aa r X i v : . [ m a t h . GN ] J un Counting overweight spaces
Gerald Kuba
1. Introduction
Write | M | for the cardinal number (the size ) of a set M and define c := | R | = 2 ℵ . Weuse κ, λ, µ throughout to stand for infinite cardinal numbers. As usual, w ( X ) denotesthe weight of a topological space X . Naturally, w ( X ) ≤ | X | and | X | ≤ w ( X ) for everyinfinite T -space X . It is trivial that w ( X ) ≤ | X | for every infinite, first countable space X and well-known (see [2, 3.3.6]) that w ( X ) ≤ | X | for every compact Hausdorff space X .Furthermore, w ( X ) ≥ | X | for every infinite, scattered T -space X (see Lemma 1 below).According to the title, we are concerned with topological spaces X satisfying the strictinequality w ( X ) > | X | . While the extreme case w ( X ) = 2 | X | is of natural interest, toinvestigate the case | X | < w ( X ) < | X | is reasonable in view of the following remarkablefact.(1.1) It is consistent with
ZFC set theory that µ < λ implies µ < λ and that for everyregular κ there exist precisely κ cardinals λ with κ < λ < κ . (A short explanation why (1.1) is true is given in Section 2.) For fundamental enumerationtheorems about spaces X with w ( X ) ≤ | X | see [3], [5], [6], [7], [8], [9], [11]. However, itwould be artificial to avoid an overlap with these enumeration theorems and hence in thefollowing we include the case w ( X ) = | X | . The benefit of this inclusion is that we willalso establish several new enumeration theorems about spaces X with w ( X ) = | X | . Ashort proof of the following basic estimate is given in the next section.(1.2) If θ is an infinite cardinal and F is a family of mutually non-homeomorphic infinite T -spaces such that max {| X | , w ( X ) } ≤ θ for every X ∈ F then |F | ≤ θ . For abbreviation let us call a Hausdorff space X almost discrete if and only if X \ { x } isa discrete subspace of X for some x ∈ X . Recall that a space is perfectly normal when itis normal and every closed set is a G δ -set. Note that every subspace of a perfectly normalspace is perfectly normal. Recall that a normal space is strongly zero-dimensional if andonly if for every closed set A and every open set U ⊃ A there is an open-closed set V with A ⊂ V ⊂ U . Our first goal is to prove the following enumeration theorem. Theorem 1. If κ ≤ λ ≤ κ then there exist λ mutually non-homeomorphic scattered,strongly zero-dimensional, hereditarily paracompact, perfectly normal spaces X with | X | = κ and w ( X ) = λ . In case that λ ≤ µ < λ for some µ it can be accomplished thatall these spaces are also almost discrete. Moreover, it can be accomplished that all thesespaces are almost discrete and extremally disconnected in case that λ = 2 µ for some µ (which includes the case λ = 2 κ ). Since every scattered Hausdorff space is totally disconnected, the following theorem is anoteworthy counterpart of Theorem 1. For abbreviation, let us call a space X almostmetrizable if and only if X is perfectly normal and X \ { x } is metrizable for some x ∈ X .In view of Lemma 3 in Section 3, almost metrizable space are hereditarily paracompact.1 heorem 2. If c ≤ κ ≤ λ ≤ κ then there exist λ mutually non-homeomorphicpathwise connected, locally pathwise connected, almost metrizable spaces of size κ andweight λ . The restriction c ≤ κ in Theorem 2 is inevitable because if X is an infinite, pathwise connected Hausdorff space then X is arcwise connected (see [2, 6.3.12.a]) and hence c = | [0 , | ≤ | X | . However, for infinite, connected Hausdorff spaces X the restriction c ≤ | X | is not justified and we can prove the following theorem. Note that, by applying (1.1) for κ = ℵ , the existence of c infinite cardinals κ < c is consistent with ZFC. Theorem 3. If κ < c and κ ≤ λ ≤ κ then there exist λ mutually non-homeomorphicconnected and locally connected Hausdorff spaces of size κ and weight λ . In particular, upto homeomorphism there exist precisely c countably infinite, connected, locally connectedHausdorff spaces and precisely c countably infinite, connected, locally connected, secondcountable Hausdorff spaces. No space provided by Theorem 3 is completely regular because, naturally, every completelyregular space of size smaller than c and greater than 1 is totally disconnected. Moreover,every countably infinite, regular space is totally disconnected (see [2, 6.2.8]). The connected spaces provided by Theorem 3 are totally pathwise disconnected since they are Hausdorffspaces of size smaller than c . Therefore the following counterpart of Theorem 2 is worthmentioning. Theorem 4. If c ≤ κ ≤ λ ≤ κ then there exist λ mutually non-homeomorphic con-nected, totally pathwise disconnected, nowhere locally connected, almost metrizable spacesof size κ and weight λ .
2. Some explanations and preparations
Referring to Jech’s profound text book [4], a proof of (1.1) can be carried out as follows.Define in G¨odel’s universe L for every regular cardinal κ a cardinal number θ ( κ ) by θ ( κ ) := min { µ | µ = ℵ µ ∧ cf µ = κ + } . Then |{ λ | κ < λ < θ ( κ ) }| = θ ( κ ) holds inevery generic extension of L. By applying Easton’s theorem [4, 15.18] one can create anEaston universe E generically extending L such that the continuum function κ κ = κ + in L is changed into κ κ = g ( κ ) in E with g ( κ ) = θ ( κ ) for every regular cardinal κ .So in E we have |{ λ | κ < λ < κ }| = 2 κ for every regular κ . By definition, in E wehave 2 α < β whenever α, β are regular cardinals with α < β . Therefore and in viewof [4] Theorem 5.22 and [4] Exercise 15.12, if µ is singular in E then 2 µ is a successorcardinal in E while 2 κ is a limit cardinal in E for every regular κ in E. Consequently, inE we have 2 µ < λ whenever µ, λ are arbitrary cardinals with µ < λ .In order to verify (1.2), first of all it is clear that a topological space ( X, τ ) has a basis ofsize λ ≤ | τ | if and only if w ( X ) ≤ λ . Let S be an infinite set of size ν and let P be thepower set of S , whence | P | = 2 ν . Let µ ( ν, λ ) denote the total number of all topologies τ on S such that ( S, τ ) has a basis B of size λ . Clearly, µ ( ν, λ ) = 0 if λ > ν . For λ ≤ ν we have µ ( ν, λ ) ≤ | P | λ = max { ν , λ } . So if θ and F satisfy the assumption in(1.2) then |F | is not greater than the sum Σ of all cardinals µ ( ν, λ ) with ( ν, λ ) runningthrough the set Q := { κ | κ ≤ θ } . Thus from µ ( ν, λ ) ≤ θ for all ( ν, λ ) ∈ Q we deriveΣ ≤ θ and this concludes the proof of (1.2).2n the following we write down a short proof of an important fact mentioned in the previoussection. Lemma 1. If X is an infinite scattered T -space then w ( X ) ≥ | X | .Proof. Since X is infinite and T , no basis of X is finite. Assume that λ := w ( X ) < | X | and let B be a basis of X with |B| = λ . Let X ∗ denote the set of all x ∈ X such that | U | > λ for every neighborhood U of x . Then X \ X ∗ ⊂ S { U ∈ B | | U | ≤ λ } andhence | X \ X ∗ | ≤ λ . Consequently, X ∗ = ∅ and if x ∈ X ∗ and U is a neighborhood of x then | X ∗ ∩ U | > λ (since | U | > λ ). Therefore, the nonempty set X ∗ is dense in itselfand hence the space X is not scattered, q.e.d. In order to settle the case 2 κ = 2 λ in Theorems 1 and 2 and 4 we will apply the followingtwo enumeration theorems about metrizable spaces. Note that, other than in the modelE which proves (1.1), for κ < λ ≤ κ we can rule out 2 κ = 2 λ only in case that λ = 2 κ .(Thus the following two propositions can be ignored if Theorems 1 and 2 and 4 are onlyread as enumeration theorems about spaces X of maximal possible weights | X | .)Let X + Y denote the topological sum of two Hausdorff spaces X and Y . (So X + Y is a space S such that S = ˜ X ∪ ˜ Y for disjoint open subspaces ˜ X , ˜ Y of S where ˜ X ishomeomorphic to X and ˜ Y is homeomorphic to Y .) If Y = ∅ then we put X + Y = X . Proposition 1.
For every κ there is a family H κ of mutually non-homeomorphicscattered, strongly zero-dimensional metrizable spaces of size κ such that |H κ | = 2 κ andif D is any discrete space (including the case D = ∅ ) then the spaces H + D and H + D are never homeomorphic for distinct H , H ∈ H κ . By Lemma 1 and since w ( Y ) ≤ | Y | for every metrizable space Y , we have w ( X ) = | X | for every X ∈ H κ . Proposition 1 can be verified by considering the spaces constructedin [9] which prove [9] Theorem 1. Because these spaces X are revealed as mutually non-homeomorphic ones by investigating the α th Cantor derivative X ( α ) for every ordinal α > X is any space and D is discrete then ( X + D ) ( α ) = X ( α ) forevery α > Proposition 2.
For every κ ≥ c there is a family P κ of mutually non-homeomorphicpathwise connected, locally pathwise connected, complete metric spaces of size and weight κ such that |P κ | = 2 κ and if H ∈ P κ then H contains a noncut point and the cut pointsof H lie dense in H .
