Re: Topologies on a Finite Set

Just following up here...

First of all, I didn't mean to be an irritant by posting this in two
places. This had scrolled back to page 5 or 6 with no replies so I
didn't figure the chances of anyone replying were very good at that
point.

Second, this problem is not from a textbook, nor is it any sort of
speculation on my part...it was just written up by a professor as an
exercise. That may explain some of the notation that is apparently
non-standard.

Now, on part C, yes, what is meant by T(P1, <1) and T(P2, <2) is the
topology generated by the basis described in part A, given the
partition P1 and partial order <1 on P1, and the partition P2 and
partial order <2 on P2.

Also, I didn't mean to indicate that the intersection of two basis
elements from part A was non-empty in the general case (even though
that IS what I said)...I meant that it is not ALWAYS empty. So I was
clear on that, thanks.

OK, so I'm wrong on part B. I don't mean to be dense, but I don't
understand what is meant by this:

Partition X with the equivalence relation x ~ y when cl {x} = cl {y}
and order P with A <= B when some x in A, y in B with x in cl {y}.

Well on second thought, as you've seemingly reversed the order,
it could be that you may want y in cl {x}.

What is cl {x}?

Any further help with B and C is appreciated, or pointing out a
reference where this argument is made in detail (I did plenty of
looking before posting this and didn't have any luck).

Thanks.

.

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