@ Appl. Gen. Topol. 21, no. 2 (2020), 201-214 doi:10.4995/agt.2020.12258 c© AGT, UPV, 2020 Closed subsets of compact-like topological spaces Serhii Bardyla a and Alex Ravsky b a Institute of Mathematics, University of Vienna, Austria. (sbardyla@yahoo.com) b Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Nat. Acad. Sciences of Ukraine, Lviv, Ukraine. (alexander.ravsky@uni-wuerzburg.de) Communicated by J. Galindo Abstract We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. We construct a pseudocom- pact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed. 2010 MSC: 54D30; 22A15. Keywords: pseudocompact space; H-closed space; semigroup of matrix units; bicyclic monoid. 1. Preliminaries In this paper all topological spaces are assumed to be Hausdorff. By ω we denote the first infinite cardinal. For ordinals α, β put α ≤ β, (α < β, resp.) iff α ⊂ β (α ⊂ β and α ∕= β , resp.). By [α, β] ([α, β), (α, β], (α, β), resp.) we denote the set of all ordinals γ such that α ≤ γ ≤ β (α ≤ γ < β, α < γ ≤ β, α < γ < β, resp.). The cardinality of a set X is denoted by |X|. Received 26 August 2019 – Accepted 29 March 2020 http://dx.doi.org/10.4995/agt.2020.12258 S. Bardyla and A. Ravsky For a subset A of a topological space X by A we denote the closure of the set A in X. A family F of subsets of a set X is called a filter if it satisfies the following conditions: (1) ∅ /∈ F; (2) If A ∈ F and A ⊂ B then B ∈ F; (3) If A, B ∈ F then A ∩ B ∈ F. A family B is called a base of a filter F if for each element A ∈ F there exists an element B ∈ B such that B ⊂ A. A filter on a topological space X is called an ω-filter if it has a countable base. A filter F is called free if ! F = ∅. A filter on a topological space X is called open if it has a base which consists of open subsets. A point x is called an accumulation point (θ-accumulation point, resp.) of a filter F if for each open neighborhood U of x and for each F ∈ F the set U ∩ F (U ∩ F , resp.) is non-empty. A topological space X is said to be • compact, if each filter has an accumulation point; • sequentially compact, if each sequence {xn}n∈ω of points of X has a convergent subsequence; • ω-bounded, if each countable subset of X has compact closure; • totally countably compact, if each sequence of X contains a subsequence with compact closure; • countably compact, if each infinite subset A ⊆ X has an accumulation point; • ω-bounded pracompact, if there exists a dense subset D of X such that each countable subset of the set D has compact closure in X; • totally countably pracompact, if there exists a dense subset D of X such that each sequence of points of the set D has a subsequence with compact closure in X; • countably pracompact, if there exists a dense subset D of X such that every infinite subset A ⊆ D has an accumulation point in X; • pseudocompact, if X is Tychonoff and each real-valued function on X is bounded; • H-closed, if each filter on X has a θ-accumulation point; • feebly ω-bounded, if for each sequence {Un}n∈ω of non-empty open sub- sets of X there is a compact subset K of X such that K ∩ Un ∕= ∅ for each n ∈ ω; • totally feebly compact, if for each sequence {Un}n∈ω of non-empty open subsets of X there is a compact subset K of X such that K ∩ Un ∕= ∅ for infinitely many n ∈ ω; • selectively feebly compact, if for each sequence {Un}n∈ω of non-empty open subsets of X, for each n ∈ ω we can choose a point xn ∈ Un such that the sequence {xn : n ∈ ω} has an accumulation point. • feebly compact, if each open ω-filter on X has an accumulation point. The interplay between some of the above properties is shown in the diagram at page 3 in [13]. