Definition:Scattered Space/Also defined as

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Definition

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


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.