A survey of cardinality bounds on homogeneous topological spaces
aa r X i v : . [ m a t h . GN ] J u l A SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUSTOPOLOGICAL SPACES
NATHAN CARLSONA
BSTRACT . In this survey we catalogue the many results of the past severaldecades concerning bounds on the cardinality of a topological space with homo-geneous or homogeneous-like properties. These results include van Douwen’sTheorem, which states | X | ≤ πw ( X ) if X is a power homogeneous Hausdorffspace [25], and its improvements | X | ≤ d ( X ) πχ ( X ) [42] and | X | ≤ c ( X ) πχ ( X ) [18] for spaces X with the same properties. We also discuss de la Vega’s The-orem, which states that | X | ≤ t ( X ) if X is a homogeneous compactum [24],as well as its recent improvements and generalizations to other settings. Thisreference document also includes a table of strongest known cardinality boundson spaces with homogeneous-like properties. The author has chosen to givesome proofs if they exhibit typical or fundamental proof techniques. Finally,a few new results are given, notably (1) | X | ≤ d ( X ) πnχ ( X ) if X is homoge-neous and Hausdorff, and (2) | X | ≤ πχ ( X ) c ( X ) qψ ( X ) if X is a regular homo-geneous space. The invariant πnχ ( X ) , defined in this paper, has the property πnχ ( X ) ≤ πχ ( X ) and thus (1) improves the bound d ( X ) πχ ( X ) for homoge-neous Hausdorff spaces. The invariant qψ ( X ) , defined in [32], has the properties qψ ( X ) ≤ πχ ( X ) and qψ ( X ) ≤ ψ c ( X ) if X is Hausdorff, thus (2) improvesthe bound c ( X ) πχ ( X ) in the regular, homogeneous setting.
1. I
NTRODUCTION
A topological space X is homogeneous if for every x, y ∈ X there exists ahomeomorphism h : X → X such that h ( x ) = y . Roughly, X is homogeneous ifthe topology at every point is “identical” to that of every other point. X is powerhomogeneous if there exists a cardinal κ such that X κ is homogeneous. Manycommonly studied spaces are homogeneous (for example, R , the unit circle, allconnected manifolds in general, and topological groups) and as such are ubiqui-tous across fields of mathematics. In particular, those homogeneous spaces thatare compact play a prominent role. In 1931 Keller [35] showed that the HilbertCube [0 , ω is homogeneous. As [0 , is not homogeneous, this was an earlyexample of a compact power homogeneous space that is not homogenous. An-other such example is the ordinal space ω + 1 , as ( ω + 1) ω is homogeneous. Theauthor refers the reader to the 2014 book chapter Topological Homogeneity byA.V. Arhangel ′ ski˘ı and J. van Mill [7] for a broad and extensive reference on thetheory of general homogeneous topological spaces.Arhangel ′ ski˘ı [1] showed in 1969 that the cardinality of any compact, first count-able space is at most c , the cardinality of the continuum R , thus answering a 50 yearold question of Alexandroff and Urysohn. Soon afterwards, in 1970, he showed that any compact, homogeneous, sequential space also has cardinality at most c [2][3]. This demonstrated that in the presence of homogeneity the first countablecondition can be relaxed to the weaker sequential condition. This might be re-garded as the first example of cardinality bound that can be improved if a space isadditionally known to be homogeneous. In the decades that followed, and in re-cent years, many well-known cardinality bounds on topological spaces have beenimproved with homogeneity, or homogeneous-like properties.The purpose of this survey is to give a thorough account of subsequent resultsconcerning the cardinality of a homogeneous topological space. While there areimportant open problems in the more general theory of cardinal functions on ho-mogeneous spaces (such as van Douwen’s Problem, which asks if the cellularity c ( X ) of a homogeneous compactum is at most c ), in this survey we confine our-selves only to cardinality considerations.Several proofs are given in this survey. The ones that are chosen were chosen fortheir illustrative nature, as they are fundamental to the theory of cardinality boundson homogeneous spaces. They were also chosen for their simplicity and elegance,in the author’s opinion. Theorems that have proofs that are more involved andcomplicated are simply cited in this survey. In addition, there are a few new proofsgiven in this paper that represent mostly minor improvements of known results.This paper is organized as follows. In § § §
4, we explore two bounds for homo-geneous Hausdorff spaces that improved van Douwen’s Theorem: d ( X ) πχ ( X ) and c ( X ) πχ ( X ) , respectively. In § § § § VAN D OUWEN ’ S T HEOREM
In 1978 Eric van Douwen [25] showed that | X | ≤ πw ( X ) for a Hausdorff ho-mogeneous space X . This was first cardinality bound for a general homogeneousspace X . Homogeneity, or homogeneous-like properties are necessary in this re-sult; for example, the non-homogeneous space βω does not satisfy this bound. Infact, van Douwen showed that this bound holds for general power homogeneousspaces using sophisticated “clustering” techniques that encode information aboutprojection maps of the form π : X κ → X . Van Douwen made extensive use ofcollections of sets invariant under homeomorphisms. As one can see from the next SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUS TOPOLOGICAL SPACES 3 theorem, his paper was primarily focused on results that imply a space is not ho-mogeneous, or not power homogeneous. His cardinality bound was indeed simplyjust a consequence of this sophisticated theorem.
