On feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

Original · EN

We study feebly compact shift-continuous T₁-topologies on the symmetric inverse semigroup Iλⁿ of finite transformations of the rank n. For any positive integer n2 and any infinite cardinal λ a Hausdorff countably pracompact non-compact shift-continuous topology on Iλⁿ is constructed. We show that for an arbitrary positive integer n and an arbitrary infinite cardinal λ for a T₁-topology τ on Iλⁿ the following conditions are equivalent: (i) τ is countably pracompact; (ii) τ is feebly compact; (iii) τ is d-feebly compact; (iv) (Iλⁿ,τ) is H-closed; (v) (Iλⁿ,τ) is Nd-compact for the discrete countable space Nd; (vi) (Iλⁿ,τ) is R-compact; (vii) (Iλⁿ,τ) is infra H-closed. Also we prove that for an arbitrary positive integer n and an arbitrary infinite cardinal λ every shift-continuous semiregular feebly compact T₁-topology τ on Iλⁿ is compact.