We start this section by recalling a few well-known definitions and introducing some related notation.A spaceXis said to be aD-space if for any neighbourhood assignment φdefined onX there is a closed discrete setD ⊂Xsuch that ∪{φ(x) :x∈ D}=X.
For any spaceX we set
• D(X) = min{|A| :X=∪AandAis aD-space for eachA∈ A }.
• ls(X) = min{|A|:X =∪AandAis left-separated for eachA∈ A }.
(Note that bothD(X)and ls(X)can be finite.)
It was shown in [70] that left-separated spaces are D-spaces, hence we have D(X) ≤ ls(X)for anyX.
In [62], M. Tkaˇcenko proved the following remarkable result: IfX is a countably com-pactT3-space withls(X)≤ωthen
(i) Xis compact, (ii) Xis scattered, (iii) Xis sequential.
It is easy to see that if in a scattered compactT2-space any countably compact sub-space is compact then it is sequential, hence (iii) immediately follows from (i) and (ii), although this is not how (iii) was proved in [62].
The aim of this section is to improve (i) and (ii) as follows:
(A) Any countably compact spaceXwithD(X)≤ωis compact.
(B) IfX is compactT2 withls(X)< N(R)thenXis scattered.
Here N(R) denotes the Novák number of the real line R, i.e. the covering number cov(M)of the idealMof all meager subsets ofR.
If X is any crowded (i.e. dense-in-itself) space and Y ⊂ X then we denote by N(Y, X)the relative Novák number ofY inX, that is the smallest number of nowhere
dense subsets of X needed to coverY. In particular, N(X) = N(X, X)is the Novák number ofX.
We should also mention that a weaker version of statement (A), in whichD(X)< ω is assumed instead ofD(X)≤ω, has been established in [20].
Similarly as in [62], we can actually prove the following higher-cardinal generaliza-tion of statement (A).
Theorem 1.14. Letκbe any infinite cardinal andXbe initiallyκ-compact withD(X)≤ κ. ThenX is actually compact.
The proof of Theorem 1.14 is based on the following lemma that may have some independent interest in itself.
Lemma 1.15. Let X be any space and Y ⊂ X its D subspace. If ρ is a regular cardinal such that X has no closed discrete subset of sizeρ(i.e. e(X)ˆ ≤ ρ), moreover U = {Uα : α ∈ ρ}is a strictly increasing open cover of X then there is a closed set Z ⊂X such thatZ∩Y =∅andZ 6⊂Uα for allα∈ρ.
Proof.If there is anα ∈ ρwithY ⊂ Uα thenZ = X −Uα is clearly as required. So assume from here on thatY 6⊂Uαfor allα∈ρ.
For every pointy ∈Y letα(y)be theminimalordinalαsuch thaty ∈Uαand then consider the neighbourhood assignmentφonY defined by
φ(y) = Uα(y).
SinceY is aD-space there is a setE ⊂Y, closed and discrete inY, such thatY ⊂φ[E].
We claim thatZ =E0, the derived set ofE, is now as required.
Indeed, Z is closed in X and Z ∩Y = ∅ as E has no limit point withinY. It remains to show thatZ 6⊂ Uαfor allα ∈ρ. Assume, indirectly, thatZ ⊂ Uαfor some α ∈ ρ. Note first that for any point y ∈ Y ∩Uα we have α(y) ≤ α, consequently φ[E ∩Uα] ⊂ Uα. On the other hand, Z = E0 ⊂ Uα implies that E −Uα is closed discrete inX, hence|E−Uα|< ρ by our assumption. But then
β = sup{α(y) :y∈E−Uα}< ρ becauseρis regular, consequently we have
Y ⊂φ[E] =φ[E∩Uα]∪φ[E−Uα]⊂Uα∪Uβ =Umax{α,β}, contradicting that no member ofU coversY.
Now, we can turn to the proof of our theorem.
Proof.It suffices to prove that for no regular cardinalρis there a strictly increasing open cover ofX of the form U = {Uα : α ∈ ρ}. For ρ ≤ κ this is clear, for X is initially
κ-compact. So assume now thatρ > κ, and assume indirectly thatU = {Uα :α ∈ ρ} is a strictly increasing open cover ofX. Note also thatX has no closed discrete subset of sizeρ > κbecauseX is initiallyκ-compact.
