Let S be a semigroup of homeomorphisms of a compact metric space M and suppose that <img border=0 id="_x0000_i1026" src="../../../../../img/revistas/cam/v22n3/a03img86.gif" align=absmiddle>is a family of subsets of S. This paper gives a characterization of the <img border=0 id="_x0000_i1027" src="../../../../../img/revistas/cam/v22n3/a03img86.gif" align=absmiddle>-chain control sets as intersection of control sets for the semigroups generated by the neighborhoods of the subsets in <img border=0 id="_x0000_i1028" src="../../../../../img/revistas/cam/v22n3/a03img86.gif" align=absmiddle>. We also study the behavior of <img border=0 id="_x0000_i1029" src="../../../../../img/revistas/cam/v22n3/a03img86.gif" align=absmiddle>-chain control sets on principal bundles and their associated bundles.
semigroups; control sets; chain control sets; principal bundles; associated bundles