3. Almost discrete and almost metrizable spaces
In accordance with [13], a space is completely normal when every subspace is normal. (In[2] such spaces are called hereditarily normal .) Lemma 2. If X is a Hausdorff space and z ∈ X such that X \ { z } is a discrete subspaceof X then X is scattered and completely normal and strongly zero-dimensional.Proof. Put Y := X \ { z } . Since Y is a discrete and open subspace of X , everynonempty subset of X contains an isolated point, whence X is scattered. Let A, B ⊂ X with A ∩ B = A ∩ B = ∅ . If z A ∪ B then A, B ⊂ Y and hence A ⊂ U and B ⊂ V with the two disjoint open sets U = A and V = B . Assume z ∈ A ∪ B and,say, z ∈ A . Then z B and hence B ⊂ Y . Thus A ⊂ ˜ U and B ⊂ ˜ V with the two3isjoint open sets ˜ U = X \ B and ˜ V = B . So X is completely normal. Finally, let A ⊂ X be closed. If z A then A is open. If z ∈ A and U is an open neighborhoodof A then U is closed since X \ U ⊂ Y . So every closed subset of X has a basis ofopen-closed neighborhoods and hence X is strongly zero-dimensional, q.e.d. Lemma 3. If Z is a regular space such that Z \ { z } is paracompact for some z ∈ Z then Z is paracompact.Proof. Let U be an open cover of Z . Trivially, U ∗ := { U \ { z } | U ∈ U } is an opencover of the paracompact open subspace P = Z \ { z } of Z . Hence we can find an opencover V ∗ of P which is a locally finite refinement of U ∗ . Fix one set U z ∈ U with z ∈ U z and choose a closed neighborhood C of z in the regular space Z such that C ⊂ U z .Now put V := { V ∗ \ C | V ∗ ∈ V ∗ } ∪ { U z } . Clearly, V is an open cover of Z which isa refinement of U . If z = x ∈ Z then some neighborhood of x meets only finitely manymembers of V ∗ and hence only finitely many members of V . And C is a neighborhoodof z which meets V ∈ V if and only if V = U z . Therefore, the cover V is locally finitein Z and hence Z is paracompact, q.e.d. Since metrizability implies paracompactness and since the union of two G δ -sets is a G δ -set,from Lemma 2 and Lemma 3 we derive the following two corollaries. Corollary 1.
Let X be a Hausdorff space and z ∈ X such that X \ { z } is a discretesubspace of X and { z } is a G δ -set in X . Then the almost discrete space X is hereditarilyparacompact and perfectly normal. Corollary 2.
Let X be a regular space and z ∈ X such that the subspace X \ { z } ismetrizable and { z } is a G δ -set in X . Then X is hereditarily paracompact and perfectlynormal and hence almost metrizable.
4. The single filter topology
Let
X, z be as in Lemma 2 and consider the family U of all open neighborhoods of thepoint z . Since { x } is open in X whenever z = x ∈ X , the family U coincides with theneighborhood filter at z in the space X . Consequently, U ∗ := { U \ { z } | U ∈ U } isthe power set of X \ { z } if z is isolated in X or, equivalently, if X is discrete. And U ∗ is a filter on the set X \ { z } if z is a limit point of X or, equivalently, if the discretesubspace X \ { z } is dense in X . Since X is Hausdorff, it is plain that T U ∗ = ∅ .Conversely, let Y be an infinite set and z Y and let F be a filter on the set Y . Definea topology τ [ F ] on the set X := Y ∪ { z } by declaring U ⊂ X open if and only if either z U or U = { z } ∪ F for some F ∈ F . It is plain that this is a correct definitionof a topology on the set X . Furthermore, Y is a discrete and open and dense subspaceof ( X, τ [ F ]) , whence { z } is closed in X . It is plain that ( X, τ [ F ]) is a Hausdorff spaceif and only if the filter F is free, i.e. T F = ∅ . So by Lemma 2 the almost discretespace ( X, τ [ F ]) is hereditarily paracompact and scattered and strongly zero-dimensionalfor every free filter F on Y .For abbreviation throughout the paper let us call a filter F ω -free if and only if T A = ∅ for some countable A ⊂ F . In view of Corollary 1 the following statement is evident.(4.1) If F is a filter on Y then ( X, τ [ F ]) is almost discrete and perfectly normal if andonly if F is ω -free. If F is a free filter on Y then the almost discrete space ( X, τ [ F ]) is extremallydisconnected if and only if F is an ultrafilter.Proof. Firstly let F be a free ultrafilter. Let U ⊂ X be open. If U = U then U isopen. So assume U = U . Then U = U ∪ { z } and z U since z is the only limit pointin X . Thus U ⊂ Y and z is a limit point of U . Hence every open neighborhood of z meets U . In other words, F ∩ U = ∅ for every F ∈ F . Consequently, U ∈ F since F is an ultrafilter. Thus U = U ∪ { z } is open in X , whence ( X, τ [ F ]) is extremallydisconnected. Secondly, let F be a free filter and assume that ( X, τ [ F ]) is extremallydisconnected. Let A ⊂ Y , whence A is open in X . If A = A then X \ A is open andhence Y \ A lies in F . If A = A then A = { z } ∪ A is open and hence A lies in F .This reveals F as an ultrafilter, q.e.d.Remark. If | Y | = ℵ and F is a free ultrafilter on Y then τ [ F ] is the well-known singleultrafilter topology (see Example 114 in [13].)For a filter F on Y let χ ( F ) denote the least possible size of a filter base which generates F . Trivially, χ ( F ) ≤ |F | ≤ | Y | . The notation χ ( · ) corresponds with the obvious factthat χ ( F ) is the character of z in ( X, τ [ F ]) . (The character χ ( a, A ) of a point a ina space A is the smallest possible size of a local basis at a in the space A .) Therefore,since { y } is open in ( X, τ [ F ]) for every y ∈ Y , we obtain:(4.3) If F is a free filter on Y then the weight of ( X, τ [ F ]) is max {| Y | , χ ( F ) } . Proposition 3. If | Y | = κ ≤ λ ≤ κ then there exist λ ω -free filters F on Y suchthat χ ( F ) = λ .Remark. The cardinal 2 λ in Proposition 3 is best possible. Indeed, let Y be an infiniteset of size κ and let λ ≥ κ . Since a filter base on Y is a subset of the power set of Y ,there are at most 2 λ filter bases B on Y with |B| = λ . Hence Y cannot carry morethan 2 λ filters F with χ ( F ) = λ . Proof of Proposition 3.
Assume | Y | = κ ≤ λ ≤ κ and let A be a family of subsets of Y such that |A| = 2 κ and(4.4) If D , E 6 = ∅ are disjoint finite subfamilies of A then T D 6⊂ S E . A construction of such a family A is elementary, see [4, 7.7]. However, this is not enoughfor our purpose. In view of the property ω -free , we additionally have to make sure that thefamily A also contains a countably infinite family A ω such that T A ω = ∅ . By applyingLemma 8 in Section 11 for µ = ℵ we can assume that such a family A ω ⊂ A exists.Now put A λ := { H | A ω ⊂ H ⊂ A ∧ |H| = λ } . Clearly, | A λ | = (2 κ ) λ = 2 λ . By virtue of (4.4), if for H ∈ A λ we put B H := { H ∩ · · · ∩ H n | n ∈ N ∧ H , ..., H n ∈ H } then ∅ 6∈ B H and hence B H is a filter base on Y . For every H ∈ A λ let F [ H ] denotethe filter on Y generated by B H . Clearly, |B H | = |H| = λ for every H ∈ A λ .5he filter F [ H ] is ω -free because A ω ⊂ F [ H ] by definition. Furthermore, (4.4) impliesthat for distinct families H , H ∈ A λ the filters F [ H ] and F [ H ] must be distinct. Sothe family { F [ H ] | H ∈ A λ } consists of 2 λ ω -free filters on Y .It remains to verify that χ ( F [ H ]) = λ for every H ∈ A λ . Assume indirectly that for some H ∈ A λ we have χ ( F [ H ]) = λ and hence χ ( F [ H ]) < λ . (Clearly χ ( F [ H ]) ≤ λ since |B H | = |H| = λ .) Choose a filter base B on Y which generates the filter F [ H ] such that |B| < λ . Since B ⊂ F [ H ] and F [ H ] is generated by the filter base B H , we can choosefor every B ∈ B a finite set H B ⊂ H such that B ⊃ T H B . Put U := S B ∈B H B .Then U ⊂ H and |U | ≤ |B| < λ . Consequently,
H \ U 6 = ∅ . Choose any set A ∈ H \ U .Then A ∈ F [ H ] and hence we can find a set B ∈ B with A ⊃ B . Then A ⊃ T H B and hence A ∈ H B by virtue of (4.4). But then A ∈ U in contradiction with choosing A in H \ U , q.e.d. Proposition 3 can be improved in the important case λ = 2 κ as follows. Proposition 4.