Theorem 2.1 (van Douwen [25], 1978) . If the space X admits a continuous maponto a Hausdorff space Y with | Y | > πw ( Y ) , then no power of X is homogeneousin each of the following cases: (a) f is open or is a retraction and d ( X ) ≤ πw ( Y ) , (b) f is perfect, X is regular, and d ( X ) ≤ πw ( Y ) , or (c) X is compact Hausdorff and w ( X ) ≤ πw ( Y ) . As d ( X ) ≤ πw ( X ) for any space X , we have the following corollary, whichwe will refer to as van Douwen’s Theorem. Corollary 2.2 (van Douwen [25], 1978) . If X is power homogeneous then | X | ≤ πw ( X ) . In fact, it also follows from Theorem 2.1 that no power of βω \ ω is homoge-neous, answering a question of Murray Bell. (Frolik [30] had previously shownthis space is not homogeneous in ZFC).While van Douwen’s work was groundbreaking and answered important ques-tions, it turns out that proofs that | X | ≤ πw ( X ) when X is homogeneous are,by comparison, straightforward. In this survey we’ll see several proofs that implythis result (see Theorems 2.3, 3.3, and 4.1). Furthermore, the full version of vanDouwen’s Theorem, in the power homogeneous case, has been improved upon indifferent directions in subsequent decades. See Theorems 3.5, 3.6, and 4.5.Corollary 2.2 has a straightforward improvement in the homogeneous case byconsidering the group H ( X ) of autohomeomorphisms on a space X . This wasshown by Frankiewicz. The proof we give here is adapted from that given in [33],2.38. The author considers this proof to involve basic techniques that are typicallyused when considering bounds on the cardinality of a homogeneous space and thegroup H ( X ) . Theorem 2.3 (Frankiewicz [29], 1979) . If X is Hausdorff then | H ( X ) | ≤ πw ( X ) .Proof. Let κ = πw ( X ) and let B be a π -base for X such that | B | ≤ κ . We showthat map φ : H ( X ) → P ( B ) B defined by φ ( h )( B ) = { C ∈ B : C ⊆ h [ B ] } is one-to-one. Suppose we have f, g ∈ H ( X ) such that f = g . Then thereexists x ∈ X such that f ( x ) = g ( x ) . Let U and V be disjoint neighborhoodsof f ( x ) and g ( x ) , respectively. As x ∈ f ← [ U ] ∩ g ← [ V ] , there exists B ∈ B such that B ⊆ f ← [ U ] ∩ g ← [ V ] . Thus f [ B ] ⊆ U and g [ B ] ⊆ V . There exists C ∈ B such that C ⊆ f [ B ] , and thus C is not a subset of g [ B ] . This shows φ ( f )( B ) = φ ( g )( B ) and φ ( f ) = φ ( g ) . We conclude that φ is one-to-one andtherefore | H ( X ) | ≤ | P ( B ) B | ≤ (2 κ ) κ = 2 κ . (cid:3) As it is easily seen that | X | ≤ | H ( X ) | if X is homogeneous, the homogeneouscase of 2.2 follows. Years later, in 2008, an improved bound for | H ( X ) | was givenby the author and Ridderbos using the Erd¨os-Rado theorem from partition theory. NATHAN CARLSON
Theorem 2.4 (C., Ridderbos [18], 2008) . If X is Hausdorff then | H ( X ) | ≤ c ( X ) πχ ( X ) sd ( X ) . The separation degree sd ( X ) in the cardinal inequality above is defined asfollows. We say that a subset Z of X separates a subset G of H ( X ) , if for all f, g ∈ G with f = g there is some z ∈ Z with f ( z ) = g ( z ) . sd ( X ) is defined by sd ( X ) = min {| Z | : Z separates H ( X ) } . It is always the case that sd ( X ) ≤ d ( X ) ,and thus 2.4 is a logical improvement of 2.3.3. T HE CARDINALITY BOUND d ( X ) πχ ( X ) The proof of Theorem 2.3 gives a straightforward way to demonstrate van Douwen’sTheorem in the homogeneous case. A few years later, in 1981, Ismail [32] gaveanother relatively simple proof with a slightly stronger conclusion that used the no-tion of a q -pseudo base. For a point x in a space X , a family B of nonempty opensubsets of X is a q -pseudo base of x in X if for each y ∈ X such that y = x , thereis a subfamily C of B such that x ∈ S C and y / ∈ S C . Ismail defined the q -pseudocharacter of x in X by qψ ( x, X ) = min {| B | : B is a q -pseudo base of x in X } and the q -pseudo character of X by qψ ( X ) = sup { qψ ( x, X ) : x ∈ X } . It wasshown in [32] that if X is Hausdorff then qψ ( X ) ≤ | X | , qψ ( X ) ≤ πχ ( X ) , andthat qψ ( X ) ≤ ψ c ( X ) .Recall a set U in a space X is regular open if U = intclU and that RO ( X ) de-notes the collection of regular open subsets of X . Ismail showed the following fun-damental result, for which we provide a proof. Integral to the proof is the fact thatin a homogeneous space X if one fixes a point p ∈ X there exist homeomorphisms h x : X → X such that h x ( p ) = x for each x ∈ X . These homeomorphisms playan important role in most proofs of cardinality bounds on homogeneous spaces.The proof exhibits how these homeomorphisms interact with the invariant family RO ( X ) . Upon examination, the proof does not require the Hausdorff property,despite the fact that Ismail listed that property as an hypothesis. Theorem 3.1 (Ismail [32], 1981) . If X is a homogeneous space, then | X | ≤| RO ( X ) | qψ ( X ) .Proof. Fix a point p ∈ X and let B be a q-pseudo base at x in X such that | B | = qψ ( X ) . Without loss of generality we can assume that B ⊆ RO ( X ) , for otherwisewe could consider the q-pseudo base { intB : B ∈ B } ⊆ RO ( X ) , which has thesame cardinality as B . For all x ∈ X , there exists a homeomorphism h x : X → X such that h x ( p ) = x .Define a function φ : X → RO ( X ) B by φ ( x )( B ) = h x [ B ] . (Note that if B ∈ RO ( X ) then h x [ B ] ∈ RO ( X ) ). We show φ is one-to-one. Let x, y ∈ X such that x = y . Then h ← x ( y ) = p . As B is a q-pseudo base at p , there exists C ⊆ B such that p ∈ S C and h ← x ( y ) / ∈ S C . Therefore, y = h y ( p ) ∈ S { h y [ C ] : C ∈ C } and y / ∈ S { h x [ C ] : C ∈ C } . It follows that there exists a C ∈ C ⊆ B such that h x [ C ] = h y [ C ] , and thus φ ( x )( C ) = φ ( y )( C ) and φ ( x ) = φ ( y ) . This shows φ isone-to-one and | X | ≤ (cid:12)(cid:12) RO ( X ) B (cid:12)(cid:12) ≤ | RO ( X ) | qψ ( X ) . (cid:3) To see that the above result is logically stronger than Theorem 2.2 in the ho-mogeneous case, recall that | RO ( X ) | ≤ d ( X ) for an arbitrary space X (see, for SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUS TOPOLOGICAL SPACES 5 example, 2.6d in [33]) and that πw ( X ) = d ( X ) πχ ( X ) ≥ d ( X ) qψ ( X ) if X isHausdorff. Corollary 3.2 (Ismail [32], 1981) . If X is a homogeneous Hausdorff space, then | X | ≤ | RO ( X ) | πχ ( X ) . It was noted independently by de la Vega [23, Theorem 