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 202 Closed subsets of compact-like topological spaces Remark 1.1. H-closed topological spaces have few different equivalent defini- tions. For a topological space X the following conditions are equivalent: • X is H-closed; • if X is a subspace of a Hausdorff topological space Y , then X is closed in Y ; • each open filter on X has an accumulation point; • for each open cover F = {Fα}α∈A of X there exists a finite subset B ⊂ A such that ∪α∈BFα = X. H-closed topological spaces in terms of θ-accumulation points were investigated in [7, 16, 17, 19, 21, 22, 28, 29]. Also observe that each H-closed space is feebly compact. In this paper we investigate closed subsets (subsemigroups, resp.) of compact- like topological spaces (semigroups, resp.). We prove that each topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of ω×ω-matrix units cannot be embedded into a topo- logical semigroup which is a weakly H-closed topological space. We show that each topological space is a closed subspace of some ω-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. Also, we construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup, providing a pos- itive solution of [4, Problem 7.2]. 2. Closed subspaces of compact-like topological spaces The productivity of compact-like properties is a known topic in general topol- ogy. According to Tychonoff’s theorem, a (Tychonoff) product of a family of compact spaces is compact, On the other hand, there are two countably com- pact spaces whose product is not feebly compact (see [10], the paragraph before Theorem 3.10.16). The product of a countable family of sequentially compact spaces is sequentially compact [10, Theorem 3.10.35]. But already the Cantor cube Dc is not sequentially compact (see [10], the paragraph after Example 3.10.38). On the other hand some compact-like properties are also preserved by products, see [27, § 3-4] (especially Theorem 3.3, Proposition 3,4, Example 3.15, Theorem 4.7, and Example 4.15), [26, § 5], and [13, Sec. 2.3]. Proposition 2.1. A product of any family of feebly ω-bounded spaces is feebly ω-bounded. Proof. Let X = " {Xα : α ∈ A} be a product of a family of feebly ω-bounded spaces and let {Un}n∈ω be a family of non-empty open subsets of the space X. For each n ∈ ω let Vn be a basic open set in X which is contained in Un. For each n ∈ ω and α ∈ A let Vn,α = πα(Vn) where by πα we denote the projection on Xα. For each α ∈ A there exists a compact subset Kα of Xα, intersecting each Vn,α. Then the set K = " {Kα : α ∈ A} is a compact subset of X intersecting each Vn ⊂ Un. □ c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 203 S. Bardyla and A. Ravsky A non-productive compact-like properties still can be preserved by products with more strong compact-like spaces. For instance, a product of a countably compact space and a countably compact k-space or a sequentially compact space is countably compact, and a product of a pseudocompact space and a pseudocompact k-space or a sequentially compact Tychonoff space is pseudo- compact (see [10, Sec. 3.10]). Proposition 2.2. A product X × Y of a countably pracompact space X and a totally countably pracompact space Y is countably pracompact. Proof. Let D be a dense subset of X such that each infinite subset of D has an accumulation point in X and F be a dense subset of Y such that each sequence of points of the set F has a subsequence contained in a compact set. Then D×F is a dense subset of X×Y . So to prove that the space X×Y is countably pracompact it suffices to show that each sequence {(xn, yn)}n∈ω of points of D×F has an accumulation point. Taking a subsequence, if needed, we can assume that a sequence {yn}n∈ω is contained in a compact set K. Let x ∈ X be an accumulation point of a sequence {xn}n∈ω and B(x) be the family of neighborhoods of the point x. For each U ∈ B(x) put YU = {yn | xn ∈ U}. Then {YU | U ∈ B(x)} is a centered family of closed subsets of a compact space K, so there exists a point y ∈ ! {YU | U ∈ B(x)}. Clearly, (x, y) is an accumulation point of the sequence {(xn, yn)}n∈ω. □ Proposition 2.3. A product X × Y of a selectively feebly compact space X and a totally feebly compact space Y is selectively feebly compact. Proof. Let {Un}n∈ω be a sequence of open subsets of X×Y . For each n ∈ ω pick a non-empty open subsets U1n of X and U 2 n of Y such that U 1 n×U2n ⊂ Un. Taking a subsequence, if needed, we can assume that that there exists