Let $\langle M_i:i<\theta\rangle$ be an increasing chain of Banach spaces, where each $M_i$ has density character $\mu$ (i.e.,the mininum cardinality of a dense subset of $M_i$ is $\mu$). Let $B_i\subset M_i$ be a dense subset of $M_i$ of cardinality $B_i$. Notice that $\bigcup_{i<\theta}B_i$ is a dense subset of $\overline{\bigcup_{i<\theta}M_i}=:M$, so $densitycharacter(M)\le \mu$. Is it possible to prove that $densitycharacter(M)=\mu$? Thank you.
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closed as too localized by Bill Johnson, Andreas Blass, George Lowther, Matthew Daws, Ryan Budney Nov 3 '11 at 22:05This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 


If $X$ is any metric space and $Y$ is any subspace of $X$ then $dc(Y) \leq dc(X)$. 

