# Mathematical objects

<https://en.wikipedia.org/wiki/Mathematical_object>

Mathematical object primitive set number space structure axiom theorem lemma theory conjecture

**Set** is is an unordered collection of distinct mathematical objects that share a common property. Despite the fact that sets are the basis of mathematics, with nearly all mathematical objects resembling some kind of a set, set remains undefined, it is taken as a mathematical primitive.

**Mathematical structure** on a set is an additional structural object that, in some manner, attaches (or relates) to that set, endowing it with some extra meaning or significance. Structure-preserving relations map structures in domain to equivalent structures in codomain: **homomorphisms** preserve algebraic structures, **homeomorphisms** preserve topological structures, **diffeomorphisms** preserve differential structures.

## Mathematical space

**Mathematical space** is a universal set (universe) with an added structure. Math deals with many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, probability spaces, it doesn't define the notion of "space" itself; it is accepted as a mathematical primitive.

## List of mathematical objects

concept notion element collection member set multiset (mset, bag, list, bunch, heap, sample, weighted set, collection, suite) ordered set list sequence