scattered space



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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.


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

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


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

  • Discrete spaces are scattered.


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


Given a space XX, using the derivative :2 X2 X\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 AAA\subseteq\partial A. Sierpiński (1927) studies the abstract properties of dense-in-itself and corresponding scattered subsets relative to arbitrary monotone maps 2 X2 X2^X\to 2^X.


  • 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)

Revised on June 16, 2016 06:45:13 by Thomas Holder (