If the latter is true only for closed subspaces, then the property is called weakly hereditary or closed-hereditary.
For example, second countability and metrisability are hereditary properties.
Hereditary properties occur throughout combinatorics and graph theory, although they are known by a variety of names.
[2] Equivalently, a hereditary property is preserved by the removal of vertices.
is called hereditary if it is closed under induced subgraphs.
Sometimes the term "hereditary" has been defined with reference to graph minors; then it may be called a minor-hereditary property.
The Robertson–Seymour theorem implies that a minor-hereditary property may be characterized in terms of a finite set of forbidden minors.
The term "hereditary" has been also used for graph properties that are closed with respect to taking subgraphs.
[3] In such a case, properties that are closed with respect to taking induced subgraphs, are called induced-hereditary.
The most important result from this area is the unique factorization theorem.
[4] There is no consensus for the meaning of "monotone property" in graph theory.
[8] Some authors choose to resolve this by using the term increasing monotone for properties preserved under the addition of some object, and decreasing monotone for those preserved under the removal of the same object.
For instance, a family that is closed under taking matroid minors may be called "hereditary".
In planning and problem solving, or more formally one-person games, the search space is seen as a directed graph with states as nodes, and transitions as edges.
This notion can trivially be extended to more discriminating partitions by instead of properties, considering aspects of states and their domains.
Then no covering is possible any more, because the difference between number of uncovered white fields and the number of uncovered black fields is 2, and adding a domino tile (which covers one white and one black field) keeps that number at 2.
[9] In model theory and universal algebra, a class K of structures of a given signature is said to have the hereditary property if every substructure of a structure in K is again in K. A variant of this definition is used in connection with Fraïssé's theorem: A class K of finitely generated structures has the hereditary property if every finitely generated substructure is again in K. See age.
Recursive definitions using the adjective "hereditary" are often encountered in set theory.
Thus the notion of hereditary set is interesting only in a context in which there may be urelements.
[10][11] A property (of a set) is thus said to be hereditary if it is inherited by every subset.
For example, being well-ordered is a hereditary property, and so it being finite.