ON COUNTABLY COMPACT BRANDT λ 0 -EXTENSIONS OF TOPOLOGICAL SEMIGROUPS OLEG GUTIK AND KATERYNA PAVLYK All topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [1, 2, 3]. A topological space X is called countably compact if any countable open cover of X contains a finite subcover [3]. A topological (inverse ) semigroup is a Hausdorff topological space with a continuous binary associative operation (and inversion). Let S be a semigroup with zero and λ be a cardinal  1. On the set B λ (S)= λ × S × λ ∪{0} we define the semigroup operation as follows (α, a, β) · (γ,b,δ)=  (α, ab, δ), if β = γ ; 0, if β = γ, and (α, a, β) · 0=0 · (α, a, β)=0 · 0=0, for all α,β,γ,δ ∈ λ and a, b ∈ S. If S = S 1 then the semigroup B λ (S) is called the Brandt λ-extension of the semigroup S [4]. Obviously, J = {0}∪{(α, O,β) |O is the zero of S} is an ideal of B λ (S). We put B 0 λ (S)= B λ (S)/J and we shall call B 0 λ (S) the Brandt λ 0 -extension of the semigroup S with zero [5]. Let S be some class of topological monoids with zero. Let λ be any cardinal  1, and (S, τ ) ∈S . Let τ B be a topology on B 0 λ (S) such that ( B 0 λ (S),τ B ) ∈S and τ B | (α,S,α) = τ for some α ∈ λ. Then ( B 0 λ (S),τ B ) is called a topological Brandt λ 0 -extension of (S, τ ) in S . If S coincides with the class of all topological monoids with zero, then ( B 0 λ (S),τ B ) is called a topological Brandt λ 0 -extension of (S, τ ) [5]. A topological Brandt λ 0 -extension ( B 0 λ (S),τ B ) is called compact (resp., count- ably compact ) if the topological space ( B 0 λ (S),τ B ) is compact (resp., countably compact) [6]. Gutik and Repovˇ s in [6] describe the structure of compact topologi- cal Brandt λ 0 -extensions of topological monoids with zero. Theorem 1. A topological Brandt λ 0 -extension B 0 λ (S) of a topological monoid (S, τ ) with zero is countably compact if and only if the cardinal λ  1 is finite and (S, τ ) is a countably compact topological semigroup. Moreover, for any countably compact topological monoid (S, τ ) with zero and for any finite cardinal λ  1 there exists an unique countably compact topological Brandt λ 0 -extension ( B 0 λ (S),τ B ) and the topology τ B generated by the base B B =  {B B (t) | t ∈ B 0 λ (S)}, where: (i) B B (t)= {(α, U (s) \{0 S },β) | U (s) ∈B S (s)}, when t =(α, s, β) is a non- zero element of B 0 λ (S), α, β ∈ λ and B S (s) is a base of the topology τ at s ∈ S; (ii) B B (0) = {{0}∪  α,β∈λ (α, U (0 S ) \{0 S },β) | U (0 S ) ∈B S (0 S )}, where 0 is zero of B 0 λ (S), and B S (s) is a base of the topology τ at the point s ∈ S. Theorem 2. Every countably compact topological Brandt λ 0 -extension ( B 0 λ (S),τ B ) of a topological inverse semigroup (S, τ ) is a topological inverse semigroup. Let S and T be topological monoids with zeros. Let CHom 0 (S, T ) be a set of all continuous homomorphisms σ : S → T such that (0 S )σ =0 T . We put E top 1 (S, T )= {e ∈ E(T ) | there exists σ ∈ CHom 0 (S, T ) such that (1 S )σ = e} and define the family H top 1 (S, T )= {H(e) | e ∈ E top 1 (S, T )}, where by H(e) we denote the maximal subgroup with the unity e in the semigroup T . We shall say 1