g-2 – Scattered Spaces

Publisher Summary A topological space (all topological spaces are assumed to be Hausdorff) is called scattered (dispersed) if its every non-empty subspace has an isolated point. A space is scattered if and only if (iff) it is right separated, so |X| ≤ (X) for scattered spaces. A scattered space is hereditarily disconnected. Locally compact scattered spaces are totally disconnected and so they are zero-dimensional, because in a compact space, the component and the quasi-component of a point are the same. A zero-dimensional scattered space need not be normal. Different variations of scatteredness are illustrated in this chapter—such as a topological space is C-scattered if for every closed subspace F (not null) there is a point x with a compact neighborhood contained in F. The notion of C-scatteredness is a simple simultaneous generalization of scatteredness and of local compactness. A rim-scattered space has a base each of whose elements has a scattered boundary. A space X is called σ -scattered if X = ⋃{Xn: n ∈ ω}, where each Xn is scattered. A space is Gω-scattered if every subspace contains a point that is a (relative) Gδ. A space is N-scattered iff nowhere dense subsets are all scattered.