On an infinite set of size κ there exist precisely κ ω -free ultrafilters F such that χ ( F ) = 2 κ .Proof. Let Y be a set of size κ . As in the previous proof let A be a family of subsets of Y such that |A| = 2 κ and (4.4) holds. (Here we need not consider A ω ⊂ A .)Let A denote the family of all subfamilies G of A such that |G| = 2 κ . Clearly, | A | = 2 κ . Now for every G ∈ A define W [ G ] := G ∪ { Y \ T H | H ⊂ G ∧ |H| ≥ ℵ } ∪ { Y \ A | A ∈ A \ G } . A moment’s reflection suffices to see that (4.4) implies that W ∩ · · · ∩ W n = ∅ whenever W , ..., W n ∈ W [ G ] . Hence for every G ∈ A we can choose an ultrafilter U [ G ] on Y suchthat U [ G ] ⊃ W [ G ] (see [1] 7.1).If G , G ∈ A are distinct and, say, G ∈ G \ G then G ∈ W [ G ] and Y \ G ∈ W [ G ]and hence G ∈ U [ G ] and G
6∈ U [ G ] and hence the ultrafilters U [ G ] and U [ G ] aredistinct as well. Consequently, the family { U [ G ] | G ∈ A } consists of 2 κ ultrafilters on Y . All these ultrafilters are ω -free because if G ∈ A and H is a countably infinite subsetof G then by virtue of (4.4) the family H ∗ := { H \ T H | H ∈ H } is countably infiniteand it is trivial that T H ∗ = ∅ and from H ⊂ W [ G ] and Y \ T H ∈ W [ G ] we derive H ∗ ⊂ U [ G ] . (Actually, by a deep argument from set theory it is superfluous to verify that U [ G ] is ω -free, see the remark below.)Finishing the proof, we claim that χ ( U [ G ]) = 2 κ for every G ∈ A . Assume indirectlythat for G ∈ A the ultrafilter U [ G ] is generated by a filter base B with |B| < κ . Since G ⊂ U [ G ] , for every G ∈ G we have G ⊃ B for some B ∈ B . From |B| < |G| wederive the existence of a set B ∈ B and an infinite subset H ⊂ G such that H ⊃ B forevery H ∈ H . Consequently, T H ⊃ B and hence T H ∈ U [ G ] . This, however, is acontradiction since Y \ T H lies in U [ G ] by the definition of W [ G ] , q.e.d.Remark. Our proof of Proposition 4 is elementary and purely set-theoretical. There is alsoa topological but much less elementary way to prove Proposition 4. First of all, if one canprove that any set of size κ carries 2 κ ultrafilters of character 2 κ then Proposition 4must be true. Because, an ultrafilter F is free if and only if χ ( F ) > F is not ω -free then it is plain that F is σ -complete. However, the existenceof a σ -complete free ultrafilter is unprovable in ZFC! (See [4, 10.2] and [4, 10.4].) Now,6onsider the set Y of size κ equipped with the discrete topology and consider the Stone-ˇCech compactification βY of Y and its compact remainder Y ∗ = βY \ Y . So the pointsin Y ∗ are the free ultrafilters on Y and if for p ∈ Y ∗ we consider the subspace Y ∪ { p } of βY then it is clear that the character of the ultrafilter p equals χ ( p, Y ∪ { p } ) . It is anice exercise to verify that χ ( p, Y ∪ { p } ) = χ ( p, Y ∗ ) for every p ∈ Y ∗ . By embeddingan appropriate Stone space of a Boolean algebra into Y ∗ it can be proved that Y ∗ mustcontain 2 κ points p with χ ( p, Y ∗ ) = 2 κ , see 7.13, 7.14, 7.15 in [1].
5. Proof of Theorem 1
Assume µ ≤ κ ≤ λ ≤ µ and let Y be a set of size µ . Let F λ denote a family of ω -freefilters on Y such that | F λ | = 2 λ and χ ( F ) = λ for every F ∈ F λ . Such a family existsby Proposition 3. We additionally assume that if λ = 2 µ then every member of F λ is anultrafilter. This additional assumption is justified by Proposition 4.Now fix z Y and for every F ∈ F λ consider the single filter topology τ [ F ] on the set X = Y ∪ { z } as in Section 4. If µ < κ then let D be a discrete space of size κ . If µ = κ then put D = ∅ . In both cases define the space ( ˜ X, ˜ τ [ F ]) as the topological sumof D and the space ( X, τ [ F ]) . (So if µ = κ then ˜ X = X and ˜ τ [ F ] = τ [ F ] .) Clearly,˜ X is almost discrete, scattered, strongly zero-dimensional, hereditarily paracompact, andperfectly normal. Furthermore, w ( ˜ X ) = λ and | ˜ X | = κ . If λ = 2 µ then the space ˜ X isalso extremally disconnected by virtue of (4.2).Obviously, ˜ τ [ F ] = ˜ τ [ F ] whenever the filters F , F ∈ F λ are distinct. (For if F , F ∈ F λ and F ∈ F \ F then F ∪ { z } is ˜ τ [ F ]-open but not ˜ τ [ F ]-open.) Consequently, thefamily T λ := { ˜ τ [ F ] | F ∈ F λ } is of size 2 λ .We distinguish the two cases 2 λ > µ and 2 λ ≤ µ . Assume firstly that 2 λ > µ or,equivalently, that |T λ | > µ . Define an equivalence relation ∼ on T λ by τ ∼ τ if andonly if the spaces ( ˜ X, τ ) and ( ˜ X, τ ) are homeomorphic. We claim that the size of anequivalence class cannot be greater than 2 µ .This is clearly true if µ = κ because there are only 2 µ permutations on X . So assume µ < κ . If τ ∈ T λ then in the space ( ˜ X , τ ) the point z is the only limit point andevery neighborhood U of z is open-closed. As a consequence, for τ , τ ∈ T λ the spaces( ˜ X, τ ) and ( ˜ X, τ ) are homeomorphic if and only if there is a homeomorphism ϕ fromthe τ -subspace X of ˜ X onto some τ -open-closed subspace of ˜ X . Indeed, if f is ahomeomorphism from ( ˜ X, τ ) onto ( ˜ X, τ ) then put ϕ ( x ) = f ( x ) for every x ∈ X and ϕ fits since f ( z ) = z . Conversely, if ϕ is a homeomorphism from the τ -subspace X of˜ X onto some τ -open-closed subspace of ˜ X and g is any bijection from ˜ X \ X onto˜ X \ ϕ ( X ) then it is plain that a homeomorphism f from ( ˜ X, τ ) onto ( ˜ X, τ ) is definedby f ( x ) = ϕ ( x ) for x ∈ X and f ( x ) = g ( x ) for x X . (Note that | ˜ X \ X | = | ˜ X \ ϕ ( X ) | since µ < κ .) Therefore, since there are precisely κ µ mappings from X into ˜ X , the sizeof an eqivalence class in T λ cannot exceed κ µ . And from 2 < µ ≤ κ ≤ µ we derive2 µ ≤ µ µ ≤ κ µ ≤ (2 µ ) µ = 2 µ and hence κ µ = 2 µ .So the size of an equivalence class can indeed not be greater than 2 µ . Consequently, |T λ | > µ implies that the total number of all equivalence classes equals |T λ | = 2 λ . Thusby choosing one topology in each equivalence class we obtain 2 λ mutually non-equivalenttopologies τ ∈ T λ and hence the 2 λ corresponding spaces ( ˜ X, τ ) are mutually non-homeomorphic. This settles the case 2 λ > µ . In particular, we have already proved the7econd and the third statement in Theorem 1 because, under the assumption κ ≤ λ ≤ κ ,if λ = 2 µ for some µ then λ = 2 µ (and hence 2 λ > µ ) for some µ ≤ κ and if λ ≤ µ < λ and µ > κ then 2 κ ≤ µ < λ and hence 2 λ > µ ′ for µ ′ = κ .Secondly assume that 2 λ ≤ µ . Then we have 2 λ = 2 κ since µ ≤ κ ≤ λ implies2 µ ≤ κ ≤ λ . So in order to conclude the proof of Theorem 1 we assume κ ≤ λ ≤ κ =2 λ . (Then, of course, κ ≤ λ < κ = 2 λ .) Since the special case κ = λ is settled byProposition 1, we also assume κ < λ . For two spaces X and X let, again, X + X denote the topological sum of X and X . Let H κ be a family provided by Proposition 1.Due to metrizability, every space in H κ is perfectly normal and hereditarily paracompact.By considering an appropriate single filter topology on a set of size κ , we can choosea perfectly normal space Z of size κ such that for some point z ∈ Z the subspace Z \ { z } is discrete and χ ( z, Z ) = λ . (Consequently, w ( Z ) = λ .) For every space H ∈ H κ consider the topological sum H + Z . Of course, the topological sum of twoparacompact spaces is paracompact and ( H + Z ) \ { z } = H + ( Z \ { z } ) for every H ∈ H κ . Consequently, for every H ∈ H κ the space H + Z is scattered and strongly zero-dimensional and perfectly normal and hereditarily paracompact and | H + Z | = | H | = κ and w ( H + Z ) = max { w ( H ) , w ( Z ) } = max { κ, λ } = λ . Therefore, since |H κ | = 2 κ ,the case 2 λ = 2 κ in Theorem 1 is settled by showing that for two distinct (and hencenon-homeomorphic) metrizable spaces H , H ∈ H κ the two spaces H + Z and H + Z are never homeomorphic. Assume that H , H ∈ H κ and that f is a homeomorphismfrom H + Z onto H + Z . Then f ( z ) = z since w (( H i + Z ) \ { z } ) = κ < λ and χ ( z, H i + Z ) = χ ( z, Z ) = λ . Consequently, f maps ( H + Z ) \ { z } onto ( H + Z ) \ { z } .Therefore, since Z \ { z } is discrete and ( H + Z ) \ { z } = H + ( Z \ { z } ) for every H ∈ H κ ,we have H = H in view of Proposition 1.
