Scattered Space

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

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

Examples

* 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 $\R^2$ 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.

Properties

* 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 $\$. 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. Note that the T₀ assumption is necessary here. For example, if $X=\$ with the indiscrete topology, $\$ and $\$ are both scattered, but their union, $X$, is not scattered as it has no isolated point.
* Every T₁ scattered space is totally disconnected. (''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.)
* Every second countable scattered space is countable.
* Every topological space ''X'' can be written in a unique way as the disjoint union of a perfect set and a 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. (''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''. This holds in particular in any Polish space, which is the contents of the Cantor–Bendixson theorem.

Notes

References

* Engelking, Ryszard, ''General Topology'', Heldermann Verlag Berlin, 1989.
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* {{Citation , last=Willard , first=Stephen , title=General Topology , origyear=1970 , publisher=Addison-Wesley , edition=Dover reprint of 1970 , year=2004

Properties of topological spaces