3.1 Dense and Nowhere Dense Sets
Definition 3.7. Let \(X\) be a metric space. A subset \(E\) of \(X\) is said to be dense in \(X\) if \(\overline {E}=X\).
Example 3.8. The set \(\mathbb {Q}\) of all rational numbers is dense in \(\mathbb {R}\), \(\hspace {0.2cm}\overline {\mathbb {Q}}=\mathbb {Q}\cup \mathbb {I}_{rr}=\mathbb {R}\hspace {0.2cm}\). \(\mathbb {Q}\) is dense in \(\mathbb {R}\) because every
irrational number is a limit of a sequence of rational numbers.
e.g
\[\sqrt {2}=1.41421\dots =1+0.4+0.01+0.004+\dots \]
consider the sequence \(\hspace {0.2cm}\displaystyle {\Big \{1,1+\frac {2}{5},1+\frac {2}{5}+\frac {1}{100},1+\frac {2}{5}+\frac {1}{100}+\frac {4}{1000}+\dots \Big \}}\)
Definition 3.9. A subset \(E\) of \(\mathbb {R}\) is said to be nowhere dense in \(\mathbb {R}\) if \(\overline {E}\) contains no non-empty open intervals.
Note. A closed set \(E\) of \(\mathbb {R}\) is nowhere dense if \(E\) itself contains no open intervals.
Example 3.10. The set \(\mathbb {N}=\{1,2,3,\dots \}\) of positive integers is nowhere dense in \(\mathbb {R}\).
Solution. \(\mathbb {N}\) is closed in \(\mathbb {R}\): its complement is the union of the open intervals \((-\infty ,1)\) and \((n,n+1)\) for \(n\in \mathbb {N}\). So \(\overline {\mathbb {N}}=\mathbb {N}\).
Its interior is empty, since any interval \((n-r,n+r)\) contains non-integers. Hence \(\overline {\mathbb {N}}\) contains no open interval and \(\mathbb {N}\) is nowhere dense.
Example 3.11. Let \(\hspace {0.2cm}E=\Big \{1,\dfrac {1}{2},\dfrac {1}{3},\dfrac {1}{4},\dots \Big \}\) \[\overline {E}=\Big \{0,1,\frac {1}{2},\frac {1}{3},\frac {1}{4},\dots \Big \}\] \(E\) is nowhere dense because it has no non-empty open intervals.
Example 3.12. Any one-point set \(\{a\}\) in \(\mathbb {R}\) with the usual metric is nowhere dense.
Solution. A one-point set is closed, so \(\overline {\{a\}}=\{a\}\). Its interior is empty, because any interval \((a-r,a+r)\) contains points other than \(a\) and so is not contained in \(\{a\}\). A set whose closure has empty interior is nowhere dense.
The same reasoning shows any finite set is nowhere dense; but note that an infinite set can be nowhere dense too, as \(\mathbb {N}\) and the Cantor set both show, and a countable set need not be, as \(\mathbb {Q}\) shows.
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