Remote point
In general topology, a remote point is a point [math]\displaystyle{ p }[/math] that belongs to the Stone–Čech compactification [math]\displaystyle{ \beta X }[/math] of a Tychonoff space [math]\displaystyle{ X }[/math] but that does not belong to the topological closure within [math]\displaystyle{ \beta X }[/math] of any nowhere dense subset of [math]\displaystyle{ X }[/math].[1] Let [math]\displaystyle{ \R }[/math] be the real line with the standard topology. In 1962, Nathan Fine and Leonard Gillman proved that, assuming the continuum hypothesis:
There exists a point [math]\displaystyle{ p }[/math] in [math]\displaystyle{ \beta \R }[/math] that is not in the closure of any discrete subset of [math]\displaystyle{ \R }[/math] ...[2]
Their proof works for any Tychonoff space that is separable and not pseudocompact.[1]
Chae and Smith proved that the existence of remote points is independent, in terms of Zermelo–Fraenkel set theory, of the continuum hypothesis for a class of topological spaces that includes metric spaces.[3] Several other mathematical theorems have been proved concerning remote points.[4][5]
References
- ↑ 1.0 1.1 Van Douwen, Eric K. (1978). "Existence and applications of remote points". Bulletin of the American Mathematical Society 84 (1): 161–164. doi:10.1090/S0002-9904-1978-14454-1. ISSN 0002-9904.
- ↑ Fine, Nathan J.; Gillman, Leonard (1962). "Remote points in [math]\displaystyle{ \beta R }[/math]". Proceedings of the American Mathematical Society 13: 29–36. doi:10.1090/S0002-9939-1962-0143172-5.
- ↑ Chae, Soo Bong; Smith, Jeffrey H. (1980). "Remote points and G-spaces". Topology and Its Applications 11 (3): 243–246. doi:10.1016/0166-8641(80)90023-1.
- ↑ Van Mill, Jan; Van Douwen, Eric (March 1983). "Spaces without remote points". Pacific Journal of Mathematics 105 (1): 69–75. doi:10.2140/pjm.1983.105.69. https://msp.org/pjm/1983/105-1/p06.xhtml.
- ↑ Dow, Alan (1983). "Remote points in large products". Topology and Its Applications 16 (1): 11–17. doi:10.1016/0166-8641(83)90003-2. ISSN 0166-8641.
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