Definition:Scattered Space

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

Then $T$ is scattered iff it contains no non-empty subset which is dense-in-itself.

That is, $T$ is scattered iff every non-empty subset $H$ of $S$ contains at least one point which is isolated in $H$.

Alternative Definition
$T$ is scattered iff it contains no non-empty closed set which is dense-in-itself.

That is, $T$ is scattered iff every non-empty closed set $H$ of $S$ contains at least one point which is isolated in $H$.

Also see

 * Equivalence of Definitions of Scattered Space

Note
According to, a topological space 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.