Continuité d'une action d'un semilattis compact
Résumé
A net $(s_j)_{j\in J}$ on a compact semilattice is dominated (by a net $(v_j)_{j\in J}$) if it converges, if $\lim s_j=\lim v_j$ and for all $k$ there exists $m$ such that for all $j\geq m$, $s_j\leq v_k$. Now $S$ is said to have property $\scr P$ if every net in $S$ has a dominated subnet. Theorem. If $S$ has property $\scr P$ then every separately continuous action of $S$ on a compact Hausdorff space is jointly continuous. (The proof is based on a theorem of Lawson on joint continuity.) The author describes various classes of compact semilattices which have property $\scr P$. It suffices, e.g., that every point of $S$ have arbitrarily small $\bigvee$-complete neighborhoods. A compact topological lattice has this property if its opposite lattice is a Lawson semilattice. If $S$ is a compact topological semilattice and indeed a lattice in which the interval topology is Hausdorff then $S$ has property $\scr P$. ( K. H. Hofmann)