Re: Topologies on a Finite Set
- From: William Elliot <marsh@xxxxxxxxxxxxxxxxxx>
- Date: Sun, 3 Sep 2006 03:06:16 -0700
From: mathmanmeister@xxxxxxxxx
Ask-a-Topologist, as is often the case at night, is temporarily not
connecting, so I'll post here instead of there.
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
That is incorrectly stated. Order theory notation is
down A0
where down is a one character downward pointing arrow.
So what you're almost describing is \/down A0
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?
Do note that the intersection can be empty. For examle b /\ c
below, using lower to upper for <= (view monospace font)
a
/\
b c
down b /\ down c = nulset. Here's another one.
a b c <= a,b; d <= a,b
|\/| down a /\ down b = { c,d } isn't of the form down x
|/\| However { c,d } = down c \/ down d
c d
That's to give you a picture.
You want to use the full precise and detailed version:
B is a base when B covers space and
for all U,V in B, x in U /\ B,
some W in B with x in W subset U /\ B.
equivalently
for all U,V in B, some C subset B with U /\ V = \/C.
This equivalent part was demostrated with my second example.
(Make adjustments from down a to \/down A)
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?
No. It answers the question for when X is discrete topology.
It doesn't answer the question for all other topologies for X.
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.
It'd help were you to explain your notation as you go. What T(P,<=) ?
The topology generated with the base from P and <= as described above?
The hard part is to show T is an injection. The rest follows directly
from a) and b). Is this an actual problem from a text or mostly
speculation?
----
.
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- Topologies on a Finite Set
- From: mathmanmeister
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