The partially pre-ordered set of compactifications of Cp(X, Y)

• Autorzy: A. Dorantes-Aldama , R. Rojas-Hernández , Á. Tamariz-Mascarúa
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Compactifications of spaces of continuous functions; lattices; b-lattices; k-embedded subsets; Cembeddedsubsets; retracts; weakly pseudocompact spaces

• Wydawca: De Gruyter Open
• Czasopismo: Topological Algebra and its Applications
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2014-09-19

zaakceptowano

2015-04-03

online

2015-07-06

Twórcy

autor

A. Dorantes-Aldama

• Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito exterior
  s/n, Ciudad Universitaria, CP 04510, México D. F., México

autor

R. Rojas-Hernández

• Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma del Estado de México, Instituto
  Literario No. 100, Col. Centro, C.P. 50000, Toluca, Estado de México, México

autor

Á. Tamariz-Mascarúa

• Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito exterior
  s/n, Ciudad Universitaria, CP 04510, México D. F., México

Bibliografia

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Identyfikatory

DOI

10.1515/taa-2015-0002