6. Proof of Theorem 2
In order to find a natural way to prove Theorem 2 (and also Theorem 4) we give a shortproof of the following consequence of Theorem 2.(6.1) If c ≤ κ ≤ λ ≤ κ then there exist λ mutually non-homeomorphic pathwiseconnected, paracompact Hausdorff spaces of size κ and weight λ . (6.1) can easily be derived from Theorem 1 as follows. Assume c ≤ κ ≤ λ ≤ κ . By The-orem 1 there exists a family P of 2 λ mutually non-homeomorphic, totally disconnected,paracompact Hausdorff spaces X of size κ and weight λ . For every X ∈ F let Q ( X )denote the quotient space of X × [0 ,
1] by its closed subspace X × { } . The quotientspace Q ( X ) can be directly defined as follows. Consider the product space X × [0 , p X × [0 ,
1[ and put Q ( X ) := { p } ∪ ( X × [0 , U of Q ( X ) open if and only if U \ { p } is open in the product space X × [0 ,
1[ and p ∈ U implies that ( U \ { p } ) ∪ ( X × { } ) is open in the space X × [0 ,
1] . One can picture Q ( X ) as a cone with apex p and all rulings { p } ∪ ( { x } × [0 , x ∈ X ) homeomorphicto the unit interval [0 ,
1] . By [2, 5.1.36] and [2, 5.1.28] both X × [0 ,
1] and X × [0 ,
1[ areparacompact. Consequently, Q ( X ) is a regular space and hence Q ( X ) is paracompact inview of Lemma 3. It is evident that Q ( X ) is pathwise connected. Trivially, |Q ( X ) | = κ .Unfortunately we can be sure that w ( Q ( X )) = λ for every X ∈ P only if λ = 2 κ . (Since |Q ( X ) | = κ , we have w ( Q ( X )) ≤ κ . On the other hand, w ( Q ( X )) ≥ w ( Q ( X ) \ { p } ) =8 ( X × [0 , w ( X ) = λ .) The problem with the weight is that if µ is the character ofthe apex p then w ( Q ( X )) = max { w ( X × [0 , , µ } = max { λ, µ } . But we cannot rule out λ < µ if λ < κ . Of course, if X ∈ P is compact then µ = ℵ and hence w ( Q ( X )) = λ (but also λ ≤ | X | = κ ). Fortunately, we can make the character of the apex countable alsoby harshly reducing the filter of the neighborhoods of p . Let Q ∗ ( X ) be defined as the cone Q ( X ) but with the (only) difference that U ⊂ { p } ∪ ( X × [0 , p if and only if U \ { p } is open in X × [0 ,
1[ and U ⊃ X × [ t,
1[ for some t ∈ [0 ,
1[ .Now we have χ ( p, Q ∗ ( X )) = ℵ and hence w ( Q ∗ ( X )) = w ( X ) for every X ∈ P . Ofcourse, Q ∗ ( X ) is pathwise connected. By the same arguments as for Q ( X ) , the space Q ∗ ( X ) is regular and paracompact. Finally, the spaces Q ( X ) ( X ∈ P ) are mutuallynon-homeomorphic because every X ∈ P can be recovered (up to homeomorphism) from Q ( X ) . Indeed, since X is totally disconected, if Z is the set of all z ∈ Q ( X ) such that Q ( X ) \ { z } remains pathwise connected then it is evident that Z = X × { } and hence Z is homeomorphic with X . This concludes the proof of (6.1).In the following proof of Theorem 2 we will also work with cones but we cannot use thecones Q ( X ) or Q ∗ ( X ) because it is evident that if X is not discrete then neither Q ( X )nor Q ∗ ( X ) is locally connected. Furthermore, by virtue of Corollary 2 and since { p } is a G δ -set in the space Q ∗ ( X ) , the cone Q ∗ ( X ) is almost metrizable if and only if X is metrizable. (But then w ( Q ∗ ( X )) = w ( X ) = κ .) Consequently, Q ∗ ( X ) is locallyconnected and almost metrizable if and only if X is discrete. Now the clue in the followingproof of Theorem 2 is to consider Q ∗ ( S ) for one discrete spaces S of size (and weight) κ and to reduce the topology of Q ∗ ( S ) in 2 λ ways such that the weight κ of Q ∗ ( S ) isincreased to λ and that 2 λ non-homeomorphic spaces as desired are obtained. First ofall we need a lemma. Lemma 4. If n ∈ N and A is a topological space and a ∈ A and A , ..., A n aremetrizable, closed subspaces of A and A = A ∪ · · · ∪ A n and A i ∩ A j = { a } whenever ≤ i < j ≤ n then the space A is metrizable.Proof. Assume n ≥ ≤ i ≤ n then A i \{ a } = A \ S j = i A j is an open subsetof A . Furthermore, if a ∈ U i ⊂ A i and U i is open in the subspace A i for 1 ≤ i ≤ n then U ∪ · · · ∪ U n is an open subset of the space A . (Because if V i is an open subset of A with U i = V i ∩ A i for 1 ≤ i ≤ n then U ∪ · · · ∪ U n = ( V ∩ · · · ∩ V n ) ∪ S ni =1 ( V i ∩ ( A i \ { a } ) .)For 1 ≤ i ≤ n consider A i equipped with a suitable metric d i . Define a mapping from A × A into R in the following way. If x, y ∈ A i for some i then put d ( x, y ) = d i ( x, y ) .If x ∈ A i and y ∈ A j for distinct i, j then put d ( x, y ) = d i ( x, a ) + d j ( y, a ) . Of course, d is a metric on the set A . (One may regard A as a hedgehog with body a and spines A , ..., A n .) By considering the open neighborhoods of the point a in the space A weconclude that the topology generated by the metric d coincides with the topology of thespace A , q.e.d. Now we are ready to prove Theorem 2. Assume c ≤ κ < λ ≤ κ . (We ignore the case κ = λ because this case is covered by Proposition 2.) Let S be a discrete space of size κ and F an ω -free filter on S with χ ( F ) = λ . Consider the metrizable product space S × [0 ,
1[ and fix p S × [0 ,
1[ and define a topological space Φ[ F ] in the followingway. The points in the space Φ[ F ] are the elements of { p } ∪ ( S × [0 , U of { p } ∪ ( S × [0 , U \ { p } is open in the product space S × [0 ,
1[ and secondly the point p lies in U only if9 S × [ t, ∪ ( F × [0 , ⊂ U for some t ∈ [0 ,
1[ and some F ∈ F .It is plain that this is a correct definition of a topological space such that the subspaceΦ[ F ] \ { p } is identical with the product space S × [0 ,
1[ . Similarly as above we pictureΦ[ F ] as a cone with apex p and the rulings { p } ∪ ( { x } × [0 , x ∈ X ) homeomorphicto the unit interval [0 ,
1] . (Obviously, the topology of Φ[ F ] is strictly coarser than thetopology of the cone Q ∗ ( S ) .) It is straightforward to verify that Φ[ F ] is a regular space.Hence by Corollary 2 the space Φ[ F ] is almost metrizable. (Since F is ω -free and [0 ,
1] issecond countable, it is clear that { p } is a G δ -set.) Since the subspace { p } ∪ ( { s } × [ t, F ] is a homeomorphic copy of the compact unit interval [0 ,
1] for every s ∈ S andevery t ∈ [0 ,
1[ and since S is discrete, it is clear that Φ[ F ] is pathwise connected andlocally pathwise connected. Trivially, | Φ[ F ] | = κ .Clearly, if B is a filter base on S generating the filter F then (cid:8) { p } ∪ (( S \ F ) × ]1 − − n , ∪ ( F × [0 , (cid:12)(cid:12) n ∈ N , F ∈ B (cid:9) is a local basis at p in the space Φ[ F ] . Conversely, if U p is a local basis at p and ifwe choose for every U ∈ U p a real number t U ∈ [0 ,
1[ and a set F U ∈ F such that( S × [ t U , ∪ ( F U × [0 , ⊂ U then { F U | U ∈ U p } is a filter base on S generatingthe filter F . Consequently, χ ( p, Φ[ F ]) = χ ( F ) . Therefore, since w ( S × [0 , κ , wehave w (Φ[ F ]) = χ ( F ) = λ .Now consider the pathwise connected, locally pathwise connected, amost metrizable spaceΦ[ F ] for each of the 2 λ ω -free filters F on S with χ ( F ) = λ . Since the size of eachspace is κ and the weight of each space is λ , by the same arguments about the size ofequivalence classes as in the proof of Theorem 1 (for µ = κ ), the statement in Theorem 2is true in case that 2 λ > κ because it is evident that the topologies of the spaces Φ[ F ]and Φ[ F ] are distinct topologies on the set { p } ∪ ( S × [0 , F and F aredistinct ω -free filters on S .Now assume 2 λ = 2 κ and let P κ be a family as provided by Proposition 2. Choose one ω -free filter F on S with χ ( F ) = λ and consider the space Φ[ F ] . Note that x ∈ Φ[ F ]is a noncut point of Φ[ F ] if and only if x = ( s,
0) for some s ∈ S . For every H ∈ P κ create a space X ( H ) in the following way. Consider the compact unit square [0 , andchoose a point a ∈ [0 , . (Clearly, a is a noncut point of [0 , . Note also that noconnected open subset of [0 , has cut points.) Choose a noncut point a in Φ[ F ] anda noncut point a in H . Finally, let X ( H ) be the quotient of the topological sum of thethree spaces [0 , and Φ[ F ] and H by the subspace { a , a , a } . Roughly speaking, X ( H ) is created by sticking together the three spaces so that the three points a , a , a are identified. It is clear that X ( H ) is pathwise connected and locally pathwise connectedand regular and | X ( H ) | = κ and w ( X ( H )) = λ .There is precisely one point b ∈ X ( H ) with χ ( b, X ( H )) = λ . This point b correspondswith the point p ∈ Φ[ F ] . By virtue of Lemma 4 for n = 3 the subspace X ( H ) \ { b } of X ( H ) is metrizable. Consequently, if H ∈ P κ then X ( H ) is almost metrizable. The 2 κ spaces X ( H ) ( H ∈ P κ ) are mutually non-homeomorphic because each H ∈ P κ can berecovered from X ( H ) as follows.Since cut points in H resp. in Φ[ F ] lie dense and since [0 , has no cut points, there isprecisely one point q in X ( H ) such that every neighborhood of q contains two nonempty10onnected open sets U , U where U has no cut points and where U has cut points.(This point q must be the point obtained by identifying the three points a , a , a .) Thesubspace X ( H ) \{ q } has precisely three components and every component of X ( H ) \{ q } is homeomorphic either with Φ[ F ] \{ a } or with H \{ a } or with [0 , \{ a } . Therefore,precisely one component is not metrizable. (If s ∈ S then the space Φ[ F ] \ { ( s, } is notmetrizable since it has no countable local basis at p .) The two metrizable componentsof X ( H ) \ { q } can be distinguished by the observation that one component has infinitelymany cut points while the other component has no cut points. If M is a metrizablecomponent of X ( H ) \ { q } which has cut points then the subspace M ∪ { q } of X ( H ) ishomeomorphic with H , q.e.d.
7. Proof of Theorem 3
Lemma 5.
