Remote point

In general topology, a remote point is a point ${\displaystyle p}$ that belongs to the Stone–Čech compactification ${\displaystyle \beta X}$ of a Tychonoff space ${\displaystyle X}$ but that does not belong to the topological closure within ${\displaystyle \beta X}$ of any nowhere dense subset of ${\displaystyle X}$.^[1]

Let ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle p}$ in ${\displaystyle \beta \mathbb {R} }$ that is not in the closure of any discrete subset of ${\displaystyle \mathbb {R} }$ ...^[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. 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.
2. Fine, Nathan J.; Gillman, Leonard (1962). "Remote points in ${\displaystyle \beta R}$". Proceedings of the American Mathematical Society. 13: 29–36. doi:10.1090/S0002-9939-1962-0143172-5.
3. 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.
4. 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.
5. 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.