a compact subset K of the space Y intersecting each set U2n, n ∈ ω. Since X is selectively feebly compact, for each n ∈ ω we can choose a point xn ∈ U1n such that a sequence {xn}n∈ω has an accumulation point x ∈ X. For each n ∈ ω pick a point yn ∈ U2n ∩ K. Then (xn, yn) ∈ U1n×U2n ⊂ Un. Let B(x) be the family of neighborhoods of the point x in X. For each U ∈ B(x) put YU = {yn | xn ∈ U}. Then {YU | U ∈ B(x)} is a centered family of closed subsets of the compact space K, so there exists a point y ∈ ! {YU | U ∈ B(x)}. Clearly, (x, y) is an accumulation point of the sequence {(xn, yn)}n∈ω. □ An extension of a space X is a Hausdorff space Y containing X as a dense subspace. Extensions of topological spaces were investigated in [8, 18, 23, 24, 25]. A class C of spaces is called extension closed provided each extension of each space of C belongs to C. If Y is a space, a class C of spaces is Y - productive provided X×Y ∈ C for each space X ∈ C. It is well-known or easy to check that each of the following classes of spaces is extension closed: countably pracompact, ω-bounded pracompact, totally countably pracompact, feebly compact, selectively feebly compact, and feebly ω-bounded. Since [0, ω1) endowed with the order topology is ω-bounded and sequentially compact, each c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 204 Closed subsets of compact-like topological spaces of these classes is [0, ω1)-productive by Proposition 2.2, [13, Proposition 3], [13, Proposition 2], [9, Lemma 4.2], Proposition 2.3, and Proposition 2.1, respectively. Next we introduce a construction which helps us to construct a pseudo- compact topological semigroup which contains the bicyclic monoid as a closed subsemigroup providing a positive answer to [4, Problem 7.2]. Let X and Y be topological spaces such that there exists a continuous in- jection f : X → Y . Then by EfY (X) we denote the subset [0, ω1]×Y \ {(ω1, y) | y ∈ Y \ f(X)} of a product [0, ω1]×Y endowed with a topology τ which is defined as follows. A subset U ⊂ EY (X) is open if it satisfies the following conditions: • for each α < ω1, if (α, y) ∈ U then there exist β < α and an open neighborhood Vy of y in Y such that (β, α]×Vy ⊂ U; • if (ω1, f(x)) ∈ U then there exist α < ω1, an open neighborhood Vf(x) of f(x) in Y and an open neighborhood Wx of x in X, such that f(Wx) ⊂ Vf(x) and (α, ω1)×Vf(x) ∪ {ω1}×f(Wx) ⊂ U. Remark that {ω1}×f(X) is a closed subset of E f Y (X) homeomorphic to X. Proposition 2.4. Let X be a topological space which admits a continuous injection f into a space Y and C be any extension closed, [0, ω1)-productive class of spaces. If Y ∈ C then EfY (X) ∈ C. Proof. Let Y ∈ C. Since C is [0, ω1)-productive, [0, ω1) × Y ∈ C. A space E f Y (X) is an extension of the space [0, ω1) × Y providing that E f Y (X) ∈ C. □ If a space X is a subspace of a topological space Y and id is the identity embedding of X into Y , then by EY (X) we denote the space E id Y (X). It is easy to see that EY (X) is a subspace of a product [0, ω1]×Y which implies that if Y is Tychonoff then so is EY (X). Proposition 2.5. Let X be a subspace of a pseudocompact space Y . Then EY (X) is pseudocompact and contains a closed copy of X. Proof. The argument above implies that EY (X) is Tychonoff. Fix any con- tinuous real valued function f on EY (X). Observe that the dense subspace [0, ω1)×Y of EY (X) is pseudocompact. Then the restriction of f on the subset [0, ω1)×Y is bounded, i.e., there exist reals a, b such that f([0, ω1)×Y ) ⊂ [a, b]. Then f−1[a, b] is closed in EY (X) and contains the dense subset [0, ω1) × Y witnessing that f−1[a, b] = EY (X). Hence the space EY (X) is pseudocom- pact. □ Embeddings into countable compact and ω-bounded topological spaces were investigated in [2, 3]. A family A of countable subsets of a set X is called almost disjoint if for each A, B ∈ A the set A ∩ B is finite. Given a property P , an almost disjoint family A is called P-maximal if each element of A has the property P and for c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 205 