Does this property have a name?

Question

When I have a set of numbers such as $\Bbb Z$ and say I have some $n\in \Bbb Z$, if I put some lower and upper bounds on $n$, that is $a>n>b$, then there are finite options for $n$.

Does this property, or sets that posses this property have a name?

Answer 1

An ordering with this property is called locally finite.

A total locally finite ordering will be either finite or order-isomorphic to one of $(\mathbb Z,{<})$, $(\mathbb N,{<})$, or $(\mathbb N,{>})$. There are more possibilities for a locally finite partial ordering, though.