1.14] and Ridderbos [41,Proposition 2.2.7] that | RO ( X ) | can be replaced in 3.2 by the invariant d ( X ) .The author considers the proof of this result to be elegant, representative, and afundamental model for more sophisticated related cardinality bounds on spaceswith homogeneous-like properties. One might consider this result the analogue ofthe cardinality bound | X | ≤ d ( X ) χ ( X ) for a general Hausdorff space X and, infact, involves a simpler one-to-one map argument using the homeomorphisms h x .We give this proof below and the reader should view it as fundamental in the theoryof cardinality bounds on homogeneous spaces. Theorem 3.3 (de la Vega [23], 2005, and Ridderbos [41]) . If X is homogeneousand Hausdorff then | X | ≤ d ( X ) πχ ( X ) .Proof. Fix a point p ∈ X and a local π -base B at p consisting of non-empty opensets such that | B | ≤ πχ ( X ) . Let D be a dense subset of X such that | D | = d ( X ) .For all x ∈ X let h x : X → X be a homeomorphism such that h x ( p ) = x .For all x ∈ X and B ∈ B , h x [ B ] is a non-empty open set and thus there exists d ( x, B ) ∈ h x [ B ] ∩ D . Define φ : X → D B by φ ( x )( B ) = d ( x, B ) .We show φ is one-to-one. Let x = y ∈ X . Separate x and y by disjoint opensets U and V , respectively.Then p ∈ h ← x [ U ] ∩ h ← y [ V ] , an open set. There exists B ∈ B such that B ⊆ h ← x [ U ] ∩ h ← y [ V ] . It follows that φ ( x )( B ) = d ( x, B ) ∈ h x [ B ] ⊆ U and φ ( y )( B ) = d ( y, B ) ∈ h y [ B ] ⊆ V . Thus φ ( x )( B ) = φ ( y )( B ) and φ ( x ) = φ ( y ) . This shows φ is one-to-one and | X | ≤ | D | | B | ≤ d ( X ) πχ ( X ) . (cid:3) It was shown in [17] that the semiregularization X s of a space X is homoge-neous if X is homogeneous. (See, for example, [39] for a thorough discussion ofthe semiregularlization of a space). Using results in [17], Theorem 3.3, and thefact that d ( X s ) ≤ RO ( X ) , we have | X | = | X s | ≤ d ( x s ) πχ ( X s ) ≤ RO ( X ) πχ ( X ) .Therefore Ismail’s result 3.2 above follows from 3.3. However, Theorem 3.1 andTheorem 3.3 appear to be incomparable.Theorem 3.3 has a minor but interesting improvement by replacing πχ ( X ) witha smaller cardinal function the author will call πnχ ( X ) . We define πnχ ( X ) asfollows. For a point x in a space X , we define a local π -network at x to be acollection N of sets (not necessarily open) such that if x ∈ U and U is open, thenthere exists N ∈ N such that N ∈ U . Denote by πnχ ( x, X ) the least infinitecardinal κ such that x has a local π -network N of cardinality κ and χ ( N, X ) ≤ κ for all N ∈ N . Define πnχ ( X ) = sup { πnχ ( x, X ) : x ∈ X } . Observe that πnχ ( X ) ≤ πχ ( X ) .The following result appears to be new in the literature. The proof also involvesthe construction of a one-to-one map, however there is another “layer” in this con-struction above and beyond what is done in the proof of Theorem 3.3. NATHAN CARLSON
Theorem 3.4. If X is homogeneous and Hausdorff then | X | ≤ d ( X ) πnχ ( X ) .Proof. Let κ = πnχ ( X ) . As in the proof of Theorem 3.3, fix p ∈ X and for every x ∈ X fix a homeomorphism h x : X → X such that h x ( p ) = x . There exists alocal π -network N at p such that | N | ≤ κ and χ ( N, X ) ≤ κ for all N ∈ N . Let D be dense in X such that | D | = d ( X ) . Let { U ( N, α ) : α < κ } be a neighborhoodbase at N for each N ∈ N .As D is dense, for all x ∈ X , for all N ∈ N , and for all α < κ , there exists apoint d ( x, N, α ) ∈ h x [ U ( N, α )] ∩ D . We define a function φ : X → ( D κ ) N by φ ( x )( N )( α ) = d ( x, N, α ) and show φ is one-to-one. Let x = y ∈ X and separate x and y by disjoint open sets U and V , respectively. Then p ∈ h ← x [ U ] ∩ h ← y [ V ] . As N is a local π -network for p , there exists N ∈ N such that N ⊆ h ← x [ U ] ∩ h ← y [ V ] .As { U ( N, α ) : α < κ } is a neighborhood base at N , there exists α < κ such that U ( N, α ) ⊆ h ← x [ U ] ∩ h ← y [ V ] . Thus, h x [ U ( N, α )] ⊆ U and h y [ U ( N, α )] ⊆ V ,showing d ( x, N, α ) = d ( y, N, α ) . It follows that φ ( x )( N )( α ) = φ ( y )( N )( α ) , φ ( x )( N ) = φ ( y )( N ) , and finally that φ ( x ) = φ ( y ) . This shows φ is one-to-one,and | X | ≤ (cid:12)(cid:12)(cid:12) ( D κ ) N (cid:12)(cid:12)(cid:12) ≤ ( | D | κ ) κ = | D | κ ≤ d ( X ) πnχ ( X ) . (cid:3) We turn now to the setting in which a space X is power homogeneous and notnecessarily homogeneous. In the full power homogeneous setting, van Douwen’sTheorem 2.2 has been improved in variety of ways using differing techniques. Inthe study of power homogeneous spaces X , information on the homogeneity of X κ for a cardinal κ , and the autohomeomorphisms on that space, must be utilizedin some way at the level of the space X . This information must be captured in sucha way as to generate inequalities involving cardinal functions on X . The projectionmaps π : X κ → X are typically, and necessarily, used in this process. In 2006,Ridderbos [42] used new techniques involving projection maps to give the firstimprovement to 2.2 in the full power homogeneous setting. Theorem 3.5 (Ridderbos [42], 2006) . If X is a power homogeneous Hausdorffspace, then | X | ≤ d ( X ) πχ ( X ) . If X κ is homogeneous then, as in any homogeneous space, after fixing a point p ∈ X κ , there exist homeomorphisms h x : X κ → X κ such that h x ( p ) = x for every x ∈ X κ . However, a critical ingredient in the proof of Theorem 3.5is demonstrating the existence of such homeomophisms with additional importantproperties relating to local π -bases in X κ and X . This is shown in Corollary 3.3 in[42].Recently, Bella and the author extended 3.5 to give a bound for the cardinalityof any open set in a power homogeneous Hausdorff space. Theorem 3.6 (Bella, C. [11], 2018) . Let X be a power homogeneous space. If D ⊆ X and U is an open set such that U ⊆ D , then | U | ≤ | D | πχ ( X ) . Recall that a subset D of a space X is θ -dense in X if U ∩ D = ∅ for everynon-empty open set U of X . The θ - density of X is defined by d θ ( X ) = min {| D | : SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUS TOPOLOGICAL SPACES 7 D is θ -dense in X } . A variation of the proof of 3.5 in the Urysohn setting wasgiven by the author in [17]. Theorem 3.7 (C. [17], 2007) . If X is power homogeneous and Urysohn then | X | ≤ d θ ( X ) πχ ( X ) .
