On pseudocompact topological Brandt λ0-extensions of semitopological monoids

Abstract

In the paper we investigate topological properties of a topological Brandt λ 0 -extension B 0 λ (S) of a semitopological monoid S with zero. In particular we prove that for every Tychonoff pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) semitopological monoid S with zero there exists a unique semiregular pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) extension B 0 λ (S) of S and establish theirs Stone-Čech and Bohr compactifications. We also describe a category whose objects are ingredients in the constructions of pseudocompact (resp., countably compact, sequentially compact, compact) topological Brandt λ 0 -extensions of pseudocompact (resp., countably compact, sequentially compact, compact) semitopological monoids with zeros.

On some functional generalizations of the regularity of topological spaces

“…Here ∅ is the unique element of the set N 0 . For the first time, the Gutik hedgehog has appeared in the paper [9] of Gutik and Pavlyk.…”

Pseudocompactness, Products, and Topological Brandt λ0 -Extensions of Semitopological Monoids

“…Since the space i∈I A (λ i × λ i ) × S i is homeomorphic to i∈I A (λ i × λ i ) × i∈I S i , Theorem 3.2.4 from [11] and Corollary 2.14 imply that the Tychonoff product i∈I A (λ i × λ i ) × S i is a countably pracompact space. Then by Theorem 2.11 and Proposition 2.3.6 of [11] the map g : i∈I A (λ i × λ i ) × S i → i∈I B 0 λ i (S i ) defined by the formula g = i∈I g i is continuous, and since by Lemma 8 from [17] every continuous image of a countably pracompact space is countably pracompact, we see that the direct product {B 0 λ i (S i ) : i ∈ I } with the Tychonoff topology is a semiregular pseudocompact semitopological semigroup.…”

“…It is easy to see that if (X, τ ) is a Hausdorff topological space then (X, τ r ) is a semiregular topological space. We observe that the space B 0 λ (S), τ S B is Hausdorff (resp., regular, Tychonoff, normal) if and only if the space (S, τ ) is Hausdorff (resp., regular, Tychonoff, normal) (see: Propositions 21 and 22 in [17]). β 1 ), .…”

“…; α n , β n ; Ψ) · (α, x, β) ⊆ Example 2.26 shows that there exists a Hausdorff non-semiregular pseudocompact topological Brandt λ 0 -extension of a Hausdorff compact topological group with adjoined isolated zero which is not a countably pracompact space. Also, Example 18 from [17] shows that there exists a Hausdorff non-semiregular pseudocompact topological Brandt λ 0 -extension of a countable Hausdorff compact topological monoid with adjoined isolated zero which is not a countably compact space. But, as a counterpart for the H-closed case or the sequentially pseudocompact case we have the following 18 we obtain that there exist at most finitely many pairs of indices (α 1 , β 1 ), .…”

Extensions of semigroups by symmetric inverse semigroups of a~bounded finite rank

“…We shall show that the space I n λ (S), τ c I is compact by induction. In the case when n = 1, Corollary 13 from [23] implies that the space I 1 λ (S), τ c I is compact. Next we shall show the step of induction:…”