a property between collectionwise normality and hereditary collectionwise normality

Consider the following property (*) of a (T1) topological space X
(which I came across as an incorrect definition of collectionwise
normality):

(*) If (F_i) is a collection of pairwise disjoint closed sets of X
such that every F_i is open in the union of the F_i, then
there exists a collection (U_i) of pairwise disjoint open
sets, with same index set, such that each F_i is a subset of
the corresponding U_i.

Or, more tersely:

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 (in the sense that
any point of the union of the A_i's has a neighborhood which meets
only the A_i to which it belongs).

Property (*) is stronger than collectionwise normality, which states
that any discrete[-in-the-whole] family of disjoint closed sets can be
separated (that is, we make the stronger assumption that the union of
any subcollection of the closed sets is closed). To emphasize the
difference, a collection of points is "discrete[-in-the-whole]" when
the set of these points is discrete and closed, whereas it is
"discrete-in-itself" when the set is discrete for the induced
topology.

On the other hand, property (*) is implied by hereditary
collectionwise normality, or, in fact, by the property that any open
subspace is collectionwise normal. (This is easy: consider the
closure of the union of the F_i's minus the union of the F_i's: this
is also the set of limit point of the union of the F_i's and it is
closed; separating the F_i's in the complement of this closed set
separates them in X.)

The analogous property for families of singletons (note: I assume all
topological spaces are T1) is clear: to say that any
discrete[-in-the-whole] collection of singletons can be separated is
called "collectionwise Hausdorff", and to say that any
discrete-in-itself collection of singletons can be separated
(analogous to (*)) is exactly equivalent to being hereditarily
collectionwise Hausdorff (or, again, to every open subspace being
collectionwise Hausdorff).

So:

- Does property (*) have a standard name? Failing that, what would be
a natural name for it?

- 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?

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

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

.

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