4. T
HE CARDINALITY BOUND c ( X ) πχ ( X ) Van Douwen’s Theorem 2.2 also has an improvement in a different direction.Theorem 3.3, coupled with the fact that d ( X ) ≤ πχ ( X ) c ( X ) for any regular space X ( ˇSapirovski˘ı [44]), shows that | X | ≤ c ( X ) πχ ( X ) for regular homogeneousspaces X . This was observed by Arhangel ′ ski˘ı in [5]. In [17], these results wemodified to show | X | ≤ c ( X ) πχ ( X ) for Urysohn homogeneous spaces X . Thisfollows from Theorem 3.7 and the fact that d θ ( X ) ≤ πχ ( X ) c ( X ) for any space X [17].As c ( X ) πχ ( X ) ≤ πw ( X ) for any space, we see that c ( X ) πχ ( X ) is an improvedbound over πw ( X ) . It is a real improvement, even in the compact case, as thecompact right topological group X , constructed under CH by Kunen [37], satisfies c ( X ) πχ ( X ) = ω and πw ( X ) = ω .The question remained open whether this bound was valid in the Hausdorff case.Using entirely different techniques, Ridderbos and the author answered this in theaffirmative in [18]. While relatively simple, it represented the first use of the Erd¨os-Rado Theorem in the proof of a cardinal inequality involving homogeneous spaces.It is related to the proof of the Hajnal-Juh´asz theorem | X | ≤ c ( X ) χ ( X ) for generalHausdorff spaces that uses the Erd¨os-Rado theorem (see [33, 2.15b]). Indeed, onemay view this result as the homogeneous analogue of the Hajnal-Juh´asz theorem.We give this proof below as our next fundamental proof. Theorem 4.1 (C., Ridderbos [18], 2008) . If X is homogeneous and Hausdorff then | X | ≤ c ( X ) πχ ( X ) .Proof. Let κ = c ( X ) πχ ( X ) . Fix a point p ∈ X and a local π -base B at p con-sisting of non-empty sets such that | B | ≤ κ . For all x ∈ X let h x : X → X be a homeomorphism such that h x ( p ) = x . Define B : [ X ] → B as follows.For all x = y , there exist disjoint open sets U ( x, y ) and V ( x, y ) containing x and y , respectively. For each x = y ∈ X the open set h ← x [ U ] ∩ h ← y [ V ] contains p .Thus there exists B ( x, y ) ∈ B such that B ( x, y ) ⊆ h ← x [ U ] ∩ h ← y [ V ] . Note that h x [ B ( x, y )] ∩ h y [ B ( x, y )] = ∅ for all { x, y } ∈ [ X ] .By way of contradiction suppose that | X | > κ . By the Erd¨os-Rado Theoremthere exists Y ∈ [ X ] κ + and B ∈ B such that B = B ( x, y ) for all x = y ∈ Y . For x = y ∈ Y we have h x [ B ] ∩ h y [ B ] = h x [ B ( x, y )] ∩ h y [ B ( x, y )] = ∅ . This showsthat C = { h x [ B ] : x ∈ Y } is a cellular family. But | C | = | Y | = κ + > c ( X ) , acontradiction. Therefore | X | ≤ κ . (cid:3) Using Ismail’s invariant qψ ( X ) , one observes that the above theorem has a im-provement in the case when the space X is additionally regular. The proof is a NATHAN CARLSON simple matter of lining up a few results, although the result appears to be new inthe literature.
Theorem 4.2. If X is regular and homogeneous, then | X | ≤ πχ ( X ) c ( X ) qψ ( X ) .Proof. Since X is regular and homogeneous, we have | X | ≤ | RO ( X ) | qψ ( X ) ≤ ( πχ ( X ) c ( X ) ) qψ ( X ) ≤ πχ ( X ) c ( X ) qψ ( X ) . The first inequality above is Theorem 3.1, and the second inequality follows fromthe inequality | RO ( X ) | ≤ πχ ( X ) c ( X ) for regular spaces (see [33] 2.37). (cid:3) This is an actual improvement over the bound c ( X ) πχ ( X ) because qψ ( X ) ≤ πχ ( X ) for a Hausdorff space X . Furthermore, it improves the cardinality bound πχ ( X ) c ( X ) ψ ( X ) for regular spaces X given by ˇSapirovski˘ı [44], as qψ ( X ) ≤ ψ ( X ) if X is regular. One may view Theorem 4.2 as the homogeneous analogue ofˇSapirovski˘ı’s result.We turn now to the case where the space X is power homogeneous. In this case,van Mill [38] first demonstrated the bound c ( X ) πχ ( X ) holds under the assumptionof compactness, using a variation of van Douwen’s clustering techniques. Theorem 4.3 (van Mill [38], 2005) . If X is a power homogeneous compactum,then | X | ≤ c ( X ) πχ ( X ) . Van Mill’s result follows in fact as a corollary to this result in the same paper: | X | ≤ w ( X ) πχ ( X ) for a power homogeneous compactum X . (Recall that later itwas shown that | X | ≤ d ( X ) πχ ( X ) for any power homogeneous Hausdorff spaceby Ridderbos (Theorem 3.5)). We will see, however, in Theorem 4.5 below that thecardinality bound c ( X ) πχ ( X ) holds for any power homogeneous Hausdorff space.Soon after van Mill’s result, Bella gave an improvement of Theorem 2.2 forregular power homogenous spaces using the cardinal function c ∗ ( X ) . Recall thatif X = Q i ∈ T X i is an arbitrary product of spaces and c ( Q i ∈ F X i ) ≤ κ for eachfinite subset F of T , then c ( X ) ≤ κ . If follows that if X is a space and λ is aninfinite cardinal then c ( X λ ) = c ∗ ( X ) . Theorem 4.4 (Bella [10], 2005) . If X is a power homogeneous T space, then | X | ≤ c ∗ ( X ) πχ ( X ) . Note that c ∗ ( X ) πχ ( X ) ≤ πw ( X ) for any space. Bella’s result is also a real im-provement of Theorem 2.2, as the same space X in [37] satisfies c ∗ ( X ) πχ ( X ) = ω and πw ( X ) = ω .In 2008 it was finally shown that c ( X ) πχ ( X ) is a bound for the cardinality of anypower homogeneous Hausdorff space. This represents a second full improvementof van Douwen’s theorem alongside Theorem 3.5. The proof of this is a sophisti-cated application of the Erd¨os-Rado theorem. Theorem 4.5 (C. and Ridderbos [18], 2008) . If X is power homogeneous andHausdorff then | X | ≤ c ( X ) πχ ( X ) . SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUS TOPOLOGICAL SPACES 9
A decade later a variation of this result was shown for spaces that are Urysohnor quasiregular. Recall that a space is quasiregular if every nonempty open setcontains a nonempty regular closed set. A collection of nonempty open sets isa
Urysohn cellular family if the closures of any two are disjoint. We define the
Urysohn cellularity of a space X as U c ( X ) = sup {| C | : C is a Urysohn cellular family } .It is clear that U c ( X ) ≤ c ( X ) for any space X . Theorem 4.6 (Bonanzinga, C., Cuzzup´e, Stavrova [15], 2018) . If X is a powerhomogeneous space that is Urysohn or quasiregular, then | X | ≤ Uc ( X ) πχ ( X ) .
