[] Topologies on a Finite Set

On Fri, 1 Sep 2006 mathmanmeister@xxxxxxxxx wrote:

I've done some looking and see that this has been asked many times
before, but I'm wondering if someone can help me understand the
relationship between this question and the partial orders on a
partition of a finite set.

The relationship between sci.math and Ask-a-Topologist, is not a
cross posting. It's a multi-posting. Oh impatient multiposter,
see your immediate replies at Ask-a-Topologist.

I am to...

a) Suppose P is a partition of a finite set X and that < is a partial
order on P. Show that the family of B = B(P, <) consisting of all sets
of the form

Union over all A <= A0 of A, with A0 a member of P

is a basis for a topology T(P, <) on X.

That sets of this form cover all of X is pretty clear to me, and I know
that if you intersect two of these things and A0 and A1, say, are
comparable, then their intersection will be the union over all A <=
min{A0, A1} of A, which is itself a basis element. The problem I'm
having here is saying something intelligent about the intersection of
two basis elements if A0 and A1 are NOT comparable. I've looked at
some simple examples and can see that, in general, the intersection is
not empty, but I'm not quite sure what to say about these non-empty
intersections. Any help?

b) Show that every topology on X is of the form T(P, <) for some
partition P on X and some partial order < on P.

It seems to me that if you choose P to be the partition containing
exactly 1 element in each subset of X, and choose < so that no elements
are comparable (not very interesting, I know), then any open set (say
{1,2,3}) is just (Union over A <= {1} - which is just {1} itself) union
(Union over A <= {2} - just {2}) union (Union over A <= {3} - just
{3}).

Sort of trivial, but does this answer the question?

c) Suppose P1, P2 are two partitions of X, and that <i is a partial
order on Pi, with i = 1, 2. Show that T(P1,<1) = T(P2, <2) if and only
if P1 = P2 and <1 = <2. Conclude the family of all topologies on X is
in one-to-one correspondence with the set of all pairs (P, <).

I'm pretty lost with this one - any help would be appreciated.


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