Countable connected spaces

Miller, Gary Glenn

doi:10.1090/S0002-9939-1970-0263005-0

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
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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.

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