There exists a second countable, countably infinite Hausdorff space H suchthat H \ E is connected and locally connected for every finite set E .Proof. Let H be the set N equipped with the coarsest topology such that if p is a primeand a ∈ N is not divisible by p then N ∩ { p + ka | k ∈ Z } is open. Referring to [13]Nr. 61, H is a locally connected Hausdorff space such that the intersection of the closuresof any two nonempty open subsets of H must be an infinite set. Therefore, if E is a finiteset then the subspace H \ E of H is connected. Since H is locally connected, H \ E islocally connected for every finite set E , q.e.d. The first step in proving Theorem 3 is a proof of the following enumeration theorem aboutcountable connected spaces.
Theorem 5.
For every λ ≤ c there exist λ mutually non-homeomorphic connected,locally connected Hausdorff spaces of size ℵ and weight λ .Proof. Let H be a connected, locally connected Hausdorff space with | H | = w ( H ) = ℵ as provided by Lemma 5. Fix e ∈ H and note that e is a noncut point in H . Put M := H \ { e } . So M is connected as well.Let S be an infinite discrete space and let F be a free filter on S with χ ( F ) ≥ | S | .Consider the product space S × M and fix p S × M and consider Ψ[ F ] := { p } ∪ ( S × M ) equipped with the following topology. A subset U of { p } ∪ ( S × M ) isopen if and only if U \ { p } is open in the product space S × M and p ∈ U implies that( S × ( V \ { e } )) ∪ ( F × M ) ⊂ U for some neighborhood V of e in H and some F ∈ F . Similarly as in the proof ofTheorem 2, Ψ[ F ] is a connected and locally connected Hausdorff space and | Ψ[ F ] | = | S | and w (Ψ[ F ]) = χ ( F ) .Now let S be the discrete Euclidean space N . If 2 λ > c then with the help of 2 λ freefilters on N with χ ( F ) = λ we can track down 2 λ mutually non-homeomorphic spacesΨ[ F ] . (Note that there are only c permutations on N and use the argument on sizes ofequivalence classes.) So it remains to settle the case 2 λ = c .Let Z be the space Ψ[ F ] for some free filter F on N with χ ( F ) = λ . So the underlyingset of Z is { p } ∪ ( N × ( H \ { e } )) and the countable Hausdorff space Z is connected andlocally connected and w ( Z ) = λ due to χ ( p, Z ) = λ . The point p is the only cut point of Z and Z \ { p } has infinitely many components. Keep in mind that | H | = w ( H ) = ℵ a ∈ H then the spaces H and H \ { a } and H \ { a, e } are connected andlocally connected. Fix b ∈ H \ { e } and consider the subset ˆ Z := { ( s, b ) | s ∈ N } of Z . Clearly, ˆ Z is closed and discrete and Z \ { z } is connected and locally connectedfor every z ∈ ˆ Z . Choose for every m ∈ N and every i ∈ { , ..., m } spaces H ( m ) i such that H ( m ) i is homeomorphic with H and H ( m ) i ∩ H ( n ) j = ∅ whenever m = n or i = j . Furthermore assume that H ( m ) i ∩ Z = ∅ for every m and every i . Let ϕ be a choice function on the class of all infinite sets, i.e. ϕ ( A ) ∈ A for every infiniteset A . Now define for every nonempty set T ⊂ N a Hausdorff space Q [ T ] as follows.Consider the topological sum Σ[ T ] of countably infinite and mutually disjoint spaceswhere the summands are Z and all spaces H ( m ) i with m ∈ T and i ∈ { , ..., m } .Define an equivalence relation on Σ[ T ] such that the non-singleton equivalence classesare precisely the sets { ( m, b ) } ∪ { ϕ ( H ( m )1 ) , ..., ϕ ( H ( m ) m ) } with m ∈ T . (Note that( m, b ) ∈ ˆ Z for every m ∈ T .) Finally, let Q [ T ] denote the quotient space of Σ[ T ] withrespect to this equivalence relation. Roughly speaking, Q [ T ] is the union of Z and allspaces H ( m ) i with m ∈ T and i ∈ { , ..., m } where for every m ∈ T the m + 1 points( m, b ) , ϕ ( H ( m )1 ) , ..., ϕ ( H ( m ) m ) are identified. We consider Z to be a subset of Q [ T ] . Onemay picture Q [ T ] as an expansion of Z created by attaching m copies of H to Z atthe point ( m, b ) ∈ ˆ Z for every m ∈ T . It is evident that Q [ T ] is a connected andlocally connected countably infinite Hausdorff space. We have w ( Q [ T ]) = λ since Z is asubspace of Q [ T ] with w ( Z ) = λ and χ ( x, Q [ T ]) = ℵ if p = x ∈ Q [ T ] . Thus the case2 λ = c is settled by verifying that two spaces Q [ T ] and Q [ T ] cannot be homeomorphicif ∅ 6 = T , T ⊂ N and T = T . This must be true because the set T ⊂ N is completelydetermined by the topology of Q [ T ] by the following observation.Let ∅ 6 = T ⊂ N . For every point x ∈ Q [ T ] let ν ( x ) denote the total number of allcomponents of the subspace Q [ T ] \ { x } . The following three statements for x ∈ Q [ T ]are evident. Firstly, ν ( x ) ≥ ℵ if and only if x = p . Secondly, 1 < ν ( x ) < ℵ if andonly if x = ( m, b ) ∈ ˆ Z for some m ∈ T . Thirdly, ν ( x ) = 1 if and only if x is anelement of the set Q [ T ] \ (( T × { b } ) ∪ { p } ) . Concerning the second statement we compute ν (( m, b )) = m + 1 for every m ∈ T . Consequently, { ν ( x ) − | x ∈ Q [ T ] ∧ ν ( x ) ∈ N } \ { } = T whenever T is one of the c non-empty subsets of N , q.e.d. Now in order to prove Theorem 3 assume ℵ ≤ κ < c and κ ≤ λ ≤ κ . Referring toTheorem 5 there is nothing more to show in case that κ = ℵ . So we also assume that κ > ℵ . Let S be a discrete space of size κ . By Proposition 3 there are 2 λ free filters F on S with χ ( F ) = λ . For each one of these filters F consider the connected andlocally connected Hausdorff space Ψ[ F ] of size κ and weight λ as defined in the previousproof. Hence in case that 2 λ > κ we can track down 2 λ filters F on S such that thecorresponding spaces Ψ[ F ] are mutually non-homeomorphic.So it remains to settle the case 2 λ = 2 κ . Choose any free filter F on S with χ ( F ) = λ and and consider the space Ψ := Ψ[ F ] of size κ and weight λ . Fix a noncut point z ∈ Ψ . Keep in mind that Ψ has precisely one cut point p and that Ψ[ F ] \ { p } hasprecisely κ and hence uncountably many components.12n view of our proof of Theorem 5 there is a family C of mutually non-homeomorphic countable Hausdorff spaces of weight κ (and hence not necessarily of weight λ ) such that |C| = 2 κ and if C ∈ C then C is connected and locally connected and contains preciselyone cut point q ( C ) such that C \ { q ( C ) } has infinitely many components. In particular,all these components are countable and ℵ is their total number.For every C ∈ C consider the topological sum Ψ + C and define an equivalence relationsuch that { z, q ( C ) } is an equivalence class and all other equivalence classes are singletons.Let Q [ C ] denote the quotient space of Ψ + C with respect to this equivalence relation.So Q [ C ] is obtained by sticking together the spaces Ψ and C at one point and this pointis the identification of z ∈ Ψ and q ( C ) ∈ C . It is clear that Q [ C ] is a connected, locallyconnected Hausdorff space of size κ and weight λ . So we are done by verifying that fordistinct C , C ∈ C the spaces Q [ C ] and Q [ C ] are never homeomorphic. This must betrue because each C ∈ C can be recovered from Q [ C ] as follows.There is a unique point ξ in Q [ C ] such that Q [ C ] \ { ξ } has precisely ℵ components.(This point ξ is the one corresponding with the equivalence class { z, q ( C ) } .) Among thesecomponents there is precisely one of uncountable size. (This component is the one whichcontains the point p ∈ Ψ .) Let K be the unique uncountable component of Q [ C ] \ { ξ } .Then Q [ C ] \ K is essentially identical, at least homeomorphic with the space C .
8. Proof of Theorem 4
Our goal is to derive Theorem 4 from Theorem 1 by using appropriate modifications ofthe cones Q ∗ ( X ) considered in Section 6. In order to accomplish this we need buildingblocks provided by the following lemma. Lemma 6.
