Hereditary property

In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of subobject depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory. In topology, a topological property is said to be hereditary if whenever a topological space has that property, then so does every subspace of it. 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. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary. Connectivity is not weakly hereditary. If P is a property of a topological space X and every subspace also has property P, then X is said to be "hereditarily P". The notion of hereditary properties occurs throughout combinatorics and graph theory, although they are known by a variety of names. For example, in the context of permutation patterns, hereditary properties are typically called permutation classes. In graph theory, a hereditary property is a property of a graph which also holds for (is "inherited" by) its induced subgraphs. Alternately, a hereditary property is preserved by the removal of vertices. A graph class is called hereditary if it is closed under induced subgraphs. Examples of hereditary graph classes are independent graphs (graphs with no edges), which is a special case (with c = 1) of being c-colorable for some number c, being forests, planar, complete, complete multipartite etc. In some cases, the term "hereditary" has been defined with reference to graph minors, but this is more properly 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.