S. Bardyla and A. Ravsky each countable subset F ⊂ X which has the property P there exists A ∈ A such that the set A ∩ F is infinite. Let F be a family of closed subsets of a topological space X. The topological space X is called • F-regular, if for any set F ∈ F and point x ∈ X \F there exist disjoint open sets U, V ⊂ X such that F ⊂ U and x ∈ V ; • F-normal, if for any disjoint sets A, B ∈ F there exist disjoint open sets U, V ⊂ X such that A ⊂ U and B ⊂ V . Given a topological space X, by Dω we denote the family of countable closed discrete subsets of X. We say that a subset A of X satisfies a property Dω iff A ∈ Dω. Theorem 2.6. Each Dω-regular topological space X can be embedded as an open dense subset into a Hausdorff countably pracompact topological space. Proof. By Zorn’s Lemma, there exists a Dω-maximal almost disjoint family A in X. Let Y = X ∪ A. We endow Y with the topology τ defined as follows. A subset U ⊂ Y belongs to τ iff it satisfies the following conditions: • if x ∈ U ∩X, then there exists an open neighborhood V of x in X such that V ⊂ U; • if A ∈ U ∩ A, then there exists a cofinite subset A ′ ⊂ A and an open set V in X such that A ′ ⊂ V ⊂ U. Observe that X is an open dense subset of Y and A is a discrete and closed subspace of Y . Since X is Dω-regular for each distinct points x ∈ X and y ∈ Y there exist disjoint open neighborhoods Ux and Uy in Y . By Proposition 2.1 from [2], each Dω-regular topological space is Dω-normal. Fix any distinct A, B ∈ A. Put A ′ = A \ (A ∩ B) and B ′ = B \ (A ∩ B). By the Dω-normality of X, there exist disjoint open neighborhoods UA′ and UB′ of A ′ and B ′ , respectively. Then the sets UA = {A} ∪ UA′ and UB = {B} ∪ UB′ are disjoint open neighborhoods of A and B, respectively, in Y . Hence the space Y is Hausdorff. Observe that the maximality of the family A implies that there exists no countable discrete subset D ⊂ X which is closed in Y . Hence each infinite subset A in X has an accumulation point in Y , that is, Y is countably pracom- pact. □ However, there exists a Hausdorff topological space which cannot be embed- ded as a dense open subset into any Hausdorff countably pracompact topolog- ical space. Example 2.7. Let τ be the usual topology on the real line R and C = {A ⊂ R : |R\A| ≤ ω}. By τ∗ we denote the topology on R which is generated by the subbase τ ∪ C. Obviously, the space R∗ = (R, τ∗) is Hausdorff. We claim that R∗ cannot be embedded as a dense open subset into any Hausdorff countably pracompact topological space. Assuming the contrary, let X be a Hausdorff countably pracompact topological space which contains R∗ as a dense open c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 206 Closed subsets of compact-like topological spaces subspace. Since X is countably pracompact there exists a dense subset Y of X such that each infinite subset of Y has an accumulation point in X. Since R∗ is open and dense in X the set Z = R∗ ∩ Y is dense in X. Moreover, it is dense in (R, τ). Fix any point z ∈ Z and a sequence {zn}n∈ω of distinct points of Z \ {z} converging to z in (R, τ). Since Z is dense in (R, τ) such a sequence exists. Observe that {zn}n∈ω is closed and discrete in R∗. So, its accumulation point x belongs to X \ R∗. Note that for each open neighborhood U of z in R∗ all but finitely many zn belongs to the closure of U. Hence x ∈ U for each open neighborhood U of z which contradicts to the Hausdorffness of X. Theorem 2.8. Each topological space can be embedded as a closed subset into a Hausdorff ω-bounded pracompact topological space. Proof. Let X be a topological space. By Xd we denote the set X endowed with a discrete topology. Let X∗ be the one point compactification of the space Xd. The unique non-isolated point of X ∗ is denoted by ∞. Put Y = [0, ω1]×X∗ \ {(ω1, ∞)}. We endow Y with a topology τ defined as follows. A subset U is open in (Y, τ) if it satisfies the following conditions: • if (α, ∞) ∈ U, then there exist β < α and a cofinite subset A of X∗ which contains ∞ such that (β, α]×A ⊂ U; • if (ω1, x) ∈ U, then there exist