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

A. Dorantes-Aldama; R. Rojas-Hernández; Á. Tamariz-Mascarúa

Topological Algebra and its Applications (2015)

  • Volume: 3, Issue: 1, page 11-25, electronic only
  • ISSN: 2299-3231

Abstract

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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.

How to cite

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A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa. "The partially pre-ordered set of compactifications of Cp(X, Y)." Topological Algebra and its Applications 3.1 (2015): 11-25, electronic only. <http://eudml.org/doc/270867>.

@article{A2015,
abstract = {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.},
author = {A. Dorantes-Aldama, R. Rojas-Hernández, Á. Tamariz-Mascarúa},
journal = {Topological Algebra and its Applications},
keywords = {Compactifications of spaces of continuous functions; lattices; b-lattices; k-embedded subsets; Cembedded subsets; retracts; weakly pseudocompact spaces; compactifications of spaces of continuous functions; $b$-lattices; $k$-embedded subsets; cembedded subsets},
language = {eng},
number = {1},
pages = {11-25, electronic only},
title = {The partially pre-ordered set of compactifications of Cp(X, Y)},
url = {http://eudml.org/doc/270867},
volume = {3},
year = {2015},
}

TY - JOUR
AU - A. Dorantes-Aldama
AU - R. Rojas-Hernández
AU - Á. Tamariz-Mascarúa
TI - The partially pre-ordered set of compactifications of Cp(X, Y)
JO - Topological Algebra and its Applications
PY - 2015
VL - 3
IS - 1
SP - 11
EP - 25, electronic only
AB - 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.
LA - eng
KW - Compactifications of spaces of continuous functions; lattices; b-lattices; k-embedded subsets; Cembedded subsets; retracts; weakly pseudocompact spaces; compactifications of spaces of continuous functions; $b$-lattices; $k$-embedded subsets; cembedded subsets
UR - http://eudml.org/doc/270867
ER -

References

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