5. C
OMPACT HOMOGENEOUS SPACES AND DE LA V EGA ’ S T HEOREM
In 1970 Arhangel ′ ski˘ı showed that the cardinality of a sequential, homoge-neous compactum is at most c [2][3]. (By compactum we mean a compact Haus-dorff space). He then asked if the sequential property can be relaxed to countablytight (see [2]). In 2006, de la Vega [24] answered this long-standing question byshowing that the cardinality of a homogeneous compactum is bounded by t ( X ) .Previously Dow [26] had shown under PFA that any compact space X of countabletightness contains a point of countable character; thus if the space is additionallyhomogeneous then | X | ≤ c .De la Vega’s original proof involved the elementary submodels technique and,in fact, showed that | X | ≤ L ( X ) t ( X ) pct ( X ) for any regular homogeneous space.(See the next section for the definition of pct ( X ) and a discussion of this and othergeneralizations of de la Vega’s Theorem). It was observed in [19] that much of thework of de la Vega’s elementary submodel proof can be replaced by a theorem ofPytkeev concerning covers by G κ -sets. If X is a space and κ an infinite cardinal,the G κ - modification X κ of X is the space formed on the underlying set X bytaking the collection of G κ -sets as a basis. Theorem 5.1 (Pytkeev [40], 1985) . Let X be a compactum and κ an infinite car-dinal. Then L ( X κ ) ≤ t ( X ) · κ . Another crucial ingredient in the proof of de la Vega’s theorem is a result of Arhangel ′ ski˘ı’sfrom [4]. Theorem 5.2 (Arhangel ′ ski˘ı [4], 1978) . Let X be a compactum and let κ = t ( X ) .There exists a non-empty G κ -set G and a set H ⊆ X such that | H | ≤ κ and G ⊆ H . Using Theorems 3.3, 5.2, and 5.1 a simplified proof of de la Vega’s Theoremwas given in [19]. We give this below as our third fundamental proof.
Theorem 5.3 (de la Vega [24], 2006) . If X is a homogeneous compactum then | X | ≤ t ( X ) .Proof. ([19]) Let κ = t ( X ) . By Theorem 5.2 there exists a non-empty G κ -setcontained in the closure of a set of size at most κ . Fix a point p ∈ G and, as inprevious proofs, we obtain homeomorphisms h x : X → X such that h x ( p ) = x for all x ∈ X . G = { h x [ G ] : x ∈ X } is a cover of X consisting of G κ -sets. There exists a family H = { H G : G ∈ G } such that G ⊆ H G and | H G | ≤ κ for all G ∈ G .By Pytkeev’s Theorem 5.1 there exists G ′ ⊆ G such that G ′ covers X and | G ′ | ≤ κ . It follows that X = S G ′ ⊆ S G ∈ G ′ H G ⊆ S G ∈ G ′ H G . Thus, H = S G ∈ G ′ H G is dense in X and | H | ≤ κ · κ = 2 κ . Therefore d ( X ) ≤ κ . By Theorem 3.3above and ˇSapirovski˘ı’s result that πχ ( X ) ≤ t ( X ) for a compact space X , wehave | X | ≤ d ( X ) πχ ( X ) ≤ (2 κ ) κ = 2 κ . (cid:3) While much of the work in this proof is done by Theorem 5.1, which itself is anelaborate closing-off argument, the homogeneity of the space is not utilized in 5.1.Instead the homogeneity is applied in two straightforward and elegant ways: firstby using Theorem 5.2 and homeomorphisms to cover the space by non-empty G κ -sets, and second through the use of Theorem 3.3.The compactness condition is necessary in de la Vega’s Theorem. Indeed, itdoes not hold for all countably compact homogeneous spaces, nor all H-closedhomogeneous spaces, as the next example from [21] shows. Theorem 5.4 (C., Porter, Ridderbos [21], 2017) . There exists a countably compact,H-closed, Urysohn, separable, countably tight, homogeneous space X such that | X | = 2 c .Proof. Let Y be the Cantor Cube c with it usual topology and let X be the count-able tightness modification of Y . That is, the closure of a set A in X is givenby cl X ( A ) = [ B ∈ [ A ] ≤ ω cl Y ( B ) .X has a finer topology than Y , demonstrating that X is not compact as compactspaces are minimal Hausdorff. However, by Theorem 4.2 in [21], X is countablycompact, H-closed, countably tight, and separable. Furthermore, since Y is thesemiregularization of X , X is also Urysohn. (See [39]). (cid:3) De la Vega’s Theorem was extended to power homogeneous compacta in [8].
