scattered space

A topological space $X$ is said to be scattered if for every closed subset $C$ of $X$ , the set of isolated points of $C$ is dense in $C$ . Equivalently, $X$ is a scattered space if no non-empty closed subset of $X$ is dense in itself: for every closed subset $C$ of $X$ , the closure of the interior of $C$ is not $C$ .

A subset of a topological space is called scattered if it is a scattered space with the subspace topology.

Every discrete space is scattered, since every singleton is open, hence isolated.

Scattered line. Let $\mathbb{R}$ be the real line equipped with the usual topology $T$ (formed by the open intervals). Let’s define a new topology $S$ on $\mathbb{R}$ as follows: a subset $A$ is open under $S$ ( $A\in S$ ) if $A=B\cup C$ , where $B$ is open under $T$ ( $B\in T$ ) and $C\subseteq\mathbb{R}-\mathbb{Q}$ , a subset of the irrational numbers. We make the following observations:

1.

$S$ is a topology on $\mathbb{R}$ which is finer than $T$
2.

$\mathbb{R}$ is a Hausdorff space under $S$ ,
3.

a singleton in $\mathbb{R}$ is clopen iff it contains an irrational number
4.

any subset of irrationals is scattered under the subspace topology of $\mathbb{R}$ under $S$

Proof.

1.

First note that every element of $T$ is an element of $S$ , so $\varnothing,\mathbb{R}\in S$ in particular. Suppose $A_{1},A_{2}\in S$ with $A_{1}=B_{1}\cup C_{1}$ and $A_{2}=B_{2}\cup C_{2}$ , where $B_{i},C_{i}$ are defined as in the setup above. Then $A_{1}\cap A_{2}=B\cup C$ , where $B=B_{1}\cap B_{2}\in T$ and $C=(C_{1}\cap B_{2})\cup((B_{1}\cup C_{1})\cap C_{2})$ is a subset of the irrationals. So $A_{1}\cap A_{2}\in S$ . If $A_{i}\in S$ with $A_{i}=B_{i}\cup C_{i}$ , then $\bigcup A_{i}=\bigcup B_{i}\cup\bigcup C_{i}\in S$ . So $S$ is a topology which is finer than $T$
2.

$\mathbb{R}$ is Hausdorff under $S$ is clear, the topological property is inherited from $T$ .
3.

First, any singleton is closed since $X$ is Hausdorff under $S$ . If $x$ is irrational, then $\{x\}$ is open (under $S$ ) as well. So $\{x\}$ is clopen. If $x$ is rational and $\{x\}\in S$ , then it is the union of a $T$ -open set $B$ and a subset $C$ of the irrationals. The only $T$ -open subset of $\{x\}$ is the empty set, so $\{x\}$ is a subset of the irrationals, a contradiction.
4.

Let $C$ is a subset of the irrational numbers. and considered the subspace topology under $S$ . Then every point $r$ of $C$ is isolated, since $\{r\}$ is the open subset of $C$ separating it from the rest. The closure of the collection of these points is clearly $C$ itself, so $C$ is scattered.

∎

The real line under the topology $S$ is called a scattered line.

Remark. Every topological space is a disjoint union of a perfect set and a scattered set.

Title	scattered space
Canonical name	ScatteredSpace
Date of creation	2013-03-22 16:42:59
Last modified on	2013-03-22 16:42:59
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 54G12
Related topic	DenseInItself
Defines	scattered
Defines	scattered set
Defines	scattered line