# Definition:Scattered Space

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

### Definition 1

A topological space $T = \left({S, \tau}\right)$ is scattered if and only if it contains no non-empty subset which is dense-in-itself.

That is, $T = \left({S, \tau}\right)$ is scattered if and only if every non-empty subset $H$ of $S$ contains at least one point which is isolated in $H$.

### Definition 2

A topological space $T = \left({S, \tau}\right)$ is scattered if and only if it contains no non-empty closed set which is dense-in-itself.

That is, $T = \left({S, \tau}\right)$ is scattered if and only if every non-empty closed set $H$ of $S$ contains at least one point which is isolated in $H$.

## Also defined as

According to 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology, a topological space $T$ is defined as scattered:

... if it contains no non-empty dense-in-itself subsets; ...

and it is immaterial whether those subsets are closed or not.

On the other hand, PlanetMath's definition specifically requires that in order for a space to be classified as scattered, only its closed subsets are required to contain one or more isolated points.

There are few other reliable definitions to be found (the concept can be found neither on Wikipedia nor even MathWorld), and when the concept is used at all, the definitions go either way.

However, it is apparent that the two definitions are equivalent, and so ultimately it does not matter which definition is used.

## Also see

• Results about scattered spaces can be found here.