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scattered space (Definition)

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 $ \lbrace x\rbrace$ is open (under $ S$) as well. So $ \lbrace x\rbrace$ is clopen. If $ x$ is rational and $ \lbrace x\rbrace\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 $ \lbrace x\rbrace$ is the empty set, so $ \lbrace x\rbrace$ 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 $ \lbrace r\rbrace$ 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.
$ \qedsymbol$
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.



"scattered space" is owned by CWoo.
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See Also: dense in-itself

Also defines:  scattered, scattered set, scattered line
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Cross-references: perfect set, disjoint union, collection, separating, open subset, point, contradiction, empty set, union, rational, closed, property, clear, Hausdorff, contains, iff, clopen, Hausdorff space, finer, irrational numbers, open intervals, usual topology, line, real, isolated, open, singleton, discrete space, subspace topology, subset, interior, closure, dense in itself, dense in, isolated points, closed subset, topological space
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This is version 3 of scattered space, born on 2007-02-20, modified 2007-02-22.
Object id is 8934, canonical name is ScatteredSpace.
Accessed 2226 times total.

Classification:
AMS MSC: 54G12 (General topology :: Peculiar spaces :: Scattered spaces)
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