Theorem 5.5 (Arhangel ′ ski˘ı, van Mill, and Ridderbos [8], 2007) . If X is a powerhomogeneous compactum then | X | ≤ t ( X ) . In 2018 Juh´asz and van Mill introduced new techniques and improved de laVega’s Theorem in the countable case. Considering compact homogenous spacesthat are σ -CT (a countable union of countably tight subspaces), they obtained thefollowing two results. Theorem 5.6 (Juh´asz, van Mill [34], 2018) . If a compactum X is the union ofcountably many dense countably tight subspaces and X ω is homogeneous, then | X | ≤ c . Theorem 5.7 (Juh´asz, van Mill [34], 2018) . If X is an infinite homogeneous com-pactum that is the union of finitely many countably tight subspaces, then | X | ≤ c . SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUS TOPOLOGICAL SPACES 11
A crucial ingredient in these theorems was a strengthening of Arhangel ′ ski˘ı’sTheorem 5.2 in the countable case. (Recall Theorem 5.2 played a central role inproving de la Vega’s Theorem). The strengthening has a deep and sophisticatedproof. A subset of a space is subseparable if it is contained in the closure of acountable set. Theorem 5.8 (Juh´asz, van Mill [34], 2018) . Every σ -CT compactum X has a non-empty subseparable G δ -set. Soon afterwards the “ X ω is homogeneous” condition in Theorem 5.6 was gen-eralized to “ X is power homogeneous” in [22]. Theorem 5.9 (C. [22], 2018) . If a power homogeneous compactum X is the unionof countably many dense countably tight subspaces, then | X | ≤ c . Motivated by the results of Juh´asz and van Mill, the author introduced a cardinalinvariant known as wt ( X ) , the weak tightness, in [22]. To define it we need thenotion of the κ -closure cl κ A of a set A in a space X for a cardinal κ . This isdefined by cl κ A = S { B : B ∈ [ A ] ≤ κ } . The weak tightness wt ( X ) of X isdefined as the least infinite cardinal κ for which there is a cover C of X such that | C | ≤ κ and for all C ∈ C , t ( C ) ≤ κ and X = cl κ C . We say that X is weaklycountably tight if wt ( X ) = ω . It is clear that wt ( X ) ≤ t ( X ) . Example 2.3 in [12]provides a straightforward example of a compact group of tightness ω such that,under ω = 2 ω , X is weakly countably tight.The condition “ X = cl κ C ” in the above definition can be difficult to workwith. The next proposition gives additional conditions under which this conditioncan be relaxed to “ C is dense in X ”. Proposition 5.10 ([22]) . Let X be a space, κ a cardinal, and C a cover of X suchthat | C | ≤ κ , and for all C ∈ C , t ( C ) ≤ κ and C is dense in X . If t ( X ) ≤ κ or π χ ( X ) ≤ κ then wt ( X ) ≤ κ . Pytkeev’s Theorem 5.1 has an improvement using wt ( X ) . Theorem 5.11 (C. [22]) . Let X be a compactum and κ an infinite cardinal. Then L ( X κ ) ≤ wt ( X ) · κ . Additionally, Bella and the author were able to give a result that amounts to avariation of both Theorem 5.2 and 5.8.
Theorem 5.12 (Bella, C. [12], 2020) . Let X be a compactum and let κ = wt ( X ) .Then there exists a non-empty closed set G ⊆ X and a C -saturated set H ∈ [ X ] ≤ κ such that G ⊆ H and χ ( G, X ) ≤ κ . Using Theorems 5.11 and 5.12, Bella and the author were able to give a fullimprovement to de la Vega’s Theorem in [12]. Recall that πχ ( X ) ≤ t ( X ) for acompactum X . Theorem 5.13 (Bella, C. [12], 2020) . If X is a homogeneous compactum then | X | ≤ wt ( X ) πχ ( X ) . Below we isolate the case of Theorem 5.13 where all cardinal invariants involvedare countable. It follows directly from Proposition 5.10 and the above. CompareCorollary 5.14 with Theorem 5.6.
Corollary 5.14.
Let X be a homogeneous compactum of countable π -characterwith a cover C such that | C | ≤ c and for all C ∈ C , C is countably tight and densein X . Then | X | ≤ c . Another corollary to Theorem 5.13 follows directly from the fact that in ancompact, T space there exists a point of countable π -character. (This is due toˇSapirovski˘ı). If the space X is additionally homogeneous then πχ ( X ) = ω . Thiscorollary has not been previously mentioned in the literature. Corollary 5.15. If X is compact, T , and homogeneous, then | X | ≤ wt ( X ) .
6. G
ENERALIZATIONS OF DE LA V EGA ’ S T HEOREM
This section is devoted to extensions of de la Vega’s Theorem; that is, results thatdirectly imply that theorem in a more generalized setting. Natural questions arise,such as, does Lindel¨of suffice instead of the compactness property? The answerto this question is no. In [19], an example of a σ -compact, homogeneous space X was constructed with the property | X | > L ( X ) πχ ( X ) t ( X ) . This shows L ( X ) t ( X ) isnot a bound for the cardinality of every Hausdorff homogeneous space.Exactly what are the necessary properties of compactness needed in this theo-rem? It turns out that one pair of necessary properties are Lindel¨of and countablepoint-wise compactness type. The point-wise compactness type pct ( X ) of a space X is the least infinite cardinal κ such that X can be covered by compact sets K such that χ ( K, X ) ≤ κ . Clearly compact spaces are of countable point-wise com-pactness type. Also, all locally compact spaces have this property. In [23], de laVega showed that | X | ≤ L ( X ) t ( X ) pct ( X ) for any regular homogeneous space, andthis bound was shown to be valid for regular power homogeneous spaces in [42].In [19], the regularity property was shown to be unnecessary. Theorem 6.1 (C., Ridderbos [19], 2012) . If X is a power homogeneous Hausdorffspace then | X | ≤ L ( X ) t ( X ) pct ( X ) . Thus, for example, the cardinality bound t ( X ) holds for all locally compact,Lindel¨of homogeneous Hausdorff spaces.The next five theorems represent slight improvements of Theorem 6.1. For acardinal κ and a space X , a subset { x α : α ≤ κ } ⊆ X is a free sequence of length κ if for every β < κ , cl X { x α : α < β } ∩ cl X { x α : α ≥ β } = ∅ . The freesequence number F ( X ) is the supremum of the lengths of all free sequences in X .It is well-known that t ( X ) = F ( X ) if X is a compactum. In addition, as F ( X ) ≤ L ( X ) t ( X ) for any space X , the following theorem improves Theorem 6.1. Theorem 6.2 (C., Porter, Ridderbos [20], 2012) . If X is a power homogeneousHausdorff space then | X | ≤ L ( X ) F ( X ) pct ( X ) . SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUS TOPOLOGICAL SPACES 13