There exists a second countable, connected, totally pathwise disconnected,nowhere locally connected, metrizable space M of size c which contains precisely onenoncut point b and where M \ { x, b } has precisely two components whenever b = x ∈ M .Proof. Let f be a function from R into R such that the graph of f is a dense and con-nected subset of the Euclidean plane R . (See [10] for a construction of such a function f .)Automatically, f is discontinuous everywhere. Let M be the intersection of [0 , ∞ [ × R and the graph of f . It is straightforward to check that M fits with b = (0 , f (0)) , q.e.d. Now we are ready to prove Theorem 4. Assume c ≤ κ ≤ λ ≤ κ and let Y = Y ( κ, λ )be a family of precisely 2 λ mutually non-homeomorphic scattered, normal spaces of size κ and weight λ such that if Y ∈ Y then for a certain finite set γ ( Y ) ⊂ Y the subspace Y \ γ ( Y ) is metrizable (and hence of weight κ ) and γ ( Y ) is a G δ -set in Y . Precisely, theset γ ( Y ) is empty when κ = λ and a singleton { y } when κ < λ . (Clearly, if γ ( Y ) = { y } then χ ( y, Y ) = λ .) If 2 λ > κ then such a family Y exists by considering the 2 λ almostdiscrete spaces provided by Theorem 1. If λ > κ and 2 λ = 2 κ then such a family Y existsin view of the construction in Section 5 which proves Theorem 1 in case that 2 λ = 2 κ . If λ = κ then such a family Y exists by Proposition 1.Let M be a metrizable space as in Lemma 6 and let b denote the noncut point of M and fix a point a ∈ M \ { b } . For an infinite, scattered, normal space X consider theproduct space X × M and fix p X × M and put K ( X ) := { p } ∪ ( X × ( M \ { b } ) .Declare a subset U of K ( X ) open if and only if U \ { p } is open in the product space13 × ( M \ { b } ) and p ∈ U implies that U contains X × ( N \ { b } ) for some neighborhood N of b in the space M . It is plain that K ( X ) is a well-defined regular space. Since M is metrizable and χ ( p, K ( X )) = ℵ , if X is metrizable then K ( X ) has a σ -locally finitebase and hence K ( X ) is metrizable.Now for Y ∈ Y consider the subspace L ( Y ) := K ( Y ) \ ( γ ( Y ) × ( M \ { a, b } )) of K ( Y ) and the subspace S ( Y ) := L ( Y ) \ ( γ ( Y ) × { a } ) of L ( Y ) . Trivially, the spaces K ( Y ) and L ( Y ) and S ( Y ) coincide if κ = λ . Furthermore the space S ( Y ) coincideswith the metrizable space K ( Y \ γ ( Y )) . Therefore and by Corollary 2, L ( Y ) is analmost metrizable space since γ ( Y ) × { a } is a G δ -set in K ( Y ) of size 0 or 1 . We have | L ( Y ) | = κ and w ( L ( Y )) = λ because if γ ( Y ) = { y } then χ (( y, a ) , L ( Y )) = λ . It isevident that S ( Y ) is connected and totally pathwise disconnected and nowhere locallyconnected. Consequently, L ( Y ) is totally pathwise disconnected and nowhere locallyconnected. And L ( Y ) is connected since the connected set S ( Y ) is dense in L ( Y ) .Finally, the spaces L ( Y ) ( Y ∈ Y ) are mutually non-homeomorphic because every Y ∈ Y can be recovered from L ( Y ) . Indeed, for x ∈ L ( Y ) let C ( x ) denote the family of allcomponents of the subspace L ( Y ) \ { x } of L ( Y ) . Then C ( x ) is an infinite set if andonly if x = p . Because the scattered space Y has infinitely many isolated points and if u ∈ Y is isolated then { u } × ( M \ { b } ) lies in C ( x ) . If u ∈ Y \ γ ( Y ) and b = v ∈ M then |C (( u, v )) | ≤ |C (( u, v )) | = 2 when u is isolated in Y ). And if γ ( Y ) = { y } then |C (( y, a )) | = 1 . Thus { p } = { x ∈ L ( Y ) | |C ( x ) | ≥ ℵ } , whence the point p canbe recovered from the space L ( Y ) . Now let C be the family of all components of thespace L ( Y ) \ { p } . Since Y is totally disconnected, the members of C are precisely thesets { u } × ( M \ { b } ) with u ∈ Y \ γ ( Y ) plus the singleton γ ( Y ) × { a } if and only if γ ( Y ) = ∅ . Naturally, the quotient space of L ( Y ) \ { p } by the equivalence relation definedvia the partition C is homeomorphic with Y for every Y ∈ Y . This concludes the proofof Theorem 4.
9. Overweight compact spaces
While w ( X ) ≤ | X | for every compact Hausdorff space X (see [2, 3.3.6]), for compactT -spaces X one cannot rule out w ( X ) > | X | and actually we can prove the followingenumeration theorem by applying Theorems 1 and 2 and 3. Theorem 6. If κ ≤ λ ≤ κ then there exist two families C , C of mutually non-homeomorphic compact T -spaces of size κ and weight λ such that |C | = |C | = 2 λ andall spaces in C are scattered, all spaces in C are connected and locally connected, and if κ ≥ c then all spaces in C are arcwise connected and locally arcwise connected. In order to prove Theorem 6 we consider T -compactifications of Hausdorff spaces. If Y is an infinite Hausdorff space with | Y | ≤ w ( Y ) then define a topological space Γ( Y )which expands Y in the following way. Put Γ( Y ) = Y ∪ { z } where z Y and declare U ⊂ Γ( Y ) open either when U is an open subset of Y or when z ∈ U and Y \ U isfinite. It is clear that in this way a topology on Γ( Y ) is well-defined such that Y is a densesubspace of Γ( Y ) . Obviously, Γ( Y ) \ { x } is open for every x ∈ Γ( Y ) and hence Γ( Y )is a T -space. Since all neighborhoods of z cover the whole space Γ( Y ) except finitelymany points, Γ( Y ) is compact. Trivially, | Γ( Y ) | = | Y | . We have w (Γ( Y )) = w ( Y ) since w ( Y ) ≥ | Y | and Y is a subspace of Γ( Y ) and, by definition, there is a local basis at z of size | Y | . 14vidently, if Y is scattered then Γ( Y ) is scattered. On the other hand it is clear that if Y is dense in itself then Γ( Y ) is connected and every neighborhood of z is connected. Soif Y is connected and locally connected then Γ( Y ) is connected and locally connected.We claim that if Y is pathwise connected then Γ( Y ) is arcwise connected. Assume thatthe Hausdorff space Y is pathwise connected and hence arcwise connected and let a ∈ Y .Of course it is enough to find an arc which connects the point a with the point z Y .Since Y is arcwise connected, we can define a homeomorphism ϕ from [0 ,
1] onto asubspace of Y such that ϕ (0) = a . Define an injective function f from [0 ,
1] into Γ( Y )via f (1) = z and f ( t ) = ϕ ( t ) for t < U be an open subset of Γ( Y ) . If z ∈ U then U \ Y is finite and thus f − ( U ) is a cofinite and hence open subset of [0 ,
1] . If z U then U is an open subset of Y and hence f − ( U ) = ϕ − ( U ) \ { } is an opensubset of [0 ,
1] . Thus the injective function f is continuous.Since every neighborhood of z contains all but finitely many points from Y , by exactly thesame arguments we conclude that if Y is locally pathwise connected then every neighbor-hood of z is an arcwise connected subspace of Γ( Y ) . Consequently, if the Hausdorff space Y is locally pathwise connected then the T -space Γ( Y ) is locally arcwise connected.The space Y can be recovered from Γ( Y ) (up to homeomorphism) provided that Y hasat least two limit points. Because then it is evident that z is the unique point x ∈ Γ( Y )such that the subspace Γ( Y ) \ { x } of Γ( Y ) is Hausdorff.By virtue of Theorem 1, for κ ≤ λ ≤ κ let Y ( κ, λ ) be a family of mutually non-homeomorphic, scattered Hausdorff spaces of size κ and weight λ such that |Y ( κ, λ ) | =2 λ . By virtue of Theorem 3, for κ < c and κ ≤ λ ≤ κ let Y ( κ, λ ) be a family ofmutually non-homeomorphic connected and locally connected Hausdorff spaces of size κ and weight λ such that |Y ( κ, λ ) | = 2 λ . By virtue of Theorem 2, for c ≤ κ ≤ λ ≤ κ let Y ( κ, λ ) be a family of mutually non-homeomorphic pathwise connected and locallypathwise connected Hausdorff spaces of size κ and weight λ such that |Y ( κ, λ ) | = 2 λ .Now put C := { Γ( Y ) | Y ∈ Y ( κ, λ ) } and C := { Γ( Y ) | Y ∈ Y i ( κ, λ ) } where i = 2when κ < c and i = 3 when κ ≥ c . Then C , C are families which prove Theorem 6.The condition κ ≥ c in Theorem 6 is inevitable since, trivially, | X | ≥ c for every infinite, arcwise connected space. There arises the question whether | X | ≥ c is inevitable forinfinite, pathwise connected T -spaces. (Of course, every finite T -space X is discrete andhence not pathwise connected when | X | ≥ -space of size ℵ does not exist (see also Proposition 5 below). So the essential questionis whether there are pathwise connected T -spaces X with ℵ < | X | < c (provided thatthere are cardinals µ with ℵ < µ < c ). The following proposition shows that there is nochance to track down such spaces X . Proposition 5.
Pathwise connected T -spaces X with ≤ | X | ≤ ℵ do not exist. It isconsistent with ZFC that |{ κ | ℵ < κ < c }| > ℵ and pathwise connected T -spaces X with ℵ < | X | < c do not exist. If X is a T -space and f : [0 , → X is continuous then { f − ( { x } ) | x ∈ X } \ {∅} is a decomposition of [0 ,
1] into precisely | f ([0 , | nonempty closed subsets. Therefore,Proposition 5 is an immediate consequence of15 roposition 6. Every partition of [0 , into at least two closed sets is uncountable. It isconsistent with ZFC that uncountably many cardinals κ with ℵ < κ < c exist while stilla partition P of [0 , into closed sets with ℵ < |P| < c does not exist. Certainly, the first statement in Proposition 6 is an immediate consequence of Sierpi´nski’stheorem [2, 6.1.27]. However, in order to prove Proposition 6 we need another approachthan in the proof of [2, 6.1.27]. (Moreover, the following proof is much easier than theproof of [2, 6.1.27].)Assume that P is a partition of [0 ,
1] into closed sets with |P| ≥ S ⊂ [0 ,
1] let ∂S denote the boundary of S in the compact space [0 ,
1] . (Notice that then ∂ [0 ,
1] = ∅ .)Put V := { ∂A | A ∈ P } and W := S V . Then ∅ 6∈ V since [0 ,
6∈ P and hence V is a partition of W with |V| = |P| . The nonempty set W is a closed subset of [0 , W = [0 , \ S { A \ ∂A | A ∈ P } since P is a partition of [0 ,
1] . We claim thatthe closed sets V ∈ V are nowhere dense in the compact metrizable space W .Let A ∈ P and assume indirectly that a is an interior point of ∂A in W . Then thereis an interval I open in the compact space [0 ,
1] with a ∈ I and I ∩ W ⊂ ∂A . Since a lies in the boundary of A , the interval I intersects [0 , \ A and hence for some B = A in the family P we have I ∩ B = ∅ . However, I ∩ ∂B = ∅ in view of ( ∂A ) ∩ ( ∂B ) = ∅ and I ∩ W ⊂ ∂A . Therefore, I ∩ B is a nonempty set which is open and closed in theconnected space I and hence I ∩ B = I contrarily with A ∩ I = ∅ and A ∩ B = ∅ .Thus V is a partition of the compact Hausdorff space W into nowhere dense subsets with |V| = |P| . Therefore |P| ≤ ℵ is impossible since W is a space of second category. Thisconcludes the proof of the first statement. Under the assumption of Martin’s Axiom (see [4,16.11]) also the weaker inequality |V| = |P| < c is impossible because it is well-known thatMartin’s axiom implies that no separable, compact Hausdorff space can be covered by lessthan c nowhere dense subsets. (Actually, Martin’s axiom is equivalent to the statementthat in every compact Hausdorff space of countable cellularity any intersection of less than c dense, open sets is dense.) Therefore, the proof of Proposition 6 is concluded by checkingthat the existence of uncountably many infinite cardinals below c is consistent with ZFCplus Martin’s Axiom. This is certainly true because by applying the Solovay-Tennenbaumtheorem [4, 16.13] there is a model of ZFC in which Martin’s Axiom holds and the identity2 ℵ = ℵ ω +1 is enforced. (If c = ℵ ω +1 then |{ κ | κ < c }| = ℵ > ℵ .) Remark.
