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From: Dave L. Renfro on 13 Jul 2005 18:04
William Elliot wrote (in part, on 4 Jul 2005 21:55:50):

> From my readings, not that I ever read this one thru
> to the end: Unification approach to the separation axioms
> between T_0 and completely Hausdorff.
> arxiv.org/abs/math.GN/9810074

Thanks for the reference. I'm usually not very interested
(and this is quite an understatement) in the fringe type
stuff that deals with semi-locally quasi-sub-open T_(2/5)
pseudo-regular Kothe-Cech spaces of the 5'th type, but this
paper seems to have aspects in it that might be of interest
to me. At least, I've printed it out and filed it away
in my topology preprint/paper notebooks.

William Elliot wrote (in part, on 4 Jul 2005 21:55:50):

>> (a) T_c(A) = union{T(B): A subset B}
>
> Should this be intersection? Were T identity function
> I, then I_c is constant map I_c(A) = X. In general,
> T(X) subset T_c(A).

Ooops, I got this wrong. In this case, it was a
typo when I converted/summarized my manuscript into
ASCII for Usenet posting (i.e. I had intersection in
my LaTex manuscript).

Butch Malahide wrote (on 4 Jul 2005 22:22:49):

> I think you meant to say that Price used a hypothesis
> *implied by* Martin's axiom.

Yep, you're right. This wasn't a typo generated when I
converted/summarized my manuscript into ASCII for Usenet
posting either -- I had this misstated in my manuscript
version also.

William Elliot wrote (in part, on 5 Jul 2005 00:42:16):

>> (e) Let X be a set and define T:P(P(X)) -> P(P(X)) by
>> T(U) = {union(k=1 to inf): A_k in U for each k}
>> union {complement of A: A in U}
>
> This is muddled. U is a collection of subsets of X.
> Is U countable? Is P(X) countable? Are {,} inaccurate?
> My Guess:
> T(U) = { \/{ A | A in U }, \/{ X\A | A in U } }
> a two set element of PP(X).

T(U) is supposed to be the collection of all sets that
can be formed by taking countable unions and complements
of sets that belong (as elements) to U. Note this is not
the same thing as the collection of all sets _generated_
by the operations of countable union and complementation.
Also, since we're considering unions of _sequences_ of
elements belonging to U, we get finite unions as well
(i.e. A union B union C is equal to A union B union C
union C union C ...).

If you still think I have some details wrong, let me
know and I'll stick a note to this effect in my manuscript
to remind me to look into this matter when I get around
to working on it at some future time. (I explain later
why I don't have time to dig very deep into certain things
right now.)

William Elliot wrote (in part, on 5 Jul 2005 00:42:16):

>> 6. MONOTONE FUNCTIONS DO NOT HAVE TO PRESERVE UNIONS
>
> nor intersections

Yep. I suppose this could (and should) have been mentioned
in passing, even though my primary focus was on building
up to the Kuratowski closure axioms using the most direct
route that I could, meaning that the presence or absence
of preserving unions is a key issue but not the presence or
absence of preserving intersections (since even Kuratowski
closure functions don't have to preserve intersections).

Keith Ramsay wrote (in part, on 5 Jul 2005 16:44:02):

>> A set is impredicatively defined if membership to it is
>> defined by using a property or a condition that involves
>> all the elements of the set being defined.
>
> I suspect you have the right idea, but I wouldn't
> put it that way. A mathematical object S is defined
> impredicatively

Thanks. I've printed out your comments and filed them
away where I'll see them when I get around to working
on this at some future time. In the near future, however,
I'll be pretty busy because I'm getting ready to move to
another state for another job. One of the reasons I've
posted a few essays recently (e.g. my June 22 post on
hypervolumes of n-balls references, my June 4 post on
separate and joint continuity, my May 13 post on
intersections of collections measurable sets) is that
I've been going through and organizing a lot of the
stuff laying around in my office for this upcoming move,
and occasionally I'll happen to see a sci.math post
just when I need to take a break from these other
activities and which gives me an idea for writing one
of these essays. This explains why I'm so late with
the present replies, by the way.

Dave L. Renfro

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