In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R). We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen: (a) Y is not locally compact, (b) X has only one non isolated point and Y is not compact. Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties: (i) X has a non-isolated point with countable character, (ii) X is not pseudocompact, (iii) X is infinite, pseudocompact and Cp(X) is normal, (iv) X is an infinite generalized ordered space. Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.
@article{bwmeta1.element.doi-10_1515_taa-2015-0002,
author = {A. Dorantes-Aldama and R. Rojas-Hern\'andez and \'A. Tamariz-Mascar\'ua},
title = {The partially pre-ordered set of compactifications of Cp(X, Y)},
journal = {Topological Algebra and its Applications},
volume = {3},
year = {2015},
zbl = {06471067},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_taa-2015-0002}
}
A. Dorantes-Aldama; R. Rojas-Hernández; Á. Tamariz-Mascarúa. The partially pre-ordered set of compactifications of Cp(X, Y). Topological Algebra and its Applications, Tome 3 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_taa-2015-0002/
[1] Arhangelskii A.V., On R-quotient mappings of spaces with a countable base, Soviet Math. Dokl., 1986, 33, 302-305.
[2] Chandler R.E., Hausdorff Compactifications, Marcel Dekker, 1976. | Zbl 0338.54001
[3] Contreras-Carreto A., Espacios de funciones continuas del tipo Cp(X, E), PhD thesis, UNAM, 2003.
[4] Contreras-Carreto A., Tamariz-Mascarúa Á., On some generalizations of compactness in spaces Cp(X, 2) and Cp(X, Z), Bol. Soc. Mat. Mexicana, 2003, 9(3), 291-308.
[5] Dorantes-Aldama A., Tamariz-Mascarúa Á., Some results on weakly pseudocompact spaces, accepted for publication in Houston Journal of Math.
[6] Eckertson R., Sums, products, and mappings of weakly pseudocompact spaces Topology Appl., 1996, 72, 149-157. | Zbl 0857.54022
[7] Engelking R.. General Topology, Sigma Series in Pure Mathematics, 6, Heldermann Verlag Berlin, 1989. | Zbl 0684.54001
[8] García-Ferreira S., García-Máynez A., On weakly-pseudocompact spaces, Houston J. Math., 1994, 20(1), 145-159. | Zbl 0809.54012
[9] Kannan V., Thrivikraman T., Lattices of Hausdorff compactifications of a locally compact space, Pacific J. Math., 1975, 57, 441-444. | Zbl 0305.06004
[10] Magill K.D. Jr., The lattice of compactifications of locally compact spaces, Proc. Lond. Math. Soc., 1968, 18(3), 231-234. [Crossref]
[11] Porter J., Woods R.G., Extensions and Absolutes of Hausdorff Spaces, Springer-Verlag, 1988. | Zbl 0652.54016
[12] Thrivikraman T., On compactifications of Tychonoff spaces, Yokohama Math. J., 1972, 20, 99-106. | Zbl 0259.54021
[13] Thrivikraman T., On the lattice of compactifications, J. Lond. Math. Soc. (2), 1972, 4, 711-717. [Crossref] | Zbl 0233.54013
[14] Ünlu Y., Lattices of compactifications of Tychonoff spaces, General Topology and its Applications, 1978, 9, 41-57. | Zbl 0389.54015
[15] Visliseni J., Flachsmeyer J.. The power and the structure of the lattice of all compact extensions of a completely regular space, Doklady, 1965, 165(2), 1423-1425. | Zbl 0142.20901