α < ω1 and an open (in X) neighbor- hood V of x such that (α, ω1]×V ⊂ U. It is easy to check that the space (Y, τ) is Hausdorff. Observe that the subset [0, ω1)×X∗ ⊂ Y is open, dense and ω-bounded. Hence Y is ω-bounded pracom- pact. Finally, note that the subset {ω1}×X ⊂ Y is closed and homeomorphic to X. □ Next we introduce a construction which helps us to prove that any space can be embedded as a closed subspace into an H-closed topological space. Denote the subspace {1 − 1/n | n ∈ N} ∪ {1} of the real line by J. Let X be a dense open subset of a topological space Y . By Z we denote the set (J×Y )\{(t, y) | y ∈ Y \X and t > 0}. By HY (X) we denote the set Z endowed with a topology defined as follows. A subset U ⊂ Z is open in HY (X) if it satisfies the following conditions: • for each x ∈ X if (t, x) ∈ U, then there exist open neighborhoods Vt of t in J and Vx of x in X such that Vt×Vx ⊂ U; • for each y ∈ Y \X if (0, y) ∈ U, then there exists an open neighborhood Vy of y in Y such that {0}×(Vy \ X) ∪ (J \ {1})×(Vy ∩ X) ⊂ U. Obviously, the space HY (X) is Hausdorff and the subset {(1, x) | x ∈ X} ⊂ HY (X) is closed and homeomorphic to X. Proposition 2.9. If Y is an H-closed topological space, then HY (X) is H- closed. Proof. Fix an arbitrary filter F on HY (X). One of the following three cases holds: c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 207 S. Bardyla and A. Ravsky (1) there exists t ∈ J \ {1} such that for each F ∈ F there exists y ∈ Y such that (t, y) ∈ F ; (2) for each F ∈ F there exists x ∈ X such that (1, x) ∈ F ; (3) for every t ∈ J there exists F ∈ F such that (t, y) /∈ F for each y ∈ Y . Consider case (1). For each F ∈ F put Ft = F ∩ ({t}×X ∪ {0}×(Y \ X)). Clearly, a family Ft = {Ft | F ∈ F} is a filter on {t}×X∪{0}×(Y \X). Observe that for each t ∈ J \ {1} the subspace {t}×X ∪ {0}×(Y \ X) is homeomorphic to Y and hence is H-closed. Then there exists a θ-accumulation point z ∈ {t}×X ∪ {0}×(Y \ X) of the filter Ft. Obviously, z is a θ-accumulation point of the filter F. Consider case (2). For each F ∈ F put F0 = {(0, x) | (1, x) ∈ F}. Clearly, the family F0 = {F0 | F ∈ F} is a filter on the H-closed space {0}×Y . Hence there exists y ∈ Y such that (0, y) is a θ-accumulation point of the filter F0. If y ∈ X, then (1, y) is a θ-accumulation point of the filter F. If y ∈ Y \X, then we claim that (0, y) is a θ-accumulation point of the filter F. Indeed, let U be any open neighborhood of the point (y, 0). There exists an open neighborhood Vy of y in Y such that V = {0}×(Vy \X)∪(J \{1})×(Vy ∩X) ⊂ U. Since (0, y) is a θ- accumulation point of the filter F0, V ∩F0 ∕= ∅ for each F0 ∈ F0. Fix any F ∈ F and (0, z) ∈ V ∩F0. The definition of the topology on HY (X) yields that the set {(t, z) | t ∈ J \ {1}} is contained in V . Then (1, z) ∈ {(t, z) | t ∈ J \ {1}} ⊂ V . Hence for each F ∈ F the set U ∩ F is non-empty providing that (0, y) is a θ-accumulation point of the filter F. Consider case (3). For each F ∈ F denote F ∗ = {(0, x) | there exists t ∈ I such that (t, x) ∈ F}. Let (0, y) be a θ-accumulation point of the filter F∗ = {F ∗ | F ∈ F}. If y ∈ X, then we claim that (1, y) is a θ-accumulation point of the filter F. Indeed fix any F ∈ F and an open neighborhood V of (1, y). Then there exist a positive integer n and an open neighborhood U of y in X such that {t ∈ J | t > 1−1/n}×U ⊂ V . By the assumption, there exist sets F0, . . . , Fn ∈ F such that Fi ∩ {(1 − 1/i, x) | x ∈ Y } = ∅ for every i ≤ n. Then the set H = ∩i≤nFi ∩ F belongs to F and for each (t, x) ∈ H, t > 1 − 1/n. Since (0, y) is a θ-accumulation point of the filter F∗ the set {0} × U ∩ H∗ is non- empty. Fix any (0, x) ∈ {0} × U ∩ H∗. Then there exists k > n such that (1 − 1/k, x) ∈ H ⊂ F . The definition of the topology of HY (X) implies that (1 − 1/k, x) ∈ V ∩ H ⊂ V ∩ F which implies that (1, y) is a θ-accumulation point of the filter F. If y ∈ Y \ X, then even more simple arguments show that (0, y) is a θ- accumulation point of the filter F. Hence the space HY (X) is H-closed. □ Theorem 2.10. For any Hausdorff