scattered space

• Every discrete space is scattered.
• Every ordinal number with the order topology is scattered. Indeed, every nonempty subset A contains a minimum element, and that element is isolated in A.
• A space X with the particular point topology, in particular the Sierpinski space, is scattered. This is an example of a scattered space that is not a T₁ space.
• The closure of a scattered set is not necessarily scattered. For example, in the Euclidean plane $\backslash R^2$ take a countably infinite discrete set A in the unit disk, with the points getting denser and denser as one approaches the boundary. For example, take the union of the vertices of a series of n-gons centered at the origin, with radius getting closer and closer to 1. Then the closure of A will contain the whole circle of radius 1, which is dense-in-itself.

• In a topological space X the closure of a dense-in-itself subset is a perfect set. So X is scattered if and only if it does not contain any nonempty perfect set.
• Every subset of a scattered space is scattered. Being scattered is a hereditary property.
• Every scattered space X is a T₀ space. (Proof: Given two distinct points x, y in X, at least one of them, say x, will be isolated in $\backslash \{x,y\backslash \}$. That means there is neighborhood of x in X that does not contain y.)
• In a T₀ space the union of two scattered sets is scattered.See proposition 2.8 in {{cite journal | doi=10.1515/tmmp-2016-0015|title=Scattered Spaces, Compactifications and an Application to Image Classification Problem | year=2016 | last1=Al-Hajri|first1=Monerah|last2=Belaid|first2=Karim|last3=Belaid|first3=Lamia Jaafar|journal=Tatra Mountains Mathematical Publications|volume=66|pages=1–12|s2cid=199470332|doi-access=free}}{{Cite web | url=https://math.stackexchange.com/questions/3854864|title=General topology - in a $T_0$ space the union of two scattered sets is scattered}} Note that the T₀ assumption is necessary here. For example, if $X=\backslash \{a,b\backslash \}$ with the indiscrete topology, $\backslash \{a\backslash \}$ and $\backslash \{b\backslash \}$ are both scattered, but their union, $X$, is not scattered as it has no isolated point.
• Every T₁ scattered space is totally disconnected. {{pb}} (Proof: If C is a nonempty connected subset of X, it contains a point x isolated in C. So the singleton $\backslash \{x\backslash \}$ 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 $\backslash \{x\backslash \}$. This shows that every connected component of X has a single point.)
• Every second countable scattered space is countable.{{Cite web|url=https://math.stackexchange.com/questions/376116|title = General topology - Second countable scattered spaces are countable}}
• Every topological space X can be written in a unique way as the disjoint union of a perfect set and a scattered set.Willard, problem 30E, p. 219{{Cite web|url=https://math.stackexchange.com/questions/3856152|title=General topology - Uniqueness of decomposition into perfect set and scattered set}}
• Every second countable space X can be written in a unique way as the disjoint union of a perfect set and a countable scattered open set. {{pb}} (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.) {{pb}} 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.{{Cite web|url=https://math.stackexchange.com/questions/742025|title = Real analysis - is Cantor-Bendixson theorem right for a general second countable space?}} This holds in particular in any Polish space, which is the contents of the Cantor–Bendixson theorem.

{{reflist}}

• Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. {{ISBN|3-88538-006-4}}
• {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | origyear=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995 }}
• {{Citation | last=Willard | first=Stephen | title=General Topology | origyear=1970 | publisher=Addison-Wesley | edition=Dover reprint of 1970 | year=2004}}

Category:Properties of topological spaces