Assuming the axiom of choice, is there a well-ordering of the reals such that every initial segment is closed for the usual topology? If the continuum hypothesis helps, we can also assume it.


An initial segment is a set of the form $\{x : x<y\}$ for some $y$, according to the well-ordering $<$.



Well-ordering with a topological property
No. There cannot be a strictly increasing $\omega\_1$-sequence of closed sets in a topological space with a countable base. Their complements would be unions of open sets from the basis, and it is impossible to drop elements of a countable set uncountably many times. 


