Disconnection in the Alexandroff duplicate

[EN] It was demonstrated in [2] 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.Bhattacharjee, P.; Knox, ML.; Mcgovern, WW. (2021). Disconnection in the Alexandroff duplicate. Applied General Topology. 22(2):331-344. https://doi.org/10.4995/agt.2021.14602OJS331344222P. 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.2016.4521A. V. Arkhangel'skii, Topological Function Spaces, Mathematics and Its Applications, 78, Springer, Netherlands, 1992. https://doi.org/10.1007/978-94-011-2598-7G. 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-2015-0041A. 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.2007.1880R. Engelking, On functions defined on Cartesian products, Fund. Math. 59 (1966), 221-231. https://doi.org/10.4064/fm-59-2-221-231L. 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)90145-4J. van Mill, Weak P-points in Čech-Stone compactifications, Trans. Amer. Math. Soc. 273 (1982), 657-678. https://doi.org/10.2307/1999934J. L. Verner, Lonely points revisited, Comment. Math. Univ. Carolin. 54, no. 1 (2013), 105-110

The classical ring of quotients of Cc(X)C_c(X)

[EN] We construct the classical ring of quotients of the algebra of continuous real-valued functions with countable range. Our construction is a slight modification of the construction given in [M. Ghadermazi, O.A.S. Karamzadeh, and M. Namdari, On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova, to appear]. Dowker's example shows that the two constructions can be different.Bhattacharjee, P.; Knox, ML.; Mcgovern, WW. (2014). The classical ring of quotients of $C_c(X)$. Applied General Topology. 15(2):147-154. doi:http://dx.doi.org/10.4995/agt.2014.3181.SWORD147154152Hager, A. W., Kimber, C. M., & McGovern, W. W. (2005). Unique a-closure for some ℓ-groups of rational valued functions. Czechoslovak Mathematical Journal, 55(2), 409-421. doi:10.1007/s10587-005-0031-zHenriksen, M., & Woods, R. G. (2004). Cozero complemented spaces; when the space of minimal prime ideals of a C(X) is compact. Topology and its Applications, 141(1-3), 147-170. doi:10.1016/j.topol.2003.12.004Knox, M. L., & McGovern, W. W. (2008). Rigid extensions of ℓ-groups of continuous functions. Czechoslovak Mathematical Journal, 58(4), 993-1014. doi:10.1007/s10587-008-0064-1R. Levy and M. D. Rice, Normal $P$-spaces and the $G_delta$-topology, Colloq. Math. 44, no. 2 (1981), 227-240.Levy, R., & Shapiro, J. (2005). Rings of quotients of rings of functions. Topology and its Applications, 146-147, 253-265. doi:10.1016/j.topol.2003.03.003A. Mysior, Two easy examples of zero-dimensional spaces, Proc. Amer. Math. Soc. 92, no. 4 (1984), 615-617.Porter, J. R., & Woods, R. G. (1988). Extensions and Absolutes of Hausdorff Spaces. doi:10.1007/978-1-4612-3712-9Rudin, W. (1957). Continuous functions on compact spaces without perfect subsets. Proceedings of the American Mathematical Society, 8(1), 39-39. doi:10.1090/s0002-9939-1957-0085475-

Ancient hydrothermal seafloor deposits in Eridania basin on Mars

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Saturation, Yosida Covers and Epicompleteness in Compact Normal Frames

Abstract In this article the frame-theoretic account of what is archimedean for order-algebraists, and semisimple for people who study commutative rings, deepens with the introduction of J -frames: compact normal frames that are join-generated by their saturated elements. Yosida frames are examples of these. In the category of J -frames with suitable skeletal morphisms, the strongly projectable frames are epicomplete, and thereby it is proved that the monoreflection in strongly projectable frames is the largest such. That is news, because it settles a problem that had occupied the first-named author for over five years. In compact normal Yosida frames the compact elements are saturated. When the reverse is true one gets the perfectly saturated frames: the frames of ideals Idl(E), when E is a compact regular frame. The assignment E → Idl(E) is then a functorial equivalence from compact regular frames to perfectly saturated frames, and the inverse equivalence is the saturation quotient. Inevitable are the Yosida covers (of a J -frame L): coherent, normal Yosida frames of the form Idl(F), with F ranging over certain bounded sublattices of the saturation SL of L. These Yosida frames cover L in the sense that each maps onto L densely and codensely. Modulo an equivalence, the Yosida covers of L form a poset with a top Y L, the latter being characterized as the only Yosida cover which is perfectly saturated. Viewed correctly, these Yosida covers provide, in a categorical setting, another (point-free) look at earlier accounts of coherent normal Yosida frames