By D(X) ≤ κ we have X = ∪{Yν : ν ∈ κ}, where Yν is a D subspace of X for eachν ∈ κ. Using lemma1.15then we may define by a straightforward transfinite recursion onν ∈κclosed setsZν ⊂Xsuch that for eachν ∈ κwe haveZν∩Yν =∅, Zν 6⊂Uαfor allα∈ρ, moreoverν1 < ν2impliesZν1 ⊃Zν2. In this we make use of the fact that ifν < κand{Zη : η ∈ ν}is a decreasing sequence of closed sets inX such that∩{Zη :η ∈ν} ⊂ U for some openU ⊂ Xthen there is anη ∈νwithZη ⊂ U as well, using again the initialκ-compactness ofX.
But then, applying once more thatXis initiallyκ-compact, we conclude that
∩{Zν :ν ∈κ} 6=∅, contradicting thatX =∪{Yν :ν ∈κ}.
It should be noted that in the above result no separation axiom is needed. This is in contrast with Tkaˇcenko’s result from [62].
Let us now turn to our second statement (B). Again, we need to first give a prepara-tory result. For this we recall the cardinal functionδ(X)that was introduced in [72]:
δ(X) = sup{d(S) : S is dense in X}.
Let us note here that ifX is a compactT2-space thenδ(X) = π(X), as was shown in [33].
Lemma 1.16. Assume thatXis an arbitrary crowded topological space andY ⊂Xis its left-separated subspace. Then we have
N(Y, X)≤δ(X), consequently
N(X)≤ls(X)·δ(X).
Proof.We shall proveN(Y, X)≤δ(X)by transfinite induction on the order type of the well-ordering that left-separatesY. So assume that≺is a left-separating well-ordering ofY such that ifZ is any proper initial segment ofY, w.r.t.≺, thenN(Z, X)≤δ(X).
Let Gbe the union of all those open setsU in X for which Y (or more precisely:
U ∩Y) is dense inU. Clearly, thenY \G is nowhere dense inX andY ∩Gis dense inG. The latter then implies
d(Y ∩G)≤δ(G)≤δ(X).
On the other hand, since≺left-separatesY ∩G, any dense subset ofY ∩Gmust be cofinal inY ∩G w.r.t.≺, hence we clearly have
cf(Y ∩G,≺)≤d(Y ∩G)≤δ(X).
But any proper ≺-initial segment of Y ∩ Gmay be covered by δ(X) many nowhere dense sets, by the inductive hypothesis, hence we have
N(Y, X)≤1 +δ(X)·δ(X) = δ(X),
becaused(X)and soδ(X)is always infinite by definition. The second part now follows immediately.
Note that again absolutely no separation axiom was needed in the above result. How-ever, in the proof of the following theorem the assumption of Hausdorffness is essential.
Theorem 1.17. LetX be a compactT2-space satisfyingls(X) < N(R). ThenX must be scattered.
Proof.We actually prove the contrapositive form of this statement. So assume thatX is not scattered, then it is well-known that some closed subspace F ⊂ X admits an irreducible continuous closed mapf :F →Conto the Cantor setC.
It is also well-known and easy to check that then we have δ(F) = δ(C) = ω, moreoverN(F) = N(C) = N(R)> ω. But then from lemma1.16we conclude that
ls(X)≥ls(F) = ls(F)·ω≥N(F) =N(R).
We would like to mention that 1.16 and 1.17 were motivated by the treatment of Tkaˇcenko’s results given in [60]. We also point out that theorems1.14 and1.17 yield a slight strengthening of Tkaˇcenko’s theorem in that the T3 separation axiom may be replaced by T2 in it. This is new even in the case of left-separated spaces (i. e. the assumptionls(X) = 1) that preceded Tkaˇcenko’s result in [17].
Corollary 1.18. LetX be a countably compactT2space that satisfiesls(X)≤ω. Then Xis compact, scattered, and sequential.
We finish by formulating a couple of natural problems concerning our results.
Problem 1.19. Is the upper bound N(R) in theorem 1.17 sharp? Can it actually be replaced by the cardinality of the continuum (in ZFC, of course)?
Note that as metric or compact spaces are allD-spaces, in theorem1.17one clearly cannot replacels(X) with D(X). Also, a compact (D-)space may fail to be sequen-tial. Being left-separated, however, is clearly a hereditary property, hence left-separated spaces are actually hereditaryD-spaces. Thus the following problems may be raised.
Problem 1.20. Is a compactT2hereditaryD-space sequential? Does it contain a point of countable character?
Concerning this problem we note that it follows easily from theorem 1.14 that a compactT2-spaceXsatisfyingD(Y)≤ωfor allY ⊂X has countable tightness.