Disconnection in the Alexandroff duplicate
Keywords:extremally disconnected space, Alexandroff duplicate, P-space
It was demonstrated in  that the Alexandroff duplicate of the Čech-Stone compactification of the naturals is not extremally disconnected. The question was raised as to whether the Alexandroff duplicate of a non-discrete extremally disconnected space can ever be extremally disconnected. We answer this question in the affirmative; an example of van Douwen is significant. In a slightly different direction we also characterize when the Alexandroff duplicate of a space is a P-space as well as when it is an almost P-space.
P. Alexandrov and P. Urysohn, Memoire sur les espaces topologiques compacts, Verh. Akad. Wetensch. Amsterdam, 14 (1929), 1-96.
K. Almontashery and L. Kalantan, Results about the Alexandroff duplicate space, Appl. Gen. Topol. 17, no. 2 (2016), 117-122. https://doi.org/10.4995/agt.2016.4521
A. V. Arkhangel'skii, Topological Function Spaces, Mathematics and Its Applications, 78, Springer, Netherlands, 1992. https://doi.org/10.1007/978-94-011-2598-7
G. Bezhanishvili, N. Bezhanishvili, J. Lucero-Bryan and J. van Mill, S4.3 and hereditarily extremally disconnected spaces, Georgian Mathematical Journal 22, no. 4 (2015), 469-475. https://doi.org/10.1515/gmj-2015-0041
A. Caserta and S. Watson, The Alexandroff duplicate and its subspaces, Appl. Gen. Topol. 8, no. 2 (2007), 187-205. https://doi.org/10.4995/agt.2007.1880
R. Engelking, On functions defined on Cartesian products, Fund. Math. 59 (1966), 221-231. https://doi.org/10.4064/fm-59-2-221-231
L. Gillman and M. Jerison, Rings of Continuous Functions, Graduate Texts in Mathametics, vol. 43, Springer Verlag, Berlin-Heidelberg-New York, 1976.
E. van Douwen, Applications of maximal topologies, Topology Appl. 51 (1993), 125-139. https://doi.org/10.1016/0166-8641(93)90145-4
J. van Mill, Weak P-points in Čech-Stone compactifications, Trans. Amer. Math. Soc. 273 (1982), 657-678. https://doi.org/10.2307/1999934
J. L. Verner, Lonely points revisited, Comment. Math. Univ. Carolin. 54, no. 1 (2013), 105-110.
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