What is the name for the property that a subset of a set follows the same rules as the set?

Question

I have a set that follows a certain property and I want to say that the subsets of this set also follows the property. What is this called? I know that closure under an operation means that performing the operation on members of the set produces another member of the set, but that is not exactly the phrase I'm looking for.

For example, if given three points on a line (that's a set). Take any two points and they also lie on the line. Therefore, the subsets of the set has the same property as the set. Thus, the set has__?

Answer 1

I believe you are talking about a hereditary property. From Wikipedia:

“In mathematics, a hereditary property is a property of an object, that inherits to all its subobjects, where the term subobject depends on the context.”