Suppose a topological space $X$ does not satisfy the countable chain condition (by that I mean that there exists an uncountable family of pairwise disjoint non-empty open subsets of $X$). Does it follow that there is a sequence of open sets such that their closures form a strictly increasing sequence with respect to inclusion?

  • $\begingroup$ Do you want the sequence to have uncountable length? $\endgroup$ – user642796 Mar 21 '14 at 10:24
  • $\begingroup$ No, by a sequence I mean a countable sequence. $\endgroup$ – Bach Mar 21 '14 at 10:25

Suppose you have any space $X$ in which there are pairwise disjoint nonempty open sets $\{ U_i : i \in \mathbb{N} \}$. I claim that $\langle \bigcup_{i=1}^n U_i \rangle_{n =1}^\infty$ is as required:

Given $n \geq 1$, note that $U_{n+1} \cap \bigcup_{i=1}^n U_i = \varnothing$, and so $U_{n+1} \cap \overline{ \bigcup_{i=1}^n U_i } = \varnothing$ (since $U_{n+1}$ is open, it is a witness for each $x \in U_{n+1}$ that $x \notin \overline{ \bigcup_{i=1}^n U_i }$), which implies that $\overline{ \bigcup_{i=1}^n U_i } \subsetneq \overline{ \bigcup_{i=1}^{n+1} U_i }$.

  • $\begingroup$ So according to this, much less then not having c.c.c. should suffice, right? $\endgroup$ – Bach Mar 21 '14 at 10:37
  • $\begingroup$ @Bach: Yes. Something must much less: I just assume that there is a countable family of pairwise disjoint nonempty open sets. $\endgroup$ – user642796 Mar 21 '14 at 11:15
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    $\begingroup$ @ArthurFischer which holds in every infinite Hausdorff space. $\endgroup$ – Henno Brandsma Mar 21 '14 at 21:49

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