I'm currently studying metric spaces through Gamelin and Greene's Introduction to Topology. While studying about completeness I got stuck with this concept of nowhere dense subset. The book defines a subset Y of X is nowhere dense if cl(Y) has no interior points i.e. int(cl(Y)) is empty. It then says that "evidently Y is nowhere dense if and only if X\cl(Y) is a dense open subset of X". This is the part I don't get. How can I prove the iff relationship between this statement and the original definition of a nowhere dense set?


It follows from the relation

$$X \setminus \operatorname{int} A = \operatorname{cl} (X\setminus A),$$

"the complement of the interior is the closure of the complement". (Prove that relation, if you're not yet familiar with it.)

Thus $\overline{Y}$ has empty interior if and only if $X \setminus \overline{Y}$ is dense.

  • 1
    $\begingroup$ Note that $x \in X \setminus \operatorname{int}A$ iff it is not the case that there is an open set containing $x$ that stays inside $A$. So every open set that contains $x$ also contains points not in $A$. And this just says that $x \in \operatorname{cl}(X \setminus A)$. Hence the equality. $\endgroup$ – Henno Brandsma Mar 14 '14 at 6:25
  • $\begingroup$ that was quite simple thanks a lot!! $\endgroup$ – nomadicmathematician Mar 23 '14 at 14:01
  • $\begingroup$ @HennoBrandsma Dear Heno - thanks for your comment which really clarified things for me. Regards, $\endgroup$ – user12802 Sep 16 '15 at 21:04

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