A topological space is said to be scattered if for every closed subset of , the set of isolated points of is dense in . Equivalently, is a scattered space if no non-empty closed subset of is dense in itself: for every closed subset of , the closure of the interior of is not .
A subset of a topological space is called scattered if it is a scattered space with the subspace topology.
Scattered line. Let be the real line equipped with the usual topology (formed by the open intervals). Let’s define a new topology on as follows: a subset is open under () if , where is open under () and , a subset of the irrational numbers. We make the following observations:
is a topology on which is finer than
is a Hausdorff space under ,
a singleton in is clopen iff it contains an irrational number
any subset of irrationals is scattered under the subspace topology of under
First note that every element of is an element of , so in particular. Suppose with and , where are defined as in the setup above. Then , where and is a subset of the irrationals. So . If with , then . So is a topology which is finer than
is Hausdorff under is clear, the topological property is inherited from .
First, any singleton is closed since is Hausdorff under . If is irrational, then is open (under ) as well. So is clopen. If is rational and , then it is the union of a -open set and a subset of the irrationals. The only -open subset of is the empty set, so is a subset of the irrationals, a contradiction.
The real line under the topology is called a scattered line.
|Date of creation||2013-03-22 16:42:59|
|Last modified on||2013-03-22 16:42:59|
|Last modified by||CWoo (3771)|