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- W2888098991 abstract "For a hypergraph $H=(V,mathcal E)$, a subfamily $mathcal Csubseteq mathcal E$ is called a cover of the hypergraph if $bigcupmathcal C=bigcupmathcal E$. A cover $mathcal C$ is called minimal if each cover $mathcal Dsubseteqmathcal C$ of the hypergraph $H$ coincides with $mathcal C$. We prove that for a hypergraph $H$ the following conditions are equivalent: (i) each countable subhypergraph of $H$ has a minimal cover; (ii) each non-empty subhypergraph of $H$ has a maximal edge; (iii) $H$ contains no isomorphic copy of the hypergraph $(omega,omega)$. This characterization implies that a countable hypergraph $(V,mathcal E)$ has a minimal cover if every infinite set $Isubseteq V$ contains a finite subset $Fsubseteq I$ such that the family of edges $mathcal E_F:={Einmathcal E:Fsubseteq E}$ is finite. Also we prove that a hypergraph $(V,mathcal E)$ has a minimal cover if $sup{|E|:Einmathcal E}<omega$ or for every $vin V$ the family $mathcal E_v:={Einmathcal E:vin E}$ is finite. Applying these results to topological spaces, we define a topological space $X$ to be minicompact if each open cover of $X$ has a minimal open refinement and prove that each metacompact space $X$ is minicompact. On the other hand, every uncountable regular cardinal endowed with the interval topology is not minicompact." @default.
- W2888098991 created "2018-08-31" @default.
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- W2888098991 date "2018-08-24" @default.
- W2888098991 modified "2023-09-27" @default.
- W2888098991 title "Minimal covers of hypergraphs with applications to topological spaces" @default.
- W2888098991 hasPublicationYear "2018" @default.
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