Scattered space

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In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset. [1] [2] Equivalently, every nonempty subset A of X contains a point isolated in A.

Contents

A subset of a topological space is called a scattered set if it is a scattered space with the subspace topology.

Examples

Properties

(Proof: If C is a nonempty connected subset of X, it contains a point x isolated in C. So the singleton is both open in C (because x is isolated) and closed in C (because of the T1 property). Because C is connected, it must be equal to . This shows that every connected component of X has a single point.)
(Proof: Use the perfect + scattered decomposition and the fact above about second countable scattered spaces, together with the fact that a subset of a second countable space is second countable.)
Furthermore, every closed subset of a second countable X can be written uniquely as the disjoint union of a perfect subset of X and a countable scattered subset of X. [8] This holds in particular in any Polish space, which is the contents of the Cantor–Bendixson theorem.

Notes

  1. Steen & Seebach, p. 33
  2. Engelking, p. 59
  3. See proposition 2.8 in Al-Hajri, Monerah; Belaid, Karim; Belaid, Lamia Jaafar (2016). "Scattered Spaces, Compactifications and an Application to Image Classification Problem". Tatra Mountains Mathematical Publications. 66: 1–12. doi: 10.1515/tmmp-2016-0015 . S2CID   199470332.
  4. https://math.stackexchange.com/questions/3854864
  5. https://math.stackexchange.com/questions/376116
  6. Willard, problem 30E, p. 219
  7. https://math.stackexchange.com/questions/3856152
  8. https://math.stackexchange.com/questions/742025

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