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$L$-topology; compactness; $\alpha $-compactness; countable $\alpha $-compactness; $\alpha $-Lindelöf property; $\alpha $-irresolute map; $\alpha $-continuous map
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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