Linear continuum

Remarks

The real line $\R$ is a linear continuum: clause 1 is the completeness of $\R$, and for clause 2 one takes $z = (x + y)/2$. The integers $\Z$ satisfy clause 1, but they fail clause 2, since there is no integer strictly between $1$ and $2$. The rationals $\Q$ satisfy clause 2 but fail clause 1: the set $\set{q \in \Q : q^2 < 2}$ is bounded above but has no least upper bound in $\Q$. The square $[0, 1] \times [0, 1]$ with the dictionary order is a linear continuum that is not order isomorphic to an interval of $\R$. A linear continuum and each of its intervals are connected in the order topology.