There is an interesting observation concerning compactness and the Hausdorffseparation axiom. By applying Theorem 6 and (1.2), there exist precisely c compact,countable, second countable T -spaces up to homeomorphism. If in this statement T issharpened to T then we obtain an unprovable hypothesis. Indeed, due to Mazurkiewicz andSierpi´nski [12], there exist precisely ℵ countable (and hence second countable) compactHausdorff spaces up to homeomorphism. (The hypothesis ℵ < c is irrefutable since it isa trivial consequence of (1.1).) This discrepancy of provability vanishes when uncountable compacta are counted up to homeomorphism. Indeed, by virtue of [7, Theorem 3] it can beaccomplished that in Theorem 6 for κ = λ > ℵ all spaces in the family C are Hausdorffspaces. (Note that w ( X ) = | X | for every scattered, compact Hausdorff space.)16
0. Pathwise connected, scattered spaces
Naturally, a scattered T -space is totally disconnected and hence far from being pathwiseconnected. Furthermore it is plain that no scattered space is arcwise connected. Thereforeand in view of Proposition 5 the following enumeration theorem is worth mentioning. Theorem 7. If κ ≤ λ ≤ κ then there exist two families C , L of mutually non-homeomorphic pathwise connected, scattered T -spaces of size κ and weight λ such that |C| = |L| = 2 λ and all spaces in C are compact and if κ ≤ c or κ < λ then all spacesin L are locally pathwise connected. The existence of the family C in Theorem 7 can be derived from Theorem 1 in view ofthe following considerations. Let X be an infinite Hausdorff space. Fix b X and definea topology on the set B ( X ) = X ∪ { b } by declaring U ⊂ B ( X ) open when either U = B ( X ) or U is an open subset of X . (Then { b } is closed and B ( X ) is the onlyneighborhood of b .) Obviously, B ( X ) is a compact T -space and b is a limit point ofevery nonempty subset of X = B ( X ) \ { b } . It is trivial that | B ( X ) | = | X | and clearthat w ( B ( X )) = w ( X ) . For any pair x, y of distinct points in B ( X ) define a function f from [0 ,
1] into B ( X ) via f ( t ) = x when t < and f ( ) = b and f ( t ) = y when t > . It is plain that f is continuous, whence B ( X ) is pathwise connected. Obviously,if X is scattered then B ( X ) is scattered. Finally, the space X can be recovered from B ( X ) since a singleton { a } is closed in B ( X ) if and only if a = b .Unfortunately, if X is scattered and not discrete then B ( X ) is not locally connected.Fortunately, finishing the proof of Theorem 7 we can track down a family L as desiredby adopting the proofs of Theorem 3 and Theorem 5 in Section 7 line by line such that,throughout, the building block H in the definition of Φ[ F ] provided by Lemma 5 isreplaced with the space G provided by the following lemma. (In Section 7 the restriction κ < c is only for avoiding an overlap between Theorem 2 and Theorem 3 and can clearlybe expanded to κ ≤ c . The case 2 κ < λ is settled by the 2 λ spaces Ψ[ F ] of arbitrary size κ .) Lemma 7.
There exists a second countable, scattered, countably infinite T -space G suchthat G \ E is pathwise connected and locally pathwise connected for every finite set E .Proof. Let G be the set { n ∈ Z | n ≥ } equipped with divisor topology as defined in [13, ]. (A basis of the divisor topology is the family of all sets { m ∈ Z | m ≥ ∧ m | n } with n ∈ G .) In view of the considerations in [13], it is straightforward to verify that G fits, q.e.d.Remark. If i ∈ { , , } and F i is a family of mutually non-homeomorphic compactT i -spaces X with w ( X ) ≤ κ then |F i | ≤ κ is true for i = 2 . (Because any compactHausdorff space of weight at most κ is embeddable into the Hilbert cube [0 , κ and, since w ([0 , κ ) = κ and | X | = 2 κ , the compact Hausdorff space [0 , κ has precisely 2 κ closedsubspaces.) However, the estimate |F i | ≤ κ is false for i = 0 because |F | = 2 κ can beachieved for every κ . (In view of (1.2) and since max {| X | , w ( X ) } ≤ min { | X | , w ( X ) } for every infinite T -space X , 2 κ is the maximal possible cardinality.) Indeed, considerfor X = [0 , κ the compact T -space B ( X ) = X ∪ { b } of size 2 κ and weight κ definedas above. Clearly, for every nonempty S ⊂ X the subspace S ∪ { b } of B ( X ) is compact.Since X is Hausdorff and w ( X ) = κ , there are 2 | X | = 2 κ mutually non-homeomorphic17ubspaces of X and hence 2 κ mutually non-homeomorphic compact subspaces of B ( X ) .There arises the interesting question whether the estimate |F i | ≤ κ is generally true for i = 1 .
11. Counting P-spaces
A natural modification of the proof of Theorem 1 leads to a noteworthy enumerationtheorem about P -spaces. As usual (see [1]), a Hausdorff space is a P-space if and only ifany intersection of countably many open sets is open. More generally, a Hausdorff space X is a P α - space if and only if α is an infinite cardinal number and T U is an open subsetof X whenever U is a family of open subsets of X with 0 = |U | < α . So if α = ℵ thenevery Hausdorff space is a P α -space and if α = ℵ then X is a P α -space if and only if X is a P -space. Clearly, if X is a P α -space and | X | < α then X is discrete. (It is plainthat if X is a P α -space and | X | = α and α is a singular cardinal then X is discrete.)For an infinite cardinal α let us call a Hausdorff space α -normal when it is completelynormal and every closed set is an intersection of at most α open sets. So a Hausdorff spaceis perfectly normal if and only if it is ℵ -normal. It is dull to consider perfectly normal P -spaces because, trivially, a perfectly normal P -space must be discrete. More generally,if µ < α then every µ -normal P α -space is discrete. However, the enumeration problemconcerning completely normal P -spaces and α -normal P α -spaces is not trivial and can besolved under certain cardinal restrictions.As usual, κ + denotes the smallest cardinal greater than κ , whence κ + ≤ κ and ℵ =( ℵ ) + . Furthermore, for arbitrary κ, µ the cardinal number κ <µ is defined as usual (see[4]). Note that if µ ≤ κ + then κ <µ = |{ T | T ⊂ S ∧ | T | < µ }| whenever S is a setof size κ . In particular, κ < ℵ = κ and κ < ℵ = κ ℵ for every κ . Naturally, if µ = κ + then κ <µ = 2 κ . Consequently, if µ > κ then κ < κ <µ . (If κ is a cardinal number ofcofinality smaller than µ ++ then κ < κ <µ due to K¨onig’s Theorem [4, 5.14].) On theother hand, for every µ the cardinals κ satisfying κ <µ = κ form a proper class K µ such that 2 θ ∈ K µ for every cardinal θ with θ + ≥ µ and if κ ∈ K µ then the cardinalsuccessor κ + of κ also lies in K µ due to the Hausdorff formula [4, (5.22)]. In particular,the cardinals c , c + , c ++ , ... lie in K µ for µ = ℵ . Furthermore, if κ <µ = κ ≤ λ andthere are only finitely many cardinals θ with κ ≤ θ ≤ λ then λ <µ = λ . (Note, again,that κ <α = κ implies α ≤ κ .) Theorem 8.
Let α be an uncountable cardinal. Assume κ = κ <α and κ ≤ λ ≤ κ and λ <α = λ ≤ µ < λ for some µ ≤ κ with µ <α = µ . Then there exist λ mutually non-homeomorphic scattered, strongly zero-dimensional, hereditarily paracompact, α -normal P α -spaces of size κ and weight λ . In particular, for every κ with κ = κ ℵ there existprecisely κ mutually non-homeomorphic paracompact P-spaces of size κ and weight κ up to homeomorphism. As usual (see [1] and [4]), a filter F is κ - complete if and only if T A ∈ F for every
A ⊂ F with 0 = |A| < κ . Trivially, every filter is ℵ -complete. Obviously, an ω -free filteris not κ -complete for any κ > ℵ . Let us call a filter F κ -free if and only if T A = ∅ forsome A ⊂ F with |A| ≤ κ . (So a filter is ω -free if and only if it is ℵ -free.) Clearly, ifthe topology of an almost discrete space X is the single filter topology defined with a freefilter F then for every infinite cardinal α the (completely normal) space X is α -normal18f and only if F is α -free, and X is a P α -space if and only if F is α -complete. Therefore,in view of the following counterpart of Proposition 3, Theorem 8 can be easily proved bysimply adopting the proof of the case 2 λ > µ in Theorem 1 line by line while replacingthe property ω - free with α -complete and α -free throughout. Proposition 7. If | Y | = κ = κ <µ and κ ≤ λ = λ <µ ≤ κ then there exist λ µ -complete, µ -free filters F on Y such that χ ( F ) = λ . For the proof of Proposition 7 we need a lemma. This lemma also guarantees the existenceof the family A ω in the proof of Proposition 3 since κ <µ = κ for µ = ℵ . Lemma 8.