topological space X there exists an H-closed space Z which contains X as a closed subspace. Proof. For each Hausdorff topological space X there exists an H-closed space Y which contains X as a dense open subspace (see [10, Problem 3.12.6]). By c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 208 Closed subsets of compact-like topological spaces Proposition 2.9, the space HY (X) is H-closed. It remains to note that the set {(1, x) | x ∈ X} ⊂ HY (X) is closed and homeomorphic to X. □ 3. Applications for topological semigroups A set endowed with an associative binary operation is called a semigroup. A semigroup S is called an inverse semigroup, if for each element a ∈ S there exists a unique element a−1 ∈ S such that aa−1a = a and a−1aa−1 = a−1. The map which associates every element of an inverse semigroup to its inverse is called an inversion. A topological (inverse) semigroup is a Hausdorff topological space endowed with a continuous semigroup operation (and a continuous inversion, resp.). In this case the topology of the space is called (inverse, resp.) semigroup topology. A semitopological semigroup is a Hausdorff topological space endowed with a separately continuous semigroup operation. It this case the topology of the space is called shift-continuous. Let X be a non-empty set. By BX we denote the set X×X ∪ {0} where 0 /∈ X×X endowed with the following semigroup operation: (a, b) · (c, d) = # (a, d), if b = c; 0, if b ∕= c, and (a, b) · 0 = 0 · (a, b) = 0 · 0 = 0, for each a, b, c, d ∈ X. The semigroup BX is called the semigroup of X×X-matrix units. Observe that semigroups BX and BY are isomorphic iff |X| = |Y |. If a set X is infinite then the semigroup of X×X-matrix units cannot be embedded into a compact topological semigroup (see [11, Theorem 3]). In [12, Theorem 5] this result was generalized for countably compact topological semigroups. Moreover, in [6, Theorem 4.4] it was shown that for an infinite set X the semigroup BX cannot be embedded densely into a feebly compact topological semigroup. A bicyclic monoid C(p, q) is the semigroup with the identity 1 generated by two elements p and q subject to the condition pq = 1. The bicyclic monoid is isomorphic to the set ω×ω endowed with the following semigroup operation: (a, b) · (c, d) = # (a + c − b, d), if b ≤ c; (a, d + b − c), if b > c. Neither stable nor Γ-compact topological semigroups can contain a copy of the bicyclic monoid (see [1, 15]). In [14] it was proved that the bicyclic monoid does not embed into a countably compact topological inverse semi- group. Also, a topological semigroup with a feebly compact square cannot contain the bicyclic monoid [4]. On the other hand, in [4, Theorem 6.1] it was proved that there exists a Tychonoff countably pracompact topological semi- group S densely containing the bicyclic monoid. Moreover, under Martin’s Ax- iom the semigroup S is countably compact (see [4, Theorem 6.6 and Corollary 6.7]). However, it is still unknown whether there exists under ZFC a countably c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 209 S. Bardyla and A. Ravsky compact topological semigroup containing the bicyclic monoid (see [4, Problem 7.1]). Also, in [4] the following problem was posed: Problem 3.1 ([4, Problem 7.2]). Is there a pseudocompact topological semi- group S that contains a closed copy of the bicyclic monoid? Embeddings of semigroups which are generalizations of the bicyclic monoid into compact-like topological semigroups were investigated in [5, 6]. Namely, in [6] it was proved that for each cardinal λ > 1 a polycyclic monoid Pλ does not embed as a dense subsemigroup into a feebly compact topological semigroup. In [5] embeddings of graph inverse semigroups into CLP-compact topological semigroups were described. Observe that the space [0, ω1] endowed with a semigroup operation of taking minimum becomes a topological semilattice and therefore a topological inverse semigroup. Lemma 3.2. Let X and Y be semitopological (topological, topological inverse, resp.) semigroups such that there exists