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WALLMAN SUBLATTICES AND QUASI-F COVERS

Lee, BongJu;Kim, ChangIl

  • Received : 2014.01.08
  • Accepted : 2014.01.27
  • Published : 2014.06.25

Abstract

In this paper, we first will show that for any space X and any Wallman sublattice $\mathcal{A}$ of $\mathcal{R}(X)$ with $Z(X)^{\sharp}{\subseteq}\mathcal{A}$, (${\Phi}^{-1}_{\mathcal{A}}(X)$, ${\Phi}_{\mathcal{A}}$) is the minimal quasi-F cover of X if and only if (${\Phi}^{-1}_{\mathcal{A}}(X)$, ${\Phi}_{\mathcal{A}}$) is a quasi-F cover of X and $\mathcal{A}{\subseteq}\mathcal{Q}_X$. Using this, if X is a locally weakly Lindel$\ddot{o}$f space, the set {$\mathcal{A}|\mathcal{A}$ is a Wallman sublattice of $\mathcal{R}(X)$ with $Z(X)^{\sharp}{\subseteq}\mathcal{A}$ and ${\Phi}^{-1}_{\mathcal{A}}(X)$ is the minimal quasi-F cover of X}, when partially ordered by inclusion, has the minimal element $Z(X)^{\sharp}$ and the maximal element $\mathcal{Q}_X$. Finally, we will show that any Wallman sublattice $\mathcal{A}$ of $\mathcal{R}(X)$ with $Z(X)^{\sharp}{\subseteq}\mathcal{A}{\subseteq}\mathcal{Q}_X$, ${\Phi}_{\mathcal{A}_X}:{\Phi}^{-1}_{\mathcal{A}}(X){\rightarrow}X$ is $z^{\sharp}$-irreducible if and only if $\mathcal{A}=\mathcal{Q}_X$.

Keywords

quasi-F space;covering map

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Cited by

  1. WALLMAN COVERS AND STONE-ČECH COMPACTIFICATIONS OF CLOZ COVERS vol.37, pp.3, 2015, https://doi.org/10.5831/HMJ.2015.37.3.287