 (1970), 355-360
MSC:
Primary 54.20
DOI:
https://doi.org/10.1090/S0002-9939-1970-0263005-0
MathSciNet review:
0263005
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Two pathological countable topological spaces are constructed. Each is quasimetrizable and has a simple explicit quasimetric. One is a locally connected Hausdorff space and is an extension of the rationals. The other is a connected space which becomes totally disconnected upon the removal of a single point. This space satisfies the Urysohn separation property—a property between ${T_2}$ and ${T_3}$—and is an extension of the space of rational points in the plane. Both are one dimensional in the Menger-Urysohn [inductive] sense and infinite dimensional in the Lebesgue [covering] sense.
- R. H. Bing, A connected countable Hausdorff space, Proc. Amer. Math. Soc. 4 (1953), 474. MR 60806, DOI https://doi.org/10.1090/S0002-9939-1953-0060806-9 M. Brown, A countable connected Hausdorff space, Bull. Amer. Math. Soc. 59 (1953), 367. Abstract #423.
- P. Erdös, Some remarks on connected sets, Bull. Amer. Math. Soc. 50 (1944), 442–446. MR 10603, DOI https://doi.org/10.1090/S0002-9904-1944-08171-8
- Peter Fletcher, Hughes B. Hoyle III, and C. W. Patty, The comparison of topologies, Duke Math. J. 36 (1969), 325–331. MR 242107 S. P. Franklin and G. V. Krishnarao, A topological characterization of the real line, Notices Amer. Math. Soc. 16 (1969), 694. Abstract #69T-G78.
- Solomon W. Golomb, A connected topology for the integers, Amer. Math. Monthly 66 (1959), 663–665. MR 107622, DOI https://doi.org/10.2307/2309340
- Edwin Hewitt, On two problems of Urysohn, Ann. of Math. (2) 47 (1946), 503–509. MR 17527, DOI https://doi.org/10.2307/1969089
- A. M. Kirch, A countable, connected, locally connected Hausdorff space, Amer. Math. Monthly 76 (1969), 169–171. MR 239563, DOI https://doi.org/10.2307/2317265 B. Knaster and C. Kuratowski, Sur les ensembles connexes, Fund. Math. 2 (1921), 206-255.
- Joseph Martin, A countable Hausdorff space with a dispersion point, Duke Math. J. 33 (1966), 165–167. MR 192474 G. Miller, A countable Urysohn space with an explosion point, Notices Amer. Math. Soc. 13 (1966), 589. Abstract #636-48. ---, A countable locally connected quasimetric space, Notices Amer. Math. Soc. 14(1967), 720. Abstract #67T-541.
- Gary Miller and B. J. Pearson, On the connectification of a space by a countable point set, J. Austral. Math. Soc. 13 (1972), 67–75. MR 0292039
- Lester J. Norman, A sufficient condition for quasi-metrizability of a topological space, Portugal. Math. 26 (1967), 207–211. MR 248740
- Prabir Roy, A countable connected Urysohn space with a dispersion point, Duke Math. J. 33 (1966), 331–333. MR 196701
- Ronald A. Stoltenberg, On quasi-metric spaces, Duke Math. J. 36 (1969), 65–71. MR 235515 A. H. Stone, A countable, connected, locally connected Hausdorff space, Notices Amer. Math. Soc. 16 (1969), 422. Abstract #69T-D10.
- Paul Urysohn, Über die Mächtigkeit der zusammenhängenden Mengen, Math. Ann. 94 (1925), no. 1, 262–295 (German). MR 1512258, DOI https://doi.org/10.1007/BF01208659
- R. L. Wilder, A point set which has no true quasicomponents, and which becomes connected upon the addition of a single point, Bull. Amer. Math. Soc. 33 (1927), no. 4, 423–427. MR 1561394, DOI https://doi.org/10.1090/S0002-9904-1927-04395-6
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54.20
Retrieve articles in all journals with MSC: 54.20
Additional Information
Keywords:
Countable connected Hausdorff space,
dispersion point,
locally connected space,
Urysohn space,
quasimetric,
Menger-Urysohn dimension,
Lebesgue dimension,
completion of the rationals
Article copyright:
© Copyright 1970
American Mathematical Society