The cardinality of the set of open subsets and related proofs. I have been going around in circles trying to prove this things for the last week, I would really appreciate any ideas on any of the next proofs. 
Let C be the linear continuum with no endpoints and D a dense countable subset of C.Let O be the set of all the open subsets of C. Show the next things:


*

*A subset of A in C is open iff A is the union of a set of open intervals of a linear continuum and extremities in D.

*$|O| = 2^{\aleph_0}$ 

*Every well ordered strictly increasing family of open sets is countable.

*For every $\delta < \omega_1 $ there is a strictly increasing family $\langle A_\xi: \xi < \delta \rangle$ of open subsets of C.
I only know that for the second one i should find an embedding into the rationals or something similar, but i can't figure out how to do it. 

These are a handful of questions that have been asked before. Let me provide a few helpful hints:


*

*One direction is trivial; for the other direction show that every open set is the union of open intervals, and every open interval is the union of open intervals with endpoints in $D$.

*Use the fact that if $U$ is open, then $U$ is fully determined by the countable set of intervals with endpoints in $D$ which are subsets of $U$.

*If $\langle A_\xi\mid \xi<\delta\rangle$ is an increasing sequence of open sets, then by the fact from the previous hint, this defines a well-ordered family of subsets of $\Bbb N$ ordered by $\subseteq$ which has order type $\delta$. Therefore $\delta<\omega_1$ (assign $\xi$ to least natural number which appears on the $\xi$-th set for the first time).

*Since every countable linear order embeds into $\Bbb Q$ (and therefore into $D$), embed $\delta$ into $D$ and use it to construct these open sets.
