Closed subsets of compact-like topological spaces
aa r X i v : . [ m a t h . GN ] A ug CLOSED SUBSETS OF COMPACT-LIKE TOPOLOGICAL SPACES
SERHII BARDYLA AND ALEX RAVSKY
Abstract.
We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces(semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closedsubspace into an H-closed topological space. However, the semigroup of ω × ω -matrix units cannot beembedded into a topological semigroup which is a weakly H-closed topological space. We show thateach Hausdorff topological space is a closed subspace of some ω -bounded pracompact topological spaceand describe open dense subspaces of countably pracompact topological spaces. Also, we constructa pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup,providing a positive solution of a problem posed by Banakh, Dimitrova, and Gutik. Preliminaries
In this paper all topological spaces are assumed to be Hausdorff. By ω we denote the first infinitecardinal. 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 | .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 suchthat 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 T F = ∅ . A filter on a topological space X is called open if it has a base which consistsof open subsets. A point x is called an accumulation point ( θ -accumulation point , resp.) of a filter F iffor each open neighborhood U of x and for each F ∈ F the set U ∩ F ( U ∩ F , resp.) is non-empty. Atopological space X is said to be • compact , if each filter has an accumulation point; • sequentially compact , if each sequence { x n } 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 ofthe set D has compact closure in X ; • totally countably pracompact , if there exists a dense subset D of X such that each sequence ofpoints 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; Date : August 9, 2019.2010
Mathematics Subject Classification.
Primary 54D30, 22A15.
Key words and phrases.
H-closed space, countably compact space, semigroup of matrix units, bicyclic monoid.The work of the first author is supported by the Austrian Science Fund FWF (Grant I 3709 N35). • feebly ω -bounded , if for each sequence { U n } n ∈ ω of non-empty open subsets of X there is a compactsubset K of X such that K ∩ U n = ∅ for each n ∈ ω ; • totally feebly compact , if for each sequence { U n } n ∈ ω of non-empty open subsets of X there is acompact subset K of X such that K ∩ U n = ∅ for infinitely many n ∈ ω ; • selectively feebly compact , if for each sequence { U n } n ∈ ω of non-empty open subsets of X , for each n ∈ ω we can choose a point x n ∈ U n such that the sequence { x n : n ∈ ω } has an accumulationpoint. • 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 [14]. Remark 1.1.
H-closed topological spaces has few different equivalent definitions. For a topologicalspace 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 = { F α } α ∈ A of X there exists a finite subset B ⊂ A such that ∪ α ∈ B F α = X .H-closed topological spaces in terms of θ -accumulation points were investigated in [8, 17, 18, 20, 22, 23,29, 30]. Also recall 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 Hausdorff topological space can be embedded as a closedsubspace into an H-closed topological space. However, the semigroup of ω × ω -matrix units cannot beembedded into a topological semigroup which is a weakly H-closed topological space. We show thateach Hausdorff topological space is a closed subspace of some ω -bounded pracompact topological spaceand describe open dense subspaces of countably pracompact topological spaces. Also, we construct apseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup,providing a positive solution of Problem 3.1.2. Closed subspaces of compact-like topological spaces
The productivity of compact-like properties is a known topic in general topology. According toTychonoff’s theorem, a (Tychonoff) product of a family of compact spaces is compact, On the otherhand, there are two countably compact spaces whose product is not feebly compact (see [11], theparagraph before Theorem 3.10.16). The product of a countable family of sequentially compact spaces issequentially compact [11, Theorem 3.10.35]. But already the Cantor cube D c is not sequentially compact(see [11], the paragraph after Example 3.10.38). On the other hand some compact-like properties arealso preserved by products, see [28, § § § Proposition 2.1.
A product of any family of feebly ω -bounded spaces is feebly ω -bounded.Proof. Let X = Q { X α : α ∈ A } be a product of a family of feebly ω -bounded spaces and let { U n } n ∈ ω be a family of non-empty open subsets of the space X . For each n ∈ ω let V n be a basic open set in X which is contained in U n . For each n ∈ ω and α ∈ A let V n,α = π α ( V n ) where by π α we denote theprojection on X α . For each α ∈ A there exists a compact subset K α of X α , intersecting each V n,α . Thenthe set K = Q { K α : α ∈ A } is a compact subset of X intersecting each V n ⊂ U n . (cid:3) 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 -spaceor a sequentially compact space is countably compact, and a product of a pseudocompact space anda pseudocompact k -space or a sequentially compact Tychonoff space is pseudocompact (see [11, Sec.3.10]). Proposition 2.2.
