Cardinal functions of Pixley–Roy hyperspaces

Let F [ X ] be the Pixley–Roy hyperspace of a regular space X, and let F n [ X ] = { F ∈ F [ X ] :   | F | ⩽ n } . For tightness t and supertightness st, we show the following equalities:(1) [ X ] ) = sup { st ( X n ) : n ∈ N } , t ( F n [ X ] ) : n ∈ N } = sup { t ( X n ) : n ∈ N } . irst equality answers a question posed in Sakai (1983) [18]. The inequality sup { t ( X n ) :   n ∈ N } ⩽ sup { st ( X n ) : n ∈ N } is strict, indeed there is a space Z such that sup { t ( X n ) : n ∈ N } < sup { st ( X n ) : n ∈ N } . The discrete countable chain condition and weak Lindelöf property of F [ X ] are also investigated.