The invariant aL c ( X ) , the almost Lindel¨of degree with respect to closed sets , isthe smallest infinite cardinal κ such that for every closed subset C of X and everycollection U of open sets in X that cover C , there is a subcollection V of U suchthat | V | ≤ κ and { U : U ∈ V } covers C . It is clear that aL c ( X ) ≤ L ( X ) . Theorem 6.3 (C., Porter, Ridderbos [20], 2012) . If X is a power homogeneousHausdorff space then | X | ≤ aL c ( X ) t ( X ) pct ( X ) . Recently in [13], the Lindel¨of degree L ( X ) in Theorem 6.1 was replaced bythe cardinal invariant pwL c ( X ) , introduced by Bella and Spadaro in [16]. The piecewise weak Lindel¨of degree for closed sets pwL c ( X ) of X is the least infinitecardinal κ such that for every closed set F ⊆ X , for every open cover U of F , andevery decomposition { U i : i ∈ I } of U , there are families V i ∈ [ U i ] ≤ κ for every i ∈ I such that F ⊆ S { S V i : i ∈ I } . It is clear that pwL c ( X ) ≤ L ( X ) and,importantly, it can be shown that pwL c ( X ) ≤ c ( X ) . Theorem 6.4 (Bella, C. [13], 2020) . If X is homogeneous and Hausdorff then | X | ≤ pwL c ( X ) t ( X ) pct ( X ) . While it is clear that Theorem 6.4 is an improvement of Theorem 6.1, it is alsoan improvement of Theorem 6.3 as it can be shown that pwL c ( X ) ≤ aL c ( X ) .Furthermore, it is a variation of Theorem 4.1; that is, c ( X ) πχ ( X ) is a bound for thecardinality of every homogeneous Hausdorff space. This is because pwL c ( X ) ≤ c ( X ) and πχ ( X ) ≤ t ( X ) pct ( X ) for Hausdorff spaces.In [13], a consistent improvement of Theorem 6.1 was given using the linearlyLindel¨of degree lL ( X ) . A space X is linearly Lindel¨of provided that every in-creasing open cover of X has a countable subcover. More generally, we definethe linear Lindel¨of degree lL ( X ) of X as the smallest cardinal κ such that everyincreasing open cover of X has a subcover of size not exceeding κ . Equivalently, lL ( X ) ≤ κ if every open cover of X has a subcover U such that | U | has cofinalityat most κ . Theorem 6.5 (Bella, C. [13], 2020) . Assume κ < κ + ω or < κ = 2 κ . If X isHausdorff, homogeneous, and κ = lL ( X ) F ( X ) pct ( X ) , then | X | ≤ κ . Our last improvement of Theorem 6.1 gives a bound for the cardinality of anopen set in a power homogeneous space.
Theorem 6.6 (Bella, C. [11], 2018) . If X is a power homogeneous Hausdorffspace and U ⊆ X is an open set, then | U | ≤ L ( U ) t ( X ) pct ( X ) The next four results from [45], [11], and [14] represent extensions of de laVega’s Theorem in a different direction using the invariants wL ( X ) or wL c ( X ) .The weak Lindel¨of degree of a space X is the least infinite cardinal κ such thatevery open cover U of X has a subfamily V such that | V | ≤ κ and X = S V .The invariant wL c ( X ) , the weak Lindel¨of degree with respect to closed sets , is thesmallest infinite cardinal κ such that for every closed subset C of X and everycollection U of open sets in X that cover C , there is a subcollection V of U suchthat | V | ≤ κ and C ⊆ S V . It is clear that wL ( X ) ≤ wL c ( X ) ≤ aL c ( X ) . Theorem 6.7 (Spadaro, Szeptycki [45], 2018) . If X is an initially κ -compactpower homogeneous T space then | X | ≤ F ( X ) wL c ( X ) . Theorem 6.8 (Bella, C. [11], 2018) . If X is a regular power homogeneous spaceand with a π -base B such that B is Lindel¨of for all B ∈ B , then | X | ≤ wL ( X ) t ( X ) pct ( X ) . As locally compact spaces satisfy the hypotheses in Theorem 6.8, we have thefollowing corollary.
Corollary 6.9 (Bella, C. [11], 2018) . If X is a locally compact power homoge-neous space then | X | ≤ wL ( X ) t ( X ) . The above theorem indicates that the compactness condition in de la Vega’s The-orem can be replaced with another pair of conditions: locally compact and weaklyLindel¨of. It turns out that Corollary 6.9 can be given an improved conclusion. Thiswas demonstrated in [14].
Theorem 6.10 (Bella, C., Gotchev [14], 2020) . If X is a locally compact powerhomogeneous space then | X | ≤ wL ( X ) t ( X ) .
7. O
THER RESULTS
Recently it was shown in [14] that if X is an extremally disconnected spacethen c ( X ) ≤ w ( X ) πχ ( X ) . Using Theorem 4.5, the following is an immediateconsequence. Theorem 7.1 (Bella, C., Gotchev [14], 2020) . If X is power homogeneous andextremally disconnected then | X | ≤ wL ( X ) πχ ( X ) . One should regard the bound in Theorem 7.1 as being “small” as wL ( X ) and πχ ( X ) are generally thought of as small cardinal invariants. Observe that it followsfrom Theorem 7.1 that an H-closed, extremally disconnected, power homogeneousspace has cardinality at most πχ ( X ) . However, it was shown in [17] that an infiniteH-closed extremally disconnected space cannot be power homogeneous. This latterresult is an extension of a result of Kunen [36] that an infinite compact F-space isnot power homogeneous.Given a space X , the diagonal of X , denoted by ∆ X , is the set { ( x, x ) : x ∈ X } . X is said to have a regular G δ -diagonal if there exists a countable family U of open sets in X such that ∆ X = T U = T { U : U ∈ U } . A cardinality boundfor homogeneous spaces with a regular G δ -diagonal was given in [9]. Theorem 7.2 (D. Basile, Bella, Ridderbos [9], 2014) . If X is a homogeneous spacewith a regular G δ -diagonal, then | X | ≤ wL ( X ) πχ ( X ) . A notion related to homogeneity is known as countable dense homogeneity. Aseparable space X is countable dense homogeneous (CDH) if given any two count-able dense subsets D and E of X , there is a homeomorphism h : X → X suchthat h [ D ] = E . Separability is included in the definition as clearly this notionis of interest only if X has a countable dense subset. Not every CDH space ishomogeneous, however every connected CDH space is homogeneous [28].In [6] it was shown that the cardinality of a CDH space is as most c . SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUS TOPOLOGICAL SPACES 15
Theorem 7.3 (Arhangel ′ ski˘ı, van Mill [6], 2014) . The cardinality of a CDH spaceis at most c .