a continuous injective homomorphism f : X → Y . Then EfY (X) is a semitopological (topological, topological inverse, resp.) semigroup with respect to the semigroup operation inherited from a direct product of semigroups (ω1, min) and Y . Proof. We prove this lemma for the case of topological semigroups X and Y . Proofs in other cases are similar. Fix any elements (α, x), (β, y) of E f Y (X). Also, assume that β ≤ α. In the other case the proof will be similar. Fix any open neighborhood U of (β, xy) = (α, x) · (β, y). There are three cases to consider: (1) β ≤ α < ω1; (2) β < α = ω1; (3) α = β = ω1. In case (1) there exist γ < β and an open neighborhood Vxy of xy in Y such that (γ, β]×Vxy ⊂ U. Since Y is a topological semigroup there exist open neighborhoods Vx and Vy of x and y, respectively, such that Vx ·Vy ⊂ Vxy. Put U(α,x) = (γ, α]×Vx and U(β,y) = (γ, β]×Vy. It is easy to check that U(α,x) · U(β,y) ⊂ (γ, β]×Vxy ⊂ U. Consider case (2). Similarly as in case (1) there exist an ordinal γ < β and open neighborhoods Vx, Vy and Vxy of x, y and xy, respectively, such that (γ, β]×Vxy ⊂ U and Vx · Vy ⊂ Vxy. Since the map f is continuous there exists an open neighborhood Vf−1(x) of f −1(x) in X such that f(Vf−1(x)) ⊂ Vx. Put U(ω1,x) = (β, ω1)×Vx ∪ {ω1}×f(Vf−1(x)) and U(β,y) = (γ, β]×Vy. It is easy to check that U(ω1,x) · U(β,y) ⊂ (γ, β]×Vxy ⊂ U. Consider case (3). There exist ordinal γ < ω1, an open neighborhood Vxy of xy in Y and an open neighborhood Wf−1(xy) of f −1(xy) in X such that (γ, ω1)×Vxy ∪ {ω1}×f(Wf−1(xy)) ⊂ U. Since Y is a topological semigroup there exist open (in Y ) neighborhoods Vx and Vy of x and y, respectively, such that Vx · Vy ⊂ Vxy. Since the map f is continuous and X is a topological semigroup there exist open (in X) c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 210 Closed subsets of compact-like topological spaces neighborhoods Wf−1(x) and Wf−1(y) of f −1(x) and f−1(y), respectively, such that Wf−1(x) · Wf−1(y) ⊂ Wf−1(xy), f(Wf−1(x)) ⊂ Vx and f(Wf−1(y)) ⊂ Vy. Put U(ω1,x) = (γ, ω1)×Vx ∪ {ω1}×f(Wf−1(x)) and U(ω1,y) = (γ, ω1)×Vy ∪ {ω1}×f(Wf−1(y)). It is easy to check that U(ω1,x) · U(ω1,y) ⊂ U. Hence the semigroup operation in E f Y (X, τX) is continuous. □ Remark 3.3. The subsemigroup {(ω1, f(x)) | x ∈ X} ⊂ E f Y (X) is closed and topologically isomorphic to X. Proposition 2.4, Lemma 3.2 and Remark 3.3 imply the following: Proposition 3.4. Let X be a (semi)topological semigroup which admits a con- tinuous injective homomorphism f into a (semi)topological semigroup Y and C be any [0, ω1)-productive, extension closed class of spaces. If Y ∈ C then the (semi)topological semigroup E f Y (X) ∈ C and contains a closed copy of a (semi)topological semigroup X. Proposition 2.5, Lemma 3.2 and Remark 3.3 imply the following: Proposition 3.5. Let X be a subsemigroup of a pseudocompact (semi)topologi- cal semigroup Y . Then the (semi)topological semigroup EY (X) is pseudocom- pact and contains a closed copy of the (semi)topological semigroup X. By [4, Theorem 6.1], there exists a Tychonoff countably pracompact (and hence pseudocompact) topological semigroup S containing densely the bicyclic monoid. Hence Proposition 3.5 implies the following corollary which gives an affirmative answer to Problem 3.1. Corollary 3.6. There exists a pseudocompact topological semigroup which con- tains a closed copy of the bicyclic monoid. Further we will need the following definitions. A subset A of a topological space is called θ-closed if for each element x ∈ X \ A there exists an open neighborhood U of x such that U ∩ A = ∅. Observe that if a topological space X is regular then each closed subset A of X is θ-closed. A topological space X is called weakly H-closed if each ω-filter F has a θ-accumulation point in X. Generalizations of H-closed spaces were investigated by Osipov in [19, 20]. Obviously, for a topological space X the following implications hold: X is H- closed ⇒ X is weakly H-closed ⇒ X is feebly compact. Neither of the above implications can be inverted. Indeed, an arbitrary pseudocompact but not countably compact space will be an example of feebly compact space which is not weakly H-closed. The space [0, ω1) with an order topology is an example of weakly H-closed but not H-closed space. The following theorem shows that Theorem 2.10 cannot be generalized for topological semigroups. Theorem 3.7. The semigroup Bω of ω×ω-matrix units does not embed into a weakly H-closed topological semigroup. