Saturated set
Encyclopedia
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 be a surjection; a set C in the domain of p is called saturated 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in particular in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
(X, τ) is saturated if it is an intersection of open subsets
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
of X. In a T1 space
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
every set is saturated.
Saturated sets can also be defined in terms of surjections: let p be a surjection; a set C in the domain of p is called saturated 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.