Subject: Re: how to classify of prime topological spaces?
From: Kevin Iga
Date: Thu, 5 Jul 2007 10:24:31 -0700
On Jul 5, 2007, at 5:28 AM, darush wrote:
> Subject: how to classify of prime topological spaces?
> From: "darush aghababayeedehkordi"
> Date: Wed, 4 Jul 2007 04:21:36 -0700
>
> every topological space with one limit point is a prime topological
space( definition from weakipedia)
Question: do you want to classify Hausdorff spaces with this property, or
classify all spaces with or without the Hausdorff condition?
Some ideas: First, note that if X is a topological space and q is a point
in X, then q is a limit point of some subset of X if and only if {q} is
not open.
For the proof of this fact: if {q} is open, and A is a subset of X that
does not contain q, then X-{q} is a closed set containing A as a subset
that does not contain q.
Conversely, consider X-{q}. The closure of this set is either X-{q} or X.
It is X-{q} if and only if X-{q} is closed, which is true if and only if
{q} is open.
Therefore, in a space with the property you are describing, if p is the
unique limit point, then every point other than p is open. By taking
unions, we see that every subset of X that does not contain p is open.
The question remains: what are the open sets that contain p?
We can take the collection of open sets that contain p, and for each such
open set, remove p. The resulting collection is almost a topology on
X-{p}. I say "almost" because the empty set is not in the collection. If
we throw the empty set in, the result is a topology on X-{p}. It has the
additional property that no two open sets are disjoint (intersecting with
the empty set excepted). If we do not include the empty set, and use the
partial order on this collection given by subset inclusion, then the new
property says that meets exist. The Hausdorff condition is equivalent to
saying that there is no minimal element.
Kevin Iga