I'm looking on the Monotone Convergence Theorem and asking myself whether it is the property of ANY order topology induced by some total order that is dedekind complete ( basis are the open-intervals of some ordered set ), and not only the order topology induced by the natural order on the reals.
We require the concept of Monotonicity, which is filled by the ordered set. We also require the concept of convergence, which should also make sense in any order topology. Finally, we require the concept of boundedness, and i assume that also makes sense in any order topology.
So, is it true that the monotone convergence theorem can be viewed, in the most generalized form, as a property of an order topology , and more specifically, order topology induced by Dedekind Complete total order iff order topology satisfying monotone convergence theorem ?
Thanks .
If a linearly ordered space is Dedekind complete, it satisfies the monotone convergence theorem. The converse, however, is false. Let $X=\omega_1+\omega_1^*$ with its order topology, where $\omega_1^*$ is $\omega_1$ with the opposite of its usual order. Then $X$ in its order topology satisfies the monotone convergence theorem, but $\langle\omega_1,\omega_1^*\rangle$ is a Dedekind cut that does not correspond to any element of $X$.