Saturated set

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.

An alternative definition for saturated sets comes from surjections, these definitions are not equivalent: let p : X → Y 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.

References

• 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.