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Saturated set

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In mathematics, particularly in the subfields of set theory and topology, a set is said to be saturated with respect to a function if is a subset of 's domain and if whenever sends two points and to the same value then belongs to (that is, if then ). Said more succinctly, the set is called saturated if

In topology, a subset of a topological space is saturated if it is equal to an intersection of open subsets of In a T1 space every set is saturated.

Definition

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Preliminaries

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Let be a map. Given any subset define its image under to be the set: and define its preimage or inverse image under to be the set:

Given the fiber of over is defined to be the preimage:

Any preimage of a single point in 's codomain is referred to as a fiber of

Saturated sets

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A set is called -saturated and is said to be saturated with respect to if is a subset of 's domain and if any of the following equivalent conditions are satisfied:[1]

  1. There exists a set such that
    • Any such set necessarily contains as a subset and moreover, it will also necessarily satisfy the equality where denotes the image of
  2. If and satisfy then
  3. If is such that the fiber intersects (that is, if ), then this entire fiber is necessarily a subset of (that is, ).
  4. For every the intersection is equal to the empty set or to

Related to computability theory, this notion can be extended to programs. Here, considering a subset , this can be considered saturated (or extensional) if . In words, given two programs, if the first program is in the set of programs satisfying the property and two programs are computing the same thing, then also the second program satisfies the property. This means that if one program with a certain property is in the set, all programs computing the same function must also be in the set).

In this context, this notion can extend Rice's theorem, stating that:

Let be a subset such that . If is saturated, then is not recursive.

Examples

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Let be any function. If is any set then its preimage under is necessarily an -saturated set. In particular, every fiber of a map is an -saturated set.

The empty set and the domain are always saturated. Arbitrary unions of saturated sets are saturated, as are arbitrary intersections of saturated sets.

Properties

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Let and be any sets and let be any function.

If or is -saturated then

If is -saturated then where note, in particular, that no requirements or conditions were placed on the set

If is a topology on and is any map then set of all that are saturated subsets of forms a topology on If is also a topological space then is continuous (respectively, a quotient map) if and only if the same is true of

See also

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References

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  1. ^ Monk 1969, pp. 24–54.
  • 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. Vol. 93. Cambridge University Press. ISBN 0-521-80338-1.
  • Monk, James Donald (1969). Introduction to Set Theory (PDF). International series in pure and applied mathematics. New York: McGraw-Hill. ISBN 978-0-07-042715-0. OCLC 1102.
  • Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.