On feebly compact topologies on the semigroup BF1ω

Abstract

We study the Gutik-Mykhalenych semigroup B ω F 1 in the case when the family F 1 consists of the empty set and all singleton in ω . We show that B ω F 1 is isomorphic to subsemigroup B ω ↗ ω m i n of the Brandt ω -extension of the semilattice ω , m i n and describe all shift-continuous feebly compact T 1 -topologies on the semigroup B ω ↗ ω m i n . In particular, we prove that every shift-continuous feebly compact T 1 -topology τ on B ω F 1 is compact and moreover in this case the space ( B ω F 1 , τ ) is homeomorphic to the one-point Alexandroff compactification of the discrete countable space D ( ω ) .