Re: a property between collectionwise normality and hereditary collectionwise normality



I'd just like to correct a few mistakes I made (a summary of how my
questions stand is included at the end):

[Recall that property (*) is: every discrete-in-itself collection of
pairwise disjoint closed sets can be separated - where a collection of
subsets (A_i) is called "discrete-in-itself" whenever it is discrete
in the union of the A_i's.]

KP Hart in litteris <c0aae$47fdfa25$82a1d025$18090@xxxxxxxxxxxxxxxx>
scripsit:
David Madore wrote:
- Does it hold for the Tychonoff plank? This would give an example of
a non hereditarily normal (so non hereditarily collectionwise
normal) space with property (*). Failing that, is there such an
example?

Property (*) implies normality: if A and B are closed and disjoint
then {A,B} is a collection to which (*) can be applied.
So, no, the Tychonoff plank does not satisfy (*)

Actually, I thought "the Tychonoff plank" referred to the product
(\omega_1 + 1) * (\omega + 1) (that's what Steen & Seebach define it
to be). Apparently the standard terminology is that it is the product
minus the point (\omega_1, \omega). Sorry about the confusion.

I was hoping the product (\omega_1 + 1) * (\omega + 1) would be an
example of a normal but not hereditarily normal space satisfying
property (*).

But in any case my suggestion was doomed: I wrongly believed that a
finite product of ordinals would be hereditarily collectionwise
Hausdorff, and this is not the case (consider the discrete subset of
the Tychonoff-plank[-without-the-corner] consisting of points of the
form (\omega_1,n) for n in \omega, or (\alpha,\omega) for \alpha a
successor ordinal in \omega_1: this collection of points cannot be
separated). Since property (*) is stronger than being hereditarily
collectionwise Hausdorff, (\omega_1 + 1) * (\omega + 1) does not
satisfy property (*). So I don't have an example of a space
satisfying (*) without being hereditarily collectionwise normal
(question (A)).

- Is there a space which is collectionwise normal and hereditarily
collectionwise normal, yet fails to achieve property (*)?

You noted above that hCWN implies (*) and that (*) implies CWN, so the
answer seems to be `no'.

The second "normal" was a typo: what I meant to ask was: is there a
space which is collectionwise normal and hereditarily collectionwise
HAUSDORFF, yet fails to achieve property (*)? (Question (B).)

(As noted above, (\omega_1 + 1) * (\omega + 1) fails (*) but it fails
because it is not hereditarily collectionwise Hausdorff.)

........................................................

To summarize the implications:

hCWN => (*) => CWN
| |
v v
hCWH => CWH

A space which is CWN but not hCWH is given by the product
(\omega_1+1)*(\omega+1), so this shows the implication (*)=>CWN to be
strict.

A space which is hCWH but not CWN is provided by Fleissner's "George"
(W. G. Fleissner, "A normal, collectionwise Hausdorff, not
collectionwise normal space", General Topology and Appl. 6 (1976),
57-64), or a simpler one by example 2.9 of Nyikos & Porter's preprint
"Hereditarily strongly CWH and WD(\aleph_1) vis-a-vis other separation
axioms" (<URL: http://www.math.sc.edu/~nyikos/Sep1.pdf >). This shows
the implication (*)=>hCWH to be strict.

If property (*) is to be interesting, at least the following two
questions remain to be answered:

(A) Is there a space which is (*) but not hCWN?

(B) Is there a space which is CWN and hCWH but not (*)?

--
David A. Madore
(david.madore@xxxxxx,
http://www.madore.org/~david/ )
.

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