We introduce three new classes of countably pracompact spaces, consider their basic properties and relations with other compact-like spaces.

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We study the structure of inverse primitive feebly compact semitopological and topological semigroups. We find conditions when the maximal subgroup of an inverse primitive feebly compact semitopological semigroup S is a closed subset of S and describe the topological structure of such semiregular semitopological semigroups. Later we describe the structure of feebly compact topological Brandt λ 0 -extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular we show that inversion in a quasi-regular primitive inverse feebly compact topological semigroup is continuous. Also an analogue of Comfort-Ross Theorem is proved for such semigroups: a Tychonoff product of an arbitrary family of primitive inverse semiregular feebly compact semitopological semigroups with closed maximal subgroups is feebly compact. We describe the structure of the Stone-Čech compactification of a Hausdorff primitive inverse countably compact semitopological semigroup S such that every maximal subgroup of S is a topological group.

In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, ω-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) topological Brandt λ 0 i -extensions of semitopological monoids with zero. In particular we show that ifa family of Hausdorff pseudocompact topological Brandt λ 0 i -extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product {S i : i ∈ I } is a pseudocompact space then the direct product B 0 λi (S i ), τ 0 B(Si) : i ∈ I endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup. Date: September 24, 2018. 2010 Mathematics Subject Classification. Primary 22A15, 54H10. Key words and phrases. Semigroup, Brandt λ 0 -extension, semitopological semigroup, topological Brandt λ 0 -extension, pseudocompact space, countably compact space, countably pracompact space, sequentially compact space, ω-bounded space, totally countably compact space, countable pracompact space, sequential pseudocompact space.

Abstract. Given a G-space X and a non-trivial G-invariant ideal I of subsets of X, we prove that for every partition X = A 1 ∪ · · · ∪ An of X into n ≥ 2 pieces there is a piece A i of the partition and a finite set F ⊂ G of cardinality |F | ≤ φ(n + 1) := max 1