Saturated set

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In mathematics, in particular in topology, a subset of a topological space (X, τ) is saturated if it is an intersection of open subsets of X. In a T1 space every set is saturated.

Saturated sets can also be defined in terms of surjections: let p : XY be a surjection; a subset C of X is called saturated with respect to p if for every p−1(A) that intersects C, p−1(A) is contained in C. This is equivalent to the statement that p−1p(C)=C.


  • G. Gierz; K. H. Hofmann; K. Keimel; J. D. Lawson; M. Mislove & D. S. Scott (2003). "Continuous Lattices and Domains". Encyclopedia of Mathematics and its Applications. 93. Cambridge University Press. ISBN 0-521-80338-1. 
  • J. R. Munkres (2000). Topology (2nd Edition). Prentice-Hall. ISBN 0-13-181629-2.