Let Y be an infinite set of size κ and assume κ <µ = κ . Then there exists afamily A of subsets of Y such that |A| = 2 κ and A has a subfamily K of size µ with T K = ∅ and if D , E 6 = ∅ are disjoint subfamilies of A of size smaller than µ then T D is not a subset of S E .Proof. For an infinite set S put P µ ( S ) := { T | T ⊂ S ∧ | T | < µ } . Let Y be a setof size κ and assume κ <µ = κ , whence κ ≥ µ . Choose any set X of size κ . Then |P µ ( X ) | = κ <µ = κ and hence |P µ ( P µ ( X )) | = κ <µ = κ . Therefore we may identify Y with the set P µ ( X ) × P µ ( P µ ( X )) . Now for Y := P µ ( X ) × P µ ( P µ ( X )) put A [ S ] := { ( H, H ) ∈ Y | ∅ 6 = H ∩ S ∈ H } whenever S ⊂ X . Clearly, A [ S ] = ∅ if and only if S = ∅ . We observe that A [ S ] = A [ S ]whenever S , S ⊂ X are distinct. Indeed, if S , S are subsets of X and s ∈ S \ S then ( { s } , ) ∈ A [ S ] \ A [ S ] . Put A := { A [ S ] | ∅ 6 = S ⊂ X } . Then |A| = 2 κ and we claim that A is a family as desired.For 0 = | I × J | < µ let { S i | i ∈ I } and { T j | j ∈ J } be disjoint families of nonemptysubsets of X . Choose a i,j ∈ ( S i \ T j ) ∪ ( T j \ S i ) for every ( i, j ) ∈ I × J and b i ∈ S i for every i ∈ I and put V := { a i,j | i ∈ I, j ∈ J } ∪ { b i | i ∈ I } . Then | V | < µ and ∅ 6 = V ∩ S i = V ∩ T j whenever i ∈ I and j ∈ J . Hence the pair ( V, { V ∩ S i | i ∈ I } ) liesin T i ∈ I A [ S i ] but not in S j ∈ J A [ T j ] . Finally, since | H | < µ whenever ( H, H ) ∈ A [ S ] ,if K is any subfamily of { A [ { x } ] | x ∈ X } with |K| = µ then T K = ∅ , q.e.d.Remark. The previous proof is very similar to Hausdorff’s classic construction of indepen-dent sets as carried out in the proof of [4, 7.7]. However, by Hausdorff (and in [4, 7.7]) onlythe special case µ = ℵ is considered and, unfortunately, from Hausdorff’s constructionone cannot obtain ω -free resp. α -free filters in a natural way. In order to accomplish thiswe have modified the proof of [4, 7.7] in a subtle but crucial way by including the condition ∅ 6 = H ∩ S in our definition of A [ S ] . This condition guarantees that A has a subfamily K as desired and hence that the family A ω in the proof of Proposition 3 actually exists.Now in order to prove Proposition 7 let A and K be families as in Lemma 8. For everyfamily H with K ⊂ H ⊂ A and |H| = λ put B H := { T G | ∅ 6 = G ⊂ H ∧ |G| < µ } .Then ∅ 6∈ B H and thus B H is a filter base for a µ -complete filter F [ H ] . Since K ⊂ F [ H ] ,the filter F [ H ] is µ -free. Since λ <µ = λ , we have |B H | = χ ( F [ H ]) = λ by exactly thesame arguments as in the proof of Proposition 3.19 emark. Since for no cardinal κ > ℵ the existence of a κ -complete ultrafilter is provablein ZFC (see [4]), in Theorem 8 we cannot include the property extremally disconnected .While Theorem 8 modifies Theorem 1 for P -spaces, there is no pendant of Theorem 2for P -spaces because an infinite P -space is clearly not pathwise connected and, moreover,every regular P -space X is zero-dimensional. (If x ∈ U where U ⊂ X is open thenchoose open neighborhoods U n of x such that U ⊃ U n ⊃ U n ⊃ U n +1 ⊃ U n +1 for every n ∈ N . Then V := T ∞ n =1 U n = T ∞ n =1 U n is an open-closed neighborhood of x and V ⊂ U .) References [1] Comfort, W.W., and Negrepontis, S.:
The Theory of Ultrafilters.
Springer 1974.[2] Engelking, R.:
General Topology, revised and completed edition.
Heldermann 1989.[3] Hodel, R.E.:
The number of metrizable spaces.
Fund. Math. (1983), 127-141.[4] Jech, T.:
Set Theory.
Counting topologies.
Elemente d. Math. (2011), 56-62.[6] Kuba, G.: Counting metric spaces.
Arch. Math. (2011), 569-578.[7] Kuba, G.: Counting linearly ordered spaces.
Colloq. Math. (2014), 1-14.[8] Kuba, G.:
On the variety of Euclidean point sets.
Internat. Math. News (2015),23-32.[9] Kuba, G.:
Counting ultrametric spaces.
Colloq. Math. (2018), 217-234.[10] Kulpa, W.:
On the existence of maps having graphs connected and dense.
Fund. Math. (1972), 207-211.[11] F.W. Lozier and R.H. Marty, The number of continua.
Proc. Amer. Math. Soc. (1973), 271-273.[12] S. Mazurkiewicz and W. Sierpi´nski, Contribution `a la topologie des ensembles d´enom-brables.
Fund. Math. (1920), 17-27.[13] Steen, L.A., and Seebach Jr., J.A.: Counterexamples in Topology.
Dover 1995.Gerald KubaInstitute of Mathematics,University of Natural Resources and Life Sciences, 1180 Wien, Austria.
E-mail: [email protected]
This paper is published under the less humorous title
Counting spaces of excessive weights in Matematiˇcki Vesnik . (A revision of the title has been required due to the very true factthat a mathematical definition of the property overweight cannot be found anywhere.)20
PPENDIX on Compact Hausdorff Spaces
In Section 9 we refer to the following important theorem from [7].
Theorem A.
For every κ > ℵ there exist κ mutually non-homeomorphic scattered,compact, linearly ordered spaces of size κ . A proof of Theorem A has also been put on the arXiv , seeKuba, G:
Scattered and paracomapct order topologies. arXiv 2005.09368v1 (2020)Notice that every linearly ordered space is completely normal and that (cf. [1, 2.24]) everytotally disconnected (particularly, every scatterd) compact Hausdorff space is stronglyzero-dimensional.
Notice also that the size of a scattered, compact Hausdorff space alwayscoincides with its weight. In particular, 2 κ in Theorem A is maximal due to (1.2). Asalready pointed out in Section 9, in the excluded case κ = ℵ the statement in TheoremA would be true if and only if the Continuum Hypothesis were true.In the following we derive two consequences of Theorem A worth mentioning. (Notice thatalways κ κ = 2 κ and that if κ is singular then κ θ > κ for some infinite cardinal θ < κ .) Corollary A.
Let κ, θ be cardinals where either θ = 1 and κ > ℵ , or ℵ ≤ θ ≤ κ .Then there exist precisely κ totally disconnected, compact Hausdorff spaces of weight κ and size κ θ up to homeomorphism.Proof. Despite (1.2) the cardinality 2 κ is maximal due to the estimate |F | ≤ κ in theremark in Section 10. Thus it is enough to track down 2 κ mutually non-homeomorphicspaces as desired. The case that θ = 1 and κ > ℵ is settled by Theorem A. So weassume ℵ ≤ θ ≤ κ and, for the moment, κ > ℵ and we do not care whether κ θ = κ or κ θ > κ . (Of course, in view of Theorem A there is nothing to show if κ θ = κ since κ θ = κ implies κ ≥ c .)By virtue of Theorem A, we can define a family H κ of scattered, compact Hausdorffspaces of size (and weight) κ such that |H κ | = 2 κ and distinct spaces in H κ are neverhomeomorphic. Fix one space C ∈ H κ and consider the product space C θ . Clearly, C θ isa totally disconnected compact Hausdorff space of size κ θ and weight max { w ( C ) , θ } = κ . Furthermore, the space C θ is obviously dense in itself. For every H ∈ H κ consider thetopological sum H + C θ . So H + C θ is a totally disconnected compact Hausdorff spaceof weight max {| H | , w ( C θ ) } = κ and size max {| H | , | C θ |} = κ θ .For distinct and hence non-homeomorphic spaces H , H ∈ H κ the spaces H + C θ and H + C θ are not homeomorphic because every H ∈ H κ can be recovered from H + C θ in view of the obvious fact that C θ is the perfect kernel of H + C θ . To finishthe proof by settling the remaining case ℵ = θ = κ is left as a nice exercise. (UsingCantor derivatives it is not difficult to track down c mutually non-homeomorphic totallydisconnected compact subspaces of the real line.)By considering cones as in Section 6, it is straightforward to derive from Corollary A thefollowing noteworthy enumeration theorem about continua. Corollary B.
Let κ, θ be cardinal numbers where either ℵ ≤ θ ≤ κ or θ = 1 . Thenthere exist precisely κ pathwise connected, compact Hausdorff spaces of weight κ andsize max { c , κ θ } ..