Correction to “Complete normality and metrization theory of manifolds” Peter Nyikos

The claim in this article [1] that the combination of SSA + PFA^{+} is shown in [2, p. 660] to be consistent, modulo large cardinals, is incorrect. Moreover, Paul Larson has shown that the SSA is even incompatible with MA(ω_{1}), and it is not known whether the weaker Axiom S is compatible with the PFA.

Fortunately, the topological results in [1] are all consistent. In fact, the PFA is already enough to imply every statement derived from the combination of SSA + PFA^{+} in the article, and it can also be shown that the large cardinal strength of PFA is not needed. The key to these new discoveries is the following ZFC theorem of [3]:

2.3. Theorem. Let X be a space which is either hereditarily normal (abbreviated T_{5}) or hereditarily strongly cwH, for which there are a continuous π : X → ω_{1} and a stationary subset S of ω_{1} such that the fiber π^{←}{σ} is countably compact for all σ ∈ S. Then X cannot contain an infinite family of disjoint closed countably compact subspaces with uncountable π-images.

This can be combined with the results of [1] in the following way.

1. In [1, Lemma 2.5] MA(ω_{1}) is used to show that if M is a hereditarily cwH nonmetrizable manifold, then M is of Type I. That is, M is the union of a strictly ascending ω_{1}-sequence of open subspaces M_{α}(α ∈ ω_{1}) such that M_{α} has Lindelöf closure contained in all M_{β} such that β > α.

2. In [1, Lemma 2.6] it is shown how M_{α} can be chosen so that M_{α} = ⋃{M_{ξ} : ξ < α\} whenever α is a limit ordinal, and so that each point of B_{α} = \overline{M_{α}} \ M_{α} is contained in a compact, connected, infinite subset K_{α} of B_{α} so long as dim(X) > 1. [Actually, compactness of K_{α} is not needed for the new proof.]

3. The following is implicit in the proof of Lemma 2.7 in [1]:

Lemma A. If CC_{22} holds and M is a locally compact space in which every countable subset has Lindelöf closure, and S is a stationary subset of ω_{1} and {x_{α} : α ∈ S} is a subset of M, then there is a stationary subset E of S such that either:

(1) {x_{α} : α ∈ E} is a closed discrete subspace of M, or

(2) every countable subset of {x_{α} : α ∈ E} has compact closure in M.

This is used in the proof of [1, Theorem 2.7], along with the axiom (which follows from the PFA, see [4, Corollary 6.6]) that every 1st countable perfect preimage of ω_{1} contains a copy of ω_{1}. These axioms are used there to produce a copy W of ω_{1} in any hereditarily cwH nonmetrizable Type I manifold M. For any such copy W = {p_{α} : α ∈ ω_{1}} the following set is a club: C_{W} = {α : p_{α} ∈ B_{α}}. Again using CC_{22}, a stationary subset S_{1} of C_{W} is produced along with points {q_{α} : α ∈ S_{1}}, such that such that F_{1} = cℓ{q_{α} : α ∈ S_{1}} is disjoint from W and countably compact and hence closed in M, and such that both p_{α} and q_{α} are contained in a connected subset K_{α} of B_{α} for all α ∈ S_{1}.

4. Also in the proof of [1, Theorem 2.7], assuming also the normality of M, a continuous real-valued function f from M to [0, 1] is constructed which is 0 on W and 1 on F_{1}. Since K_{α} is connected and meets both W and F_{1} whenever α ∈ S_{1}, this function f takes on all intermediate values on K_{α}.

In [1] it was shown that CC_{22} follows from PFA^{+}, but it can be derived just from the PFA, as explained in [3]. Also in [1], PFA^{+} was mis-stated. Correct statements can be found in [2] and [5].

Now comes the new proof of the main theorem of [1], with altered set-theoretic hypothesis:

Main Theorem. [PFA] Every T_{5}, hereditarily cwH manifold of dimension greater than 1 is metrizable.

From each K_{α}(α ∈ S_{1}) pick a point x_{α} so that f(x_{α}) is different from all f(x_{β}), β < α. Use the fact that M is cwH and the Pressing-Down Lemma to eliminate alternative (1) of Lemma A as in the proof of Theorem 2.7 of [1]. Alternative (2) then gives a stationary subset S of S_{1} such that every countable subset of {x_{α} : α ∈ S} has compact closure in M. In particular, the closure X of {x_{α} : α ∈ S} in M is countably compact and so is X ∩ B_{α} for all α ∈ ω_{1}.

Claim. The map π : X → ω_{1} which takes X ∩ B_{α} to α is continuous.

Once the claim is proven, we get a contradiction to Theorem 2.3 above as follows. The image under f of {x_{α} : α ∈ S} is an uncountable subset of [0, 1], hence it has 𝔠-many condensation points. For each condensation point p and each countable ordinal α_{0}, there is a strictly ascending sequence of ordinals ⟨α_{n} : n ∈ ω⟩ and points x_{α_{n}} ∈ K_{α_{n}} for n > 0 such that |p − f(x_{α_{n}})| < \frac{1}{n}.

Let α = sup{α_{n} : n ∈ ω}. Since X is countably compact, there is a point of X ∩ B_{α} which is sent to p by f. Thus the sets X ∩ f^{−1}{p} are a family of 𝔠-many disjoint closed countably compact sets with uncountable π-range.

⊢ Proof of Claim. If C is any closed subset of ω_{1}, then Y_{C} = ⋃{B_{γ} : γ ∈ C} is closed in M because M \ Y_{C} falls apart into the open sets M_{γ} \ \overline{M_{δ}} where δ and γ are successive members of C. We then get a natural map π^{∗} : Y_{C} → ω_{1} taking each B_{γ} to γ. This map is continuous because the preimage of each closed set is closed. If C is the closure of S in ω_{1}, then the map π of the Claim is the restriction of π^{∗} to X. ⊣

The foregoing proof allows us to slightly weaken the hypotheses on M in the main theorem: it is enough for M to be normal and hereditarily strongly cwH. [Recall that a space is termed strongly cwH if every closed discrete subspace D expands to a discrete collection of open sets U_{d} such that U_{d} ∩ D = {d} for all d ∈ D.] This is a weakening of hypotheses because every normal, cwH space is strongly cwH. It is an open problem whether normality can be dropped from this weakening. In [3] it is proven that it can be dropped under PFA + Axiom F, but it is not known whether this combination of axioms is consistent, even modulo large cardinals.

REFERENCES

[1] P. Nyikos, “Complete normality and metrization theory of manifolds,” Top. Appl. 123 (1) (2002) 181–192.

[2] S. Shelah, Proper and Improper Forcing, Perspectives in Mathematical Logic, Springer-Verlag, 1998.

[3] P. Nyikos, “Applications of some strong set-theoretic axioms to locally compact T_{5} and hereditarily scwH spaces,” Fund. Math. (2003).

[4] A. Dow, “Set theory in topology,” in: Recent Progress in General Topology, M. Hušek and J. van Mill, eds., Elsevier, 1992, 167–197.