nLab scattered space

Idea

A scattered space is a topological space that on the scale of ‘connectivity’ is at the extreme opposite of perfect spaces.

Scattered spaces are used to provide topological models of provability logic.

Definition

A subset $A$ of a topological space is called dense in itself if each point of $A$ is a limit point of $A$.

A topological space $X$ is called scattered if $X$ doesn’t contain nonempty dense-in-itself subsets.

Example

• The Sierpinski space $X=\{a,b\}$ with topology $\{\emptyset ,\{a\}, X\}$ is scattered: whereas $b$ is a limit point of $X$, $a$ isn’t, whence $X$ fails to be dense in itself. $\{a\}$ fails to be dense in itself since, whereas $b$ is a limit point, $a$ isn’t. Finally, $\{b\}$ has no limit points at all.

• Discrete spaces are scattered.

Properties

• Every topological space $X$ is the disjoint union of a scattered subset and a perfect subset.

Remark

Given a space $X$, using the derivative $\partial : 2^X\to 2^X$ that maps a subset to the set of its limit points, the property of being dense-in-itself amounts to $A\subseteq\partial A$. Sierpiński (1927) studies the abstract properties of dense-in-itself and corresponding scattered subsets relative to arbitrary monotone maps $2^X\to 2^X$.

Related entries

• scattered topos

• perfect space

• Magari algebra

• provability logic

Reference

• L. Beklemishev, D. Gabelaia, Topological interpretations of provability logic , arXiv:1210.7317 (2012). (abstract)

• W. Sierpiński, La notion de derivée comme base d’une théorie des ensembles abstraits , Math. Ann. 97 (1927) pp.321-337. (gdz)

• S. Willard, General Topology , Addison-Wesley Reading 1970. (Dover reprint 2004, p.219)