Cardinal functions of Pixley–Roy hyperspaces

Let F[X] be the Pixley–Roy hyperspace of a regular space X, and let Fn[X]={F∈F[X]: |F|⩽n}. For tightness t and supertightness st, we show the following equalities:(1)t(F[X])=sup{st(Xn):n∈N},(2)sup{t(Fn[X]):n∈N}=sup{t(Xn):n∈N}. The first equality answers a question posed in Sakai (1983) [18]. The inequality sup{t(Xn): n∈N}⩽sup{st(Xn):n∈N} is strict, indeed there is a space Z such that sup{t(Xn):n∈N}