Re: A topological property

William Elliot <marsh@xxxxxxxxxxxxxxxxxx> wrote:
From: Marc Olschok <invalid@xxxxxxxxxxx> wrote:

Just another question out of thin air:
consider for a space X and a point x in X the set
W(X,x) := { n in N | x is in a a pseudo-connected subset of size n }
and let W(X) be the union of all W(X,x).

Can every subset of N be realised as some W(X,x) or at least some
W(X) ? Perhaps already for X=N with some suitable topology?

Yicks, time to don oxygen mask. Note, 0,1 in W(X,x).
The exclude point topology for n points will give [0,n] /\ Z

Actually, it is rather easy (indicationg that my question was not so
deep at all):

Given any set X, fix some element p and take as nontrivial open sets
thoses subsets that have p as an element. Then every subset of X
is pseudo-connected.


If a set A is, as you put it pc or politically correct (or do
you mean politically connected?) is there a in A with pc A\a?

Sometimes. E.g. spaces like the above.


The converse thing, is there a in A with A\a not pc, fails.

Not at all.
Suppose A is only pc because it has a point a, that is in the
closure of A\a. Then removing may well result in a non-pc subset.
E.g. X = { a, b, c } with {b} and {c} the only nontrivial open sets.
Then X is pc but {b,c} is not.


--
Here's recap, summary and modernization of previous discussions.
It is equivalent and generalizes the original definition
which limited discussion to finite sets.
Have you any thing to add or suggest?

Not really. I just browsed it.
It seems that the 'assembled' subsets of S are exactly the
connected subspaces of _open_ subspaces of S. This might save
some terminology.

Marc
.