I was thinking a bit more about the setting of my recent question about unions of chains of nowhere dense subsets of the reals and got stuck almost immediately on a follow-up question. I suspect that my difficulty is rooted in my profound ignorance of set theory, so I have abstracted the problem away from its original setting.
Suppose that $B$ is a family (set) of sets which is closed downward ($x \in B, y \subset x \Rightarrow y \in B$ - I don't expect this property to be essential) and closed under finite unions ($x,y \in B \Rightarrow x \cup y \in B$). Let $W$ be the family of all sets which are the union of some chain in $B$. I have two questions.
- If $x,y \in W$, does it follow that $x \cup y \in W$?
- If $C \subset W$ is a chain of sets, does it follow that $\bigcup C \in W$?
Some optional motivation: $B$ is intended to model the family of nowhere dense subsets of some topological space. If $M$ is the set of countable unions of sets from $B$ (ie. models the meagre subsets) then it is easy to see that $M$ inherits both of the abstracted closure properties of $B$ and is furthermore closed under countable unions. If we instead form the (larger) family $W$, I was wondering if something analogous holds. That is, I wondered if $W$ inherits both of the closure properties of $B$ (hence my 1st question - $W$ is obviously closed downward) and if it also picks up a new closure property reminiscent of the way in which we constructed it (hence my 2nd question).
I do not embarrass easily and am fully prepared to embrace the possibility that one or both of my queries has a laughably obvious answer! Thanks.