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 211 S. Bardyla and A. Ravsky Proof. Suppose to the contrary that Bω is a subsemigroup of a weakly H-closed topological semigroup S. By E(Bω) we denote the semilattice of idempotents of Bω. Observe that E(Bω) = {(n, n) | n ∈ ω} ∪ {0} and a · b = 0 for each distinct elements a, b ∈ E(Bω). Let F be an arbitrary free ω-filter on the set {(n, n) | n ∈ ω}. Since S is weakly H-closed, there exists a θ-accumulation point s ∈ S of the filter F. Fix any open neighborhood Up of the point p = s · s. The continuity of the semigroup operation in S yields an open neighborhood Vs of s such that Vs · Vs ⊂ Up. Since s is a θ-accumulation point of the filter F there exist distinct elements (n, n), (m, m) ∈ Vs ∩ E(Bω). Hence 0 = (n, n) · (m, m) ∈ Vs · Vs ⊂ Up which implies that 0 ∈ Up for each open neighborhood Up of p witnessing that p = 0. We claim that 0 is a θ-accumulation point of the filter F. Indeed, fix any open neighborhood U of 0. Since s · s = 0 and S is a topological semigroup there exists an open neighborhood Vs of s such that Vs · Vs ⊂ U. Observe that (n, n) = (n, n) · (n, n) ∈ Vs · Vs ⊂ U for each (n, n) ∈ Vs. Hence 0 is a θ-accumulation point of the filter F. Since the filter F was selected arbitrarily we have that 0 is a θ-accumulation point of any free ω-filter on the set {(n, n) | n ∈ ω}. Thus for each open neighborhood U of 0 the set AU = {(n | (n, n) /∈ U} is finite, because if there exists an open neighborhood U of 0 such that the set AU is infinite, then 0 is not a θ-accumulation point of the ω-filter F which has a base consisting of cofinite subsets of AU. Let F be an arbitrary free ω-filter on the set {(1, n) | n ∈ ω}. Since S is weakly H-closed there exists a θ-accumulation point s ∈ S of the filter F. We claim that s · 0 = 0. Indeed, fix any open neighborhood W of s · 0. The continuity of the semigroup operation in S yields open neighborhoods Vs of s and V0 of 0 such that Vs·V0 ⊂ W . Since the set AV0 = {(n | (n, n) /∈ V0} is finite and s is a θ-accumulation point of the filter F there exist distinct n, m ∈ ω such that (1, n) ∈ Vs and (m, m) ∈ V0. Then 0 = (1, n) · (m, m) ∈ Vs · V0 ⊂ W . Hence 0 ∈ W for each open neighborhood W of s · 0 witnessing that s · 0 = 0. Fix an arbitrary open neighborhood U of 0. Since s · 0 = 0 and S is a topological semigroup, there exist open neighborhoods Vs of s and V0 of 0 such that Vs · V0 ⊂ U. Recall that the set {n | (n, n) /∈ V0} is finite. Then (1, n) = (1, n)·(n, n) ∈ Vs·V0 ⊂ U for all but finitely many elements (1, n) ∈ Vs. Hence 0 is a θ-accumulation point of the ω-filter F. Since the filter F was selected arbitrarily, 0 is a θ-accumulation point of any free ω-filter on the set {(1, n) | n ∈ ω}. As a consequence, for each open neighborhood U of 0 the set BU = {n | (1, n) /∈ U} is finite. Similarly it can be shown that for each open neighborhood U of 0 the set CU = {n | (n, 1) /∈ U} is finite. Fix an open neighborhood U of 0 such that (1, 1) /∈ U. Since 0 = 0 · 0 the continuity of the semigroup operation implies that there exists an open neighborhood V of 0 such that V · V ⊂ U. The finiteness of the sets BV c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 212 Closed subsets of compact-like topological spaces and CV implies that there exists n ∈ ω such that {(1, n), (n, 1)} ⊂ V . Hence (1, 1) = (1, n) · (n, 1) ∈ V · V ⊂ U, which contradicts to the choice of U. □ Corollary 3.8. The semigroup of ω×ω-matrix units does not embed into a topological semigroup S whose space is H-closed. However, we have the following questions: Problem 3.9. 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