scattered space

A topological spaceMathworldPlanetmath 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 closureMathworldPlanetmathPlanetmath 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 be the real line equipped with the usual topology T (formed by the open intervals). Let’s define a new topologyMathworldPlanetmath S on as follows: a subset A is open under S (AS) if A=BC, where B is open under T (BT) and C-, a subset of the irrational numbers. We make the following observations:

  1. 1.

    S is a topology on which is finer than T

  2. 2.

    is a Hausdorff space under S,

  3. 3.

    a singleton in is clopen iff it contains an irrational number

  4. 4.

    any subset of irrationals is scattered under the subspace topology of under S

  1. 1.

    First note that every element of T is an element of S, so ,S in particular. Suppose A1,A2S with A1=B1C1 and A2=B2C2, where Bi,Ci are defined as in the setup above. Then A1A2=BC, where B=B1B2T and C=(C1B2)((B1C1)C2) is a subset of the irrationals. So A1A2S. If AiS with Ai=BiCi, then Ai=BiCiS. So S is a topology which is finer than T

  2. 2.

    is Hausdorff under S is clear, the topological property is inherited from T.

  3. 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}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 setMathworldPlanetmath, so {x} is a subset of the irrationals, a contradictionMathworldPlanetmathPlanetmath.

  4. 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 collectionMathworldPlanetmath 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 unionMathworldPlanetmath of a perfect setMathworldPlanetmath 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