A product X × Y of a countably pracompact space X and a totally countably pra-compact space Y is countably pracompact. LOSED SUBSETS OF COMPACT-LIKE TOPOLOGICAL SPACES 3
Proof.
Let D be a dense subset of X such that each infinite subset of D has an accumulation pointin X and F be a dense subset of Y such that each sequence of points of the set F has a subsequencecontained 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 { ( x n , y n ) } n ∈ ω of points of D × F hasan accumulation point. Taking a subsequence, if needed, we can assume that a sequence { y n } n ∈ ω iscontained in a compact set K . Let x ∈ X be an accumulation point of a sequence { x n } n ∈ ω and B ( x )be the family of neighborhoods of the point x . For each U ∈ B ( x ) put Y U = { y n | x n ∈ U } . Then { Y U | U ∈ B ( x ) } is a centered family of closed subsets of a compact space K , so there exists a point y ∈ T { Y U | U ∈ B ( x ) } . Clearly, ( x, y ) is an accumulation point of the sequence { ( x n , y n ) } n ∈ ω . (cid:3) Proposition 2.3.
A product X × Y of a selectively feebly compact space X and a totally feebly compactspace Y is selectively feebly compact.Proof. Let { U n } n ∈ ω be a sequence of open subsets of X × Y . For each n ∈ ω pick a non-empty opensubsets U n of X and U n of Y such that U n × U n ⊂ U n . Taking a subsequence, if needed, we canassume that that there exists a compact subset K of the space Y intersecting each set U n , n ∈ ω .Since X is selectively feebly compact, for each n ∈ ω we can choose a point x n ∈ U n such that asequence { x n } n ∈ ω has an accumulation point x ∈ X . For each n ∈ ω pick a point y n ∈ U n ∩ K . Then( x n , y n ) ∈ U n × U n ⊂ U n . Let B ( x ) be the family of neighborhoods of the point x . For each U ∈ B ( x )put Y U = { y n | x n ∈ U } . Then { Y U | U ∈ B ( x ) } is a centered family of closed subsets of a compactspace K , so there exists a point y ∈ T { Y U | U ∈ B ( x ) } . Clearly, ( x, y ) is an accumulation point of thesequence { ( x n , y n ) } n ∈ ω . (cid:3) An extension of a space X is a space Y containing X as a dense subspace. Hausdorff extensionsof topological spaces were investigated in [9, 19, 24, 25, 26]. A class C of spaces is called extensionclosed 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 eachof 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 , ω ) endowed with the order topology is ω -bounded and sequentially compact, each of these classesis [0 , ω )-productive by Proposition 2.2, [14, Proposition 2.4], [14, Proposition 2.2], [10, Lemma 4.2],Proposition 2.3, and Proposition 2.1, respectively.Next we introduce a construction which helps us to construct a pseudocompact topological semigroupwhich contains the bicyclic monoid as a closed subsemigroup providing a positive answer to Problem 3.1.Let X and Y be topological spaces such that there exists a continuous injection f : X → Y . Thenby E fY ( X ) we denote the subset [0 , ω ] × Y \ { ( ω , y ) | y ∈ Y \ f ( X ) } of a product [0 , ω ] × Y endowedwith a topology τ which is defined as follows. A subset U ⊂ E Y ( X ) is open if it satisfies the followingconditions: • for each α < ω , if ( α, y ) ∈ U then there exist β < α and an open neighborhood V y of y in Y such that ( β, α ] × V y ⊂ U ; • if ( ω , f ( x )) ∈ U then there exist α < ω , an open neighborhood V f ( x ) of f ( x ) in Y and an openneighborhood W x of x in X , such that f ( W x ) ⊂ V f ( x ) and ( α, ω ) × V f ( x ) ∪ { ω }× f ( W x ) ⊂ U .Remark that { ω }× f ( X ) is a closed subset of E fY ( 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 , ω ) -productive class of spaces. If Y ∈ C then E fY ( X ) ∈ C .Proof. Let Y ∈ C . Since C is [0 , ω )-productive, [0 , ω ) × Y ∈ C . A space E fY ( X ) is an extension of thespace [0 , ω ) × Y ∈ C providing that E fY ( X ) ∈ C . (cid:3) If a space X is a subspace of a topological space Y and id is the identity embedding of X into Y ,then by E Y ( X ) we denote the space E idY ( X ). It is easy to see that E Y ( X ) is a subspace of a product[0 , ω ] × Y which implies that if Y is Tychonoff then so is E Y ( X ). S. BARDYLA AND A. RAVSKY
Proposition 2.5.