8. Q
UESTIONS AND A T ABLE OF B OUNDS
Recall that in Theorem 2.4 it was shown that | H ( X ) | ≤ c ( X ) πχ ( X ) sd ( X ) fora Hausdorff space X . In addition, in Theorem 4.1 it was shown that | X | ≤ c ( X ) πχ ( X ) if X is Hausdorff and homogeneous. In light of these theorems, thefollowing was asked by the author and Ridderbos in [18]. Question 8.1 (C., Ridderbos, 2008) . Is | H ( X ) | ≤ c ( X ) πχ ( X ) for a Hausdorffspace X ? As it was shown in [12] that the cardinality of a homogeneous compactum is atmost wt ( X ) πχ ( X ) (Theorem 5.13), it is natural to ask if either of the two cardinalinvariants wt ( X ) and πχ ( X ) can be removed from this bound. The next twoquestions were asked in [12]. The second was additionally asked by de la Vegain [23]. These two questions appear to be quite challenging. Answering either inthe affirmative would likely require new breakthrough techniques, while counter-examples would likely be complicated and have intriguing properties. Question 8.2 (Bella, C. [12], 2020) . Is the cardinality of a homogeneous com-pactum at most wt ( X ) ? Question 8.3 (de la Vega [23], 2005, Bella, C. [12], 2020) . Is the cardinality of ahomogeneous compactum at most πχ ( X ) ? The power homogeneous case of Theorem 5.13 is also an open question.
Question 8.4 (Bella, C. [12], 2020) . Is the cardinality of a power homogeneouscompactum at most wt ( X ) πχ ( X ) ? By results of ˇSapirovski˘ı, every T compactum has a point of countable π -character. It follows from Theorem 4.1 that the cardinality of a homogeneous T compactum is at most c ( X ) . This was observed by van Mill in [38] and was provedfor power homogeneous T compacta in [43]. Note additionally that Corollary 5.15states that the cardinality of a homogeneous T compactum is at most wt ( X ) . VanMill asked if the cardinality of such spaces is in fact at most c . Question 8.5 (van Mill [38], 2005) . Is the cardinality of every T homogeneouscompactum at most c ? In light of the various cardinality bounds for homogeneous-like spaces using theweak Lindel¨of degree wL ( X ) , the following was asked in [ ? ]. Question 8.6. If X is power homogeneous and Tychonoff, is | X | ≤ wL ( X ) t ( X ) pct ( X ) ? T ABLE
1. Strongest known cardinality bounds on spaces withhomogeneous-like properties.
Bound on | X | Hypotheses on X Proved in Year Thm | RO ( X ) | qψ ( X ) homog., Hausdorff Ismail [32] 1981 3.1 d ( X ) πnχ ( X ) homog., Hausdorff (current paper) 2020 3.4 d ( X ) πχ ( X ) power homog., Hausdorff Ridderbos [42] 2006 3.5 d θ ( X ) πχ ( X ) power homog., Urysohn C. [17] 2007 3.7 πχ ( X ) c ( X ) qψ ( X ) homog., T (current paper) 2020 4.2 c ( X ) πχ ( X ) power homog., Hausdorff C., Ridderbos [18] 2008 4.5 Uc ( X ) πχ ( X ) power homog., Bonanzinga, C., 2018 4.6(Urysohn or quasiregular) Cuzzup´e, Stavrova [15] c homog. compactum that Juh´asz, van Mill [34] 2018 5.7is the union of finitely manycountably tight subspaces c power homog. compactum C. [22] 2018 5.9that is the union ofcountably many densecountably tight subspaces wt ( X ) homog., compact, T (current paper) 2020 5.15 wt ( X ) πχ ( X ) homog., compactum Bella, C. [12] 2020 5.13 L ( X ) F ( X ) pct ( X ) power homog., Hausdorff C., Porter, 2012 6.2Ridderbos [20] pwL c ( X ) t ( X ) pct ( X ) homog., Hausdorff Bella, C., [13] 2020 6.4 lL ( X ) F ( X ) pct ( X ) homog., Hausdorff Bella, C., [13] 2020 6.5( κ < κ + ω , or < κ = 2 κ κ = lL ( X ) F ( X ) pct ( X ))2 F ( X ) wL c ( X ) power homog., T , Spadaro, 2018 6.7initially κ -compact Szeptycki [45] wL ( X ) t ( X ) pct ( X ) power homog., T , Bella, C., [11] 2018 6.8 π -base B such that B is Lindel¨of for all B ∈ B wL ( X ) t ( X ) power homog., loc. compact Bella, C., Gotchev [14] 2020 6.10 wL ( X ) πχ ( X ) power homog., Bella, C., Gotchev [14] 2020 7.1extremally disconnected wL ( X ) πχ ( X ) homog, D. Basile, Bella, 2014 7.2regular G δ diagonal Ridderbos [9] c countable dense homog. Arhangel ′ ski˘ı, 2014 7.3separable van Mill [6] SURVEY OF CARDINALITY BOUNDS ON HOMOGENEOUS TOPOLOGICAL SPACES 17 R EFERENCES [1] A. V. Arhangel ′ ski˘ı, On the cardinality of bicompacta satisfying the first axiom of countability ,Soviet Math. Dokl. (1969), no. 4, 951–955.[2] A. V. Arhangel ′ ski˘ı, A survey of some recent advances in general topology, old and new prob-lems , in: Proc. International Congress of Mathematicians, Nice, 1970, Tome 2, pp. 19–26.[3] A. V. Arhangel ′ ski˘ı, The Suslin number and cardinality. Character of points in sequential bi-compacta , Soviet Math. Dokl. 11 (1970) 597–601.[4] A. V. Arhangel ′ ski˘ı, The structure and classification of topological spaces and cardinal invari-ants , Uspekhi Mat. Nauk (1978), no. 6 (204), 29–84, 272.[5] A. V. Arhangel ′ ski˘ı, Topological homogeneity. Topological groups and their continuous images
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