partial ordering in a topological space
is a preorder.
We call the specialization preorder on . If , then is called a specialization point of , and a generization point of . For any set ,
the set of all specialization points of points of is called the specialization of , and is denoted by ;
the set of all generization points of points of is called the generization of , and is denoted by .
. If is (http://planetmath.org/T0), then is a partial order.
This turns a topological space into a poset, where here is called the specialization order of the space.
Given a space, we have the following:
iff implies for any open set in .
if and , then . Since , we have , a contradiction. if , then for some closed set , we have and . But then , so that , a contradiction. ∎
, the lower set of . ( iff iff ).
|Title||partial ordering in a topological space|
|Date of creation||2013-03-22 16:35:02|
|Last modified on||2013-03-22 16:35:02|
|Last modified by||CWoo (3771)|