Let X be a subspace of a pseudocompact space Y . Then E Y ( X ) is pseudocompactand contains a closed copy of X .Proof. The above arguments imply that E Y ( X ) is Tychonoff. Fix any continuous real valued function f on E Y ( X ). Observe that the dense subspace [0 , ω ) × Y of E Y ( X ) is pseudocompact. Then the restrictionof f on the subset [0 , ω ) × Y is bounded, i.e., there exist reals a, b such that f ([0 , ω ) × Y ) ⊂ [ a, b ]. Then f − [ a, b ] is closed in E Y ( X ) and contains the dense subset [0 , ω ) × Y witnessing that f − [ a, b ] = E Y ( X ).Hence the space E Y ( X ) is pseudocompact. (cid:3) Embeddings into countable compact and ω -bounded topological spaces were investigated in [3, 4].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 , the almost disjoint family A is called P -maximal if each element of A has the property P and for 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 D ω we denote the family of countable closed discrete subsets of X .We say that a subset A of X satisfies a property D ω iff A ∈ D ω . Theorem 2.6.
Each D ω -regular topological space X can be embedded as an open dense subset into acountably pracompact topological space.Proof. By Zorn’s Lemma, there exists a D ω -maximal almost disjoint family A on 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 thefollowing 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 discrete and closed in Y . Since X is D ω -regular for each distinct points x ∈ X and y ∈ Y there exist disjoint open neighborhoods U x and U y in Y . By Proposition 2.1 from [3], each D ω -regular topological space is D ω -normal. Fix any distinct A, B ∈ A . Put A ′ = A \ ( A ∩ B ) and B ′ = B \ ( A ∩ B ). By the D ω -normality of X there existdisjoint open neighborhoods U A ′ and U B ′ of A ′ and B ′ , respectively. Then the sets U A = { A } ∪ U A ′ and U B = { B } ∪ U B ′ 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 , thatis, Y is countably pracompact. (cid:3) However, there exists a Hausdorff topological space which cannot be embedded as a dense open subsetinto countably pracompact topological spaces.
Example 2.7.
Let τ be the usual topology on the real line R and C = { A ⊂ R : | R \ A | ≤ ω } . By τ ∗ wedenote the topology on R which is generated by the subbase τ ∪ C . Obviously, the space R ∗ = ( R , τ ∗ ) isHausdorff. We claim that R ∗ cannot be embedded as a dense open subset into a countably pracompacttopological space. Assuming the contrary, let X be a countably pracompact topological space whichcontains R ∗ as a dense open 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 densein X the set Z = R ∗ ∩ Y is dense in X . Moreover, it is dense in ( R , τ ). Fix any point z ∈ Z anda sequence { z n } n ∈ ω of distinct points of Z \ { z } converging to z in ( R , τ ). Since Z is dense in ( R , τ ) LOSED SUBSETS OF COMPACT-LIKE TOPOLOGICAL SPACES 5 such a sequence exists. Observe that { z n } n ∈ ω is closed and discrete in R ∗ . So its accumulation point x belongs to X \ R ∗ . Observe that for each open neighborhood U of z in R ∗ all but finitely many z n belongs to the closure of U . Hence x ∈ U for each open neighborhood U of z which contradicts to theHausdorffness of X . Theorem 2.8.
