# Article

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Keywords:
$L$-topology; compactness; $\alpha$-compactness; countable $\alpha$-compactness; $\alpha$-Lindelöf property; $\alpha$-irresolute map; $\alpha$-continuous map
Summary:
A new form of $\alpha$-compactness is introduced in $L$-topological spaces by $\alpha$-open $L$-sets and their inequality where $L$ is a complete de Morgan algebra. It doesn’t rely on the structure of the basis lattice $L$. It can also be characterized by means of $\alpha$-closed $L$-sets and their inequality. When $L$ is a completely distributive de Morgan algebra, its many characterizations are presented and the relations between it and the other types of compactness are discussed. Countable $\alpha$-compactness and the $\alpha$-Lindelöf property are also researched.
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