Each topological space can be embedded as a closed subset into an ω -bounded pracompacttopological space.Proof. Let X be a topological space. By X d we denote the set X endowed with a discrete topology. Let X ∗ be the one point compactification of the space X d . The unique non-isolated point of X ∗ is denotedby ∞ . Put Y = [0 , ω ] × X ∗ \ { ( ω , ∞ ) } . 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 ( ω , x ) ∈ U , then there exist α < ω and an open (in X ) neighborhood V of x such that( α, ω ] × V ⊂ U .Observe that the subset [0 , ω ) × X ∗ ⊂ Y is open, dense and ω -bounded. Hence Y is ω -boundedpracompact. It is easy to see that the subset { ω }× X ⊂ Y is closed and homeomorphic to X . (cid:3) Next we introduce a construction which helps us to prove that any space can be embedded as a closedsubspace into an H-closed topological space.Denote the subspace { − /n | n ∈ N } ∪ { } of the real line by J . Let X be a dense open subset ofa topological space Y . By Z we denote the set ( J × Y ) \ { ( t, y ) | y ∈ Y \ X and t > } . By H Y ( X ) wedenote the set Z endowed with a topology defined as follows. A subset U ⊂ Z is open in H Y ( X ) if itsatisfies the following conditions: • for each x ∈ X if ( t, x ) ∈ U , then there exist open neighborhoods V t of t in J and V x of x in X such that V t × V x ⊂ U ; • for each y ∈ Y \ X if (0 , y ) ∈ U , then there exists an open neighborhood V y of y in Y such that { }× ( V y \ X ) ∪ ( J \ { } ) × ( V y ∩ X ) ⊂ U .Obviously, the space H Y ( X ) is Hausdorff and the subset { (1 , x ) | x ∈ X } ⊂ H Y ( X ) is closed andhomeomorphic to X . Proposition 2.9. If Y is an H-closed topological space, then H Y ( X ) is H-closed.Proof. Fix an arbitrary filter F on H Y ( X ). One of the following three cases holds:(1) there exists t ∈ J \ { } 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 F t = F ∩ ( { t }× X ∪ { }× ( Y \ X )). Clearly, a family F t = { F t | F ∈ F } is a filter on { t }× X ∪ { }× ( Y \ X ). Observe that for each t ∈ J \ { } the subspace { t }× X ∪ { }× ( Y \ X ) is homeomorphic to Y and hence is H-closed. Then there exists a θ -accumulationpoint z ∈ { t }× X ∪ { }× ( Y \ X ) of the filter F t . Obviously, z is a θ -accumulation point of the filter F .Consider case (2). For each F ∈ F put F = { (0 , x ) | (1 , x ) ∈ F } . Clearly, the family F = { F | F ∈ F } is a filter on the H-closed space { }× Y . Hence there exists y ∈ Y such that (0 , y ) is a θ -accumulation point of the filter F . 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 beany open neighborhood of the point ( y, V y of y in Y such that V = { }× ( V y \ X ) ∪ ( J \ { } ) × ( V y ∩ X ) ⊂ U . Since (0 , y ) is a θ -accumulation point of the filter F , V ∩ F = ∅ for each F ∈ F . Fix any F ∈ F and (0 , z ) ∈ V ∩ F . The definition of the topology on H Y ( X ) yields that the set { ( t, z ) | t ∈ J \{ }} is contained in V . Then (1 , z ) ∈ { ( t, z ) | t ∈ J \ { }} ⊂ V .Hence for each F ∈ F the set U ∩ F is non-empty providing that (0 , y ) is a θ -accumulation point of thefilter F . S. BARDYLA AND A. RAVSKY
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 a basic open neighborhood V = { t ∈ J | t > − /n }× U of (1 , y ) where n is some fixed positiveinteger and U is an open neighborhood of y in X . By the assumption, there exist sets F , . . . , F n ⊂ F such that F i ∩ { (1 /i, x ) | x ∈ Y } = ∅ for every i ≤ n . Then the set H = ∩ i ≤ n F i ∩ F belongs to F and foreach ( t, x ) ∈ H , t > − /n . Since (0 , y ) is a θ -accumulation point of the filter F ∗ the set { } × U ∩ H ∗ is non-empty. Fix any (0 , x ) ∈ { } × U ∩ H ∗ . Then there exists k > n such that (1 − /k, x ) ∈ H ⊂ F .The definition of the topology on the space H Y ( X ) implies that (1 − /k, x ) ∈ V ∩ H ⊂ V ∩ F whichimplies 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 H Y ( X ) is H-closed. (cid:3) Theorem 2.10.
For any topological space X there exists an H-closed space Z which contains X as aclosed subspace.Proof. For each Hausdorff topological space X there exists an H-closed space Y which contains X as adense open subspace (see [11, Problem 3.12.6]). By Proposition 2.9, the space H Y ( X ) is H-closed. Itremains to note that the set { (1 , x ) | x ∈ X } ⊂ H Y ( X ) is closed and homeomorphic to X . (cid:3) Applications for topological semigroups
A semigroup S is called an inverse semigroup , if for each element a ∈ S there exists a unique element a − ∈ S such that aa − a = a and a − aa − = a − . The map which associates every element of an inversesemigroup to its inverse is called an inversion .A topological (inverse) semigroup is a topological space endowed with a continuous semigroup opera-tion (and an inversion, resp.). In this case the topology of the space is called ( inverse , resp.) semigrouptopology . A semitopological semigroup is a topological space endowed with a separately continuoussemigroup operation. It this case the topology of the space is called shift-continuous .Let X be a non-empty set. By B X we denote the set X × X ∪ { } where 0 / ∈ X × X endowed with thefollowing semigroup operation:( a, b ) · ( c, d ) = (cid:26) ( a, d ) , if b = c ;0 , if b = c, and ( a, b ) · · ( a, b ) = 0 · , for each a, b, c, d ∈ X. The semigroup B X is called the semigroup of X × X -matrix units . Observe that semigroups B X and B Y are isomorphic iff | X | = | Y | .If a set X is infinite then the semigroup of X × X -matrix units cannot be embedded into a compacttopological semigroup (see [12, Theorem 3]). In [13, Theorem 5] this result was generalized for countablycompact topological semigroups. Moreover, in [7, Theorem 4.4] it was shown that for an infinite set X the semigroup B X 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 thefollowing semigroup operation:( a, b ) · ( c, d ) = (cid:26) ( 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, 16]). In [15] it was proved that the bicyclic monoid does not embed into a countably compacttopological inverse semigroup. Also a topological semigroup with a pseudocompact square cannot
LOSED SUBSETS OF COMPACT-LIKE TOPOLOGICAL SPACES 7 contain the bicyclic monoid [5]. On the other hand, in [5, Theorem 6.1] it was proved that there existsa Tychonoff countably pracompact topological semigroup S densely containing the bicyclic monoid.Moreover, under Martin’s Axiom the semigroup S is countably compact (see [5, Theorem 6.6 andCorollary 6.7]). However, it is still unknown whether there exists under ZFC a countably compacttopological semigroup containing the bicyclic monoid (see [5, Problem 7.1]). Also, in [5] the followingproblem was posed: Problem 3.1 ([5, Problem 7.2]) . Is there a pseudocompact topological semigroup S that contains aclosed copy of the bicyclic monoid?Embeddings of semigroups which are generalizations of the bicyclic monoid into compact-like topo-logical semigroups were investigated in [6, 7]. Namely, in [7] it was proved that for each cardinal λ >
1a polycyclic monoid P λ does not embed as a dense subsemigroup into a feebly compact topologicalsemigroup. In [6] were described graph inverse semigroups which embed densely into feebly compacttopological semigroups.Observe that the space [0 , ω ] endowed with a semigroup operation of taking minimum becomes atopological semilattice and therefore a topological inverse semigroup. Lemma 3.2.
Let X and Y be semitopological (topological, topological inverse, resp.) semigroups suchthat there exists a continuous injective homomorphism f : X → Y . Then E fY ( X ) is a semitopological(topological, topological inverse, resp.) semigroup with respect to the semigroup operation inherited froma direct product of semigroups ( ω , min) and Y .Proof. We prove this lemma for the case of topological semigroups X and Y . Proofs in other casesare similar. Fix any elements ( α, x ) , ( β, y ) of E fY ( X ). Also, assume that β ≤ α . In the other case theproof will be similar. Fix any open neighborhood U of ( β, xy ) = ( α, x ) · ( β, y ). There are three cases toconsider:(1) β ≤ α < ω ;(2) β < α = ω ;(3) α = β = ω .In case (1) there exist γ < β and an open neighborhood V xy of xy in Y such that ( γ, β ] × V xy ⊂ U .Since Y is a topological semigroup there exist open neighborhoods V x and V y of x and y , respectively,such that V x · V y ⊂ V xy . Put U ( α,x ) = ( γ, α ] × V x and U ( β,y ) = ( γ, β ] × V y . It is easy to check that U ( α,x ) · U ( β,y ) ⊂ ( γ, β ] × V xy ⊂ U .Consider case (2). Similarly as in case (1) there exist an ordinal γ < β and open neighborhoods V x , V y and V xy of x, y and xy , respectively, such that ( γ, β ] × V xy ⊂ U and V x · V y ⊂ V xy . Since the map f is continuous there exists an open neighborhood V f − ( x ) of f − ( x ) in X such that f ( V f − ( x ) ) ⊂ V x . Put U ( ω ,x ) = ( β, ω ) × V x ∪ { ω }× f ( V f − ( x ) ) and U ( β,y ) = ( γ, β ] × V y . It is easy to check that U ( ω ,x ) · U ( β,y ) ⊂ ( γ, β ] × V xy ⊂ U .Consider case (3). There exist ordinal γ < ω , an open neighborhood V xy of xy in Y and an openneighborhood W f − ( xy ) of f − ( xy ) in X such that ( γ, ω ) × V xy ∪ { ω }× f ( W f − ( xy ) ) ⊂ U. Since Y is a topological semigroup there exist open (in Y ) neighborhoods V x and V y of x and y ,respectively, such that V x · V y ⊂ V xy . Since the map f is continuous and X is a topological semigroupthere exist open (in X ) neighborhoods W f − ( x ) and W f − ( y ) of f − ( x ) and f − ( y ), respectively, suchthat W f − ( x ) · W f − ( y ) ⊂ W f − ( xy ) , f ( W f − ( x ) ) ⊂ V x and f ( W f − ( y ) ) ⊂ V y . Put U ( ω ,x ) = ( γ, ω ) × V x ∪{ ω }× f ( W f − ( x ) ) and U ( ω ,y ) = ( γ, ω ) × V y ∪{ ω }× f ( W f − ( y ) ). It is easy to check that U ( ω ,x ) · U ( ω ,y ) ⊂ U .Hence the semigroup operation in E fY ( X, τ X ) is continuous. (cid:3) Remark 3.3.
The subsemigroup { ( ω , f ( x )) | x ∈ X } ⊂ E fY ( X ) is closed and topologically isomorphicto X .Proposition 2.4, Lemma 3.2 and Remark 3.3 imply the following: S. BARDYLA AND A. RAVSKY
Proposition 3.4.
Let X be a (semi)topological semigroup which admits a continuous injective homo-morphism f into a (semi)topological semigroup Y and C be any [0 , ω ) -productive, extension closed classof spaces. If Y ∈ C then the (semi)topological semigroup E fY ( 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)topological semigroup Y . Thenthe (semi)topological semigroup E Y ( X ) is pseudocompact and contains a closed copy of the (semi)topo-logical semigroup X . By [5, 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 thefollowing corollary which gives a positive answer to Problem 3.1. Corollary 3.6.
There exists a pseudocompact topological semigroup which contains a closed copy of thebicyclic monoid.
Further we will need the following definitions. A subset A of a topological space is called θ -closed iffor each element x ∈ X \ A there exists an open neighborhood U of x such that U ∩ A = ∅ . Observe thatif a topological space X is regular then each closed subset A of X is θ -closed. A topological space X iscalled weakly H-closed if each ω -filter F has a θ -accumulation point in X . Weakly H-closed spaces wereinvestigated in [21]. Obviously, for a topological space X the following implications hold: X is H-closed ⇒ X is weakly H-closed ⇒ X is feebly compact. However, neither of the above implications can beinverted. Indeed, an arbitrary pseudocompact but not countably compact space will be an exampleof feebly compact space which is not weakly H-closed. The space [0 , ω ) with an order topology is anexample 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 B ω of ω × ω -matrix units does not embed into a weakly H-closed topologicalsemigroup.Proof. Suppose to the contrary that B ω is a subsemigroup of a weakly H-closed topological semigroup S . By E ( B ω ) we denote the semilattice of idempotents of B ω . Observe that E ( B ω ) = { ( n, n ) | n ∈ ω } and all maximal chains of E ( B ω ) contain two elements. Then, by [2, Theorem 2.1], the set E ( B ω ) is θ -closed in S . Let F be an arbitrary ω -filter on the set { ( n, n ) | n ∈ ω } . Since S is weakly H-closedthen there exists a θ -accumulation point s ∈ S of the filter F . Since the set E ( B ω ) is θ -closed in S weobtain that s ∈ E ( B ω ). We show that s = 0. It is sufficient to prove that s · s = 0. Fix an arbitraryopen neighborhood W of s · s . Since S is a topological semigroup there exists an open neighborhood V s of s such that V s · V s ⊂ W . Since s is a θ -accumulation point of the filter F there exists infinitely many n ∈ ω such that ( n, n ) ∈ V s . Fix two distinct elements ( n , n ) ∈ V s and ( n , n ) ∈ V s . The definitionof the semigroup operation in B ω implies that 0 = ( n , n ) · ( n , n ) ∈ V s · V s ⊂ W . Hence 0 ∈ W foreach open neighborhood W of s · s which implies that s · s = 0. Hence for each ω -filter F on the set { ( n, n ) | n ∈ ω } the only θ -accumulation point of F is 0. Thus for each open neighborhood U of 0 theset A U = { ( n | ( n, n ) / ∈ U } is finite, because if there exists an open neighborhood U of 0 such that theset A U is infinite, then 0 is not a θ -accumulation point of the ω -filter F which has a base consisting ofcofinite subsets of A U .Let F be an arbitrary 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 · W of s ·
0. The continuity of the semigroupoperation in S yields open neighborhoods V s of s and V of 0 such that V s · V ⊂ W . Since the set A V = { ( n | ( n, n ) / ∈ V } is finite and s is a θ -accumulation point of the filter F there exist distinct n, m ∈ ω such that (1 , n ) ∈ V s and ( m, m ) ∈ V . Then 0 = (1 , n ) · ( m, m ) ∈ V s · V ⊂ W . Hence 0 ∈ W for each open neighborhood W of s · s · LOSED SUBSETS OF COMPACT-LIKE TOPOLOGICAL SPACES 9
Fix an arbitrary open neighborhood U of 0. Since s · S is a topological semigroup, thereexist open neighborhoods V s of s and V of 0 such that V s · V ⊂ U . Recall that the set { n | ( n, n ) / ∈ V } is finite. Then (1 , n ) = (1 , n ) · ( n, n ) ∈ V s · V ⊂ U for all but finitely many elements (1 , n ) ∈ V s .Hence 0 is a θ -accumulation point of the ω -filter F . Since the filter F was selected arbitrarily, 0 isa θ -accumulation point of any ω -filter on the set { (1 , n ) | n ∈ ω } . As a consequence, for each openneighborhood U of 0 the set B U = { n | (1 , n ) / ∈ U } is finite.Similarly it can be shown that for each open neighborhood U of 0 the set C U = { n | ( n, / ∈ U } isfinite.Fix an open neighborhood U of 0 such that (1 , / ∈ U . Since 0 = 0 · V of 0 such that V · V ⊂ U . Thefiniteness of the sets B V and C V implies that there exists n ∈ ω such that { (1 , n ) , ( n, } ⊂ V . Hence(1 ,
1) = (1 , n ) · ( n, ∈ V · V ⊂ U , which contradicts to the choice of U . (cid:3) Corollary 3.8.
The semigroup of ω × ω -matrix units does not embed into a topological semigroup S which is an H-closed topological space. However, we have the following questions:
Question 1.
Does there exist a feebly compact topological semigroup S which contains a semigroup of ω × ω -matrix units? Question 2.
Does there exist a topological semigroup S which cannot be embedded into a feebly compacttopological semigroup T ? We remark that these questions were posed at the Lviv Topological Algebra Seminar a few years ago.
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S. Bardyla: Institute of Mathematics, Kurt G¨odel Research Center, Vienna, Austria
E-mail address : [email protected] A. Ravsky: Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Nat. Acad.Sciences